The "Timaeus"Scale

Timaeus' Soale “usica Disciplina, IV - 1950 -D.3-42 { | i Sas , D SAAN fi HAND AS ?. bud D The Fite’ J, ieci prasion D PA 4. formag vey . Fblogr MA Zand

The "Timaeus" Scale Author(s): Jacques Handschin Reviewed work(s): Source: Musica Disciplina, Vol. 4, Fasc. 1 (1950), pp. 3-42 Published by: American Institute of Musicology Verlag Corpusmusicae, GmbH Stable URL: http://www.jstor.org/stable/20531803 . Accessed: 08/04/2012 04:52 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Institute of Musicology Verlag Corpusmusicae, GmbH is collaborating with JSTOR to digitize, preserve and extend access to Musica Disciplina. http://www.jstor.org

?TIMAEUS? SCALE JACQUES HANDSCHIN III, 1866, 158 ff., in a paper im Timaeos des Piaton ?, A. Boeckh, Kleine REFERENCES: Schriften ?Ueber der Weltseele die Bildung first published J. H. Vincent A. in 1807. et extraits in Notices et autres th?que du Roi ? Sur le musical diagramme von Thimus, A. Die des manuscrits XVI, bibloth?ques, de Platon ?. harmonikale Symbolik de 2, la Biblio 176 ff., 1847, des Altertums, I 1868, 156 ff. and II 210 ft. und Aristoteles, bei Plato J. Stenzel, Zahl und Gestalt ences are made to the second edition, 1933). A. E. Taylor, J. Handschin, on Plato's Timaeus, A Commentary Der Toncharakter, 1948, 360 ff. the tone B natural the sake of simplicity B flat by b, F sharp by fis and E flat by es. (refer 136 ff. 1928, For NOTE: 1924 is indicated by h, I. INTRODUCTION is one Timaeus of by Plato dialogues as reflected by this dialogue, I think the figure of Plato, to that which is current. correspond him as a thinker phenomena, of pathetic whose pessimism here may be described as it necessarily must, power. by a Divine ble signs that, even that it is order Timaeus, 56 C, idealism One takes a hostile as perceived despising (429-347 B. C). does not exactly to look upon is accustomed latest the beauty is absent as one to the In fact, where by in Plato's the world of view the senses. last dialogues. His of objectivity contemplative a world order of conception this order to us is revealed it seems by as disorder, in a higher sense, not yet disclosed that 68 B-D, 87 C, is confident this But of touch attitude which leads, established so many we must percepti assume to us; also Plato planning order,

exist even where we are not able ratios, must by numerical symbolized has always been the property of thin to verify them. This conviction ? as idealism for the so-called kers representing ?, example, objective ? as it can not be other but it is found and Leibniz; Johannes Erigena ascribe that thinkers to whom we should not especially wise ? also with of the Objective of correlation the problem variety of thought, because and and it can not be in thinking importance a r?le to a mind ours. assigning transcending it was the merit to have done of J. Stenzel justice to that is of first the Subjective without answered 1 think of stage in Plato's evolution Socratic of moral idea believed induce him suffice to the for suffice that a constitution ? superior in the that turned to the problem in this case the cognition would ? Good the Idea to man would demonstrating to realize it. Hereupon he (in his Republic) at that first also of human community, believing and soul a with (yet in conformity the first sought its law by himself establishes which and therefore idea), Plato objective human by his last dialogues is unfortunately lost). of the individual, freedom thought presented about the Good, which lecture (and by his famous from the Starting the Idea realization of of is relative by the Constitution) (represented afterwards the Good, recognizing that a ruler would and in value be required last (in Timaeus to alter the law. At by his insight could be qualified a world idea ruled by a Divine to of the advanced Plato itself) who creator, of the ruler of the state. In this process of evo the prototype he being to archaic currents of Greek think returned and more lution Plato more As we Plato's add, may a way idea survived in Byzantium Christian doctrine, and Christian doctrine. in transcendental in the idea is really the king-philosopher between govern the the state, must according himself before of 1917 in Russia in of course without any association be derived until 1912, that earthly the inherent in to Republic, Christ. In the the ? idea Plato prostrate dialogue be understood and hailed just ruler will always by the members he has even the prophetic vision of the just who, and, as is known, many pains, Viewed will be crucified. from our side Plato is, of course, a it a is reflected a exists, further, ? who are that of after thinker with from to just is not the has with connection There of Byzantine same heavenly where Republic philosopher-king. the situation and the must power in Plato's already and reflexion f.). rule until present abnegation, a rule even assumption realm of parallel in China The as and of worldly and so in doing (Stenzel p. Ill of Old Orient; ing, going back even to the thought Hellenistic he at the same time anticipated syncretism the at all the monarch sure that community suffered having restorative

?TIMAEUS ? SCALE character. humanism, as or, This to overcome What he taking this word tries Protagoras said, idea was in a way and in its narrow present to surpass is the idea ? deifying humanity also measure in Socrates, the idea of a subjectively of ? sense of ? the man the making D Good; all of ?. goods yet connected in so far Plato, with in his higher perceived to the Socrates afterwards he early dialogues, opposed sophists. But transcended the Socratic point of view; he noted even with some plea sure the strict regulation, to which in Egypt, human affairs were sub mitted on a religious basis ? and among these affairs was music. matters in the present What connection with is that, in accordance assume and music the conceptions of late Plato, mathematics necessarily ? or ? a great in philosophical they reassume thought: importance in an number being by itself symbol of order and sound connecting the way the outer world with the sensuous. the rational with unique ceptual, the physical with the per It is indeed a peculiarity of the inner, of sound that those sound relations natural, appear most perception or fundamental, to the simplest that correspond obvious arithmetical the character sounds determine of the between (and relations between by us). Of course this correspondence single sound as perceived the character of sound relations and the form of numerical is ratios ratios independent as Ptolemy of on is realized we fact whether do or do not know those ratios; I 10): our perception is just (Harmonics crying out ?, of the fifth as soon as the ratio 3:2 with the consonance ? said it meets when the the monochord. It is therefore are exemplifications we may add, demonstrations logical that mathematical in the later works found especially and musical of Plato; and, the historian who devoted it is also logical that J. Stenzel, a to this phase of Plato's dealt so much with thought special attention, of Plato; it is true he did not concern himself the mathematical conceptions but both fields are closely Plato's with ideas about music, connected It is clear that an investigation of these mathematical in themselves. in Plato is as important passages and music. of mathematics the history and musical as for In this paper we points. as well where a scale, musical obscure present many Timaeus (35 A ff.) We must note this side of des id?es et des late Plato in La that Plato's even before Stenzel L. I have in mind his thinking. nombres d'apr?s pens?e grecque him, still that passage in as cosmical, is expounded. deal with had dissertation thrown La much light upon th?orie that excellent (1908) and les origines de l'esprit soientifiqae, Avist?te et Robin in understanding But these passages platonicienne devoted chapter

THE BUILDING OF THE WORLD SOUL; STAGE 1. AND 2. thought through a fable. He tells the World Soul by taking the Indivisible the Creator made or the Other and mixing them, the Same, and the Divisible does, Plato As he often us that God or substance into a Third What Perhaps Eleatics who be meant saw the with nection yielded ? unlimited to the Two the to see how those notions I shall only note here the Other at represents musical in his turn: he the is not the Aristoxenian of view of musical the Same with the same tone different ? posed system The the whole. on the words, from the of also gives the tone and with proposed of the two opposed the gives we could Having again mixed the or second, harmony 27) third cycle of 1 and (Ch. which the to prominence further twelfth or fifth, system) the octave means basis tone of degrees fifths ?. in which the World Soul organization. this part, He or first one the but in ancient tone in From the One so as far of repetition in the since all the variety, this interval is juxta principles and essence; then be the in par that Plato, to parallel propose the is the by interval. the with c the Other would hold the by to tendencies which Pythagorean two primary three into one of number theme is to build the double the the the Other 109-136, Timaeus or One ? I invite the 2:1 the ? One the interpreted. there are com Thus Same of con be may to then, subjected the whole, then of they analogies. the the ?, of which = and to mathematical a variation mixed these is also octave Platonic ? there essence in every reality, that = Same reconciliation and how to which question, was triple relation psychology octave and by or fifth whereas the twelfth quality, tone qualities result from the different in other Creator (and, Plato of and 15 f.). p. (see below, in Taylor's p. commentary, musical the the and tonal also meant according a time in exist a beautiful namely, our elsewhere contained identifies psychology, place, in Plato commentary interval, same the remember interpreted yielding only the point and that both This melody. there 2, with been that essence, the opposition also survey have that must to determine. thought: immovable change. Probably ? limit ? and ? unlimited of occurs learned Plutarch a musical opposition or Divisible and so easy is not prec?ateme and unchangeable ?. We assumed number ?. excellence first have substances of opposition ? which to consult primary currents motion but nothing reader who main one Pythagorean the ? limited the mentators two everywhere saw two those by the blends who Heraclitus mixture substance. may Plato his unfolds the principle a third, into then took and the one a half of the latter part of of the the triple of the first, then the double of the second, the third, the octuple of the first, and 27 times the first. Thus

cTIMAEUS? SCALE 7 seven the to each other in these ratios: 1 2 3 4 9 8 27; parts were are taken to represent musical of course, these numbers or more are taken as tone the ratios between numbers exactly, as a matter tones, intervals. As we see, these numbers stand for the first three powers of two (2, the first 2 and 3 being 8) and for those of three (3, 9 and 27), reason why and odd numbers The respectively. far as the third power and not farther is, as it has been 4 and of even Plato as goes that assumed, the limit of corporal the third power and the third dimension represent a as must commentator has the soul into bodies or, said, things proceed to that question that below, p. 31). Stenzel explains (we shall return ? does not harmony shall see that it really ?musical but we seven numbers need more than needs more. Yet four powers of 2 and or not Plato himself include three powers (p. 103), in fact already Plato's as 1 is 3 respectively, equal have been conscious to 2? and to 3?; whether of of the scale. for the musical it is important interpretation our tone in been indicated, system (and, constituting already this fact, As has may the octave between every tone system) we must distinguish sense that or 1 and in fifth 3: this 2) (ratios 3: (ratio 2:1) and the twelfth since the the former is the constitutive the latter and not interval, in reality, or multiplication all kinds of tone qua by juxtaposition al is in this respect meaningless, the octave producing ? or in different site ?. Yet we must the same quality ways position means we our as too series of not proceed see, hurriedly: juxtaposition fifth produces lities, whereas the octave thrice and thrice or descending is by ascending We shall note that our octave and the 5:2, existence the fourth of but whether twelfth, consecutive and the whole tones tone 9:8. of those noted, composed But we must and a fourth. be more like juxtaposition a scale, i. e., a yet other than the the fifth yield (or numbers) can also establish the We 8:3 intervals would the do not yet know. intervals series includes we twelfth: 4:3 of an octave posed which the now series e. g., com being, turn to something composed of ? neigh ? tones. bouring that the by stating step in this direction ? intervals, ? and the ? i. e., the up triple ? ?. He does not say expressly the twelfth by two means we have to interpose within can no that doubt be but there achieves Plato Creator filled octaves and which means, either of these mean; none the first the ? double ratios of Plato's or intervals commentators the arithmetical has understood and harmonica! it otherwise.

is known, the arithmetical outer terms taken both form the harmonical whereas (or, mean first outer term be the mean is to the difference mean must to the second, is that which quantitatively, to conform the mean between is equidistant their sum), the condition as the difference is the case, e. g., in the series 6: 4: 3). octaves and twelfths in these By dividing in it is half the two ways, the the first and between and that second we (which with obtain tones distant tones respectively from the outer by 8: of four famous numbers wrhich com 6, the composite (12: 9: the twelfth in itself the series' 4: 3: 2 and 6: 4: 3), and within the octave two a fourth bines tone by a fifth (6: 4: 3: 2, this being of 3: 2: 1 and 6: 3: 2). As we see, both double divisions as to tone distance of the ratios): (or as to the magnitude two tones distant a combination are symmetrical in either case fourth from each outward the respectively) smaller and interval is in larger interval the the middle at both tone (whole sides (fourth and the octaves and and fifth respectively). to every one In applying this double division the twelfths present in the primary series, we obtain: 13^ 18 27 / 1y8 1V2 22 */?3 4 4J4 5%6 0 I 1_I 1 1 U M_! i_L in series are written of the primary (the numbers some of the numbers in different been obtained have is the arithmetical which We could with 6. Musically tone qualities easily do mean of 2 and away with fractions 1 and of course as, e. g., VA ways, the harmonic by multiplying feature of our the main Of italics). of 3 and 1. series the whole series, although speaking is fact that it includes the five different by Plato, expressed powers we may multiply of 3 (which holds good whether it with 6 or not), that is, five different degrees of the series of twelfths or fifths, or five diffe not rent 18 and 4 H and 51/t as 27 and 13J4 to 3s, 9 and (characters), correspond to 38, 3 and 6 to 3\ 1 and 2 and 4 and 8 to 3?, 1 */ and 2 % to 3~\ Here we may constitute the pentatonic this basis, remember that five tone qualities, built up on the fourth (e. g. c d e g a). have noted, the primary series contained the fifth, already and the whole the octave and the twelfth; and tone, besides the double division ther such intervals. As we constitutive applied Among in the building musical to the octave series and the twelfth fur produces whose r?le is these it is especially the fourth of a scale, and that for the simple reason that

M?SICA tone ? distance it is, as concerns DISCIPLINA ?, the least of consonant intervals (or simple tone, we proceed if, starting ratios), ? the whole ? ?. Also 9:8 is tone which represents, steps by important ? ? al as to distance, the normal needed; step being present precisely divi series, it is further produced by the double ready in the primary is first attained that which from one as 12: 9: 8: 6 and, sion of octave it is so often defined sense, we in the latter as the difference between fifth understand and why Secon fourth. even the (minor) third, 5 x/3: 4 V2 being equal darily, we have obtained fourths 5x/3:4 to 32: 27; as we see, it is the result of the two overlapping and 6:4 V2 (or, e. g., d a and e h)f but we do not need it as a constitutive we should or adding obtained the intervals element. By juxtaposing of the octaves with it filled soul substance and was twelfths, 9:8, and the mixture of the used fourths; filled the space and 27: Thus tors have out by brackets. As has interfere with each other; and 6:4^, i. e., is a large space not filled in by fourths, outer or this and and 27 the figures 27 between 9; 8, hand there included we understand been faced with agreeing tone whole with fourths the all of the three octaves lying outside them. 18 with a fourth between twelfth, by are marked the fourths p. 8, where figures been said, two of these fourths, 5 y3:4 the other this up. the Creator above on division that his series is already conti the impression conveys by fourths (4:3); but that is not the case, as we see by the divided nuously the that saying Plato intervals, By connection. having realized the double tone interval took the whole the 4: 3, i. e., the all in this the Creator continues: Plato Now not matter too does that but still others, get contains 8:1, with which to form a definite trying 13V2:9 commenta Plato's the difficulties when the fifths idea of a scale their task easier by disre They made established statement about a frame of fourths, his instructions. Plato's implicit in his commentary tone intervals. Proclus in the whole filling in in the face of Plato's lightly when, (194 F) is really taking matters ? or to in fill in he of fourth the fourths, struction every filling speaks garding before fifth?; that the more is so much at variance tone the whole that when explicitly ? ? or diatonic mains limma the half-tone. says commentators have been because in the modern periodicity. But perhaps sense a scale in the ancient with is On as Plato, inserted the other hand to skip the frame is more or less determined conception the division latter there twice, induced Greek the re modern of" fourths by octave by fourths

?TIMAEUS ? SCALE was At any rate Boeckh has not dared to disregard that preeminent. statement of Plato; he was not only a philologist but also well acquaint as ed with Greek musical we shall see, a series of has, theory. Taylor but fourths, Yet and here, definite of not it does the question already, scale with accord with of establishing begins our doubt its details, Plato's division. a frame of fourths whether Plato or he merely stated is not had really the general easily solved a in mind principles one. I ask the reader to look at the diagram For vizualizing the situation row. This illustrates this paper and, first, at the uppermost accompanying just the series of numbers given of our octaves double partition stinguish 8, 9, 27) within this series and those which above, and which twelfths. the result presents We ought always of the to di between the primary numbers (1, 2, 3, 4, are the result of division 1 2 3/3, 4 Vi, y2, 7? (1 5% 6,13 5a, 18). The thorny question and 6:4 V2. Are we to conserve of and them, those if so, which? We fourths of being b and filled overlapping which, metabolon), the tones produce mind such a concrete would is that first not be h. in harmony different octaves of our it would and therefore both, with the interfering has done, fourths 5*/a:4 or to drop one that Plato had imagine ? ? scale the Greek modulating could in similarly Yet detail of as Boeckh I am not and this: there (e. g., at all (systema as a b c d and h c d e), sure that Plato had in is even another shall see, b and as we in mind detail which h coexist in to several of the possible scale according variants, not seem logical to have them, apart from that, in the same octave and produced that ar However, by another method. fourths conclusively, any more gument does not rule out the overlapping than the argument in accordance with Proclus, urged by Taylor (p. 142) i. e., that ? mean the a-b, the apotome; as to this I should observe ? scale does not the coexistence of b and h in a ?modulating that both tones are concretely connected. I think we must say mo apotome while Plato that the coexistence of b and h as neighbours would produce ? b-h as distinct ? diatonic > half-tone from the normal does not mention that we shall drop one of the competing destly sake ?f simplicity; and in this case I think it will 5y,:4, it is not as the always in the present latter fits better in the octave fourths be only rather for the than 6:4^ and ? though periodicity; realized in music and we may perhaps not fully realize it case ? octave is in the idea of a scale. inherent periodicity

the tone consequence in framework. the support In In lost have its position as a of point to I am already hesitating this point I ought whether an at at of the good intention illustration, arriving I go on and now it is the question Plato may have meant, at fact But with continue. least, of what of having fourths and diagram, are 4 lA will everywhere. this time at lows I ask the reader to look at our Again I, II and III where some of the possibilities outlined. most to that of the uppermost closely adapted disposition row row would be I- In fact here all the tones of the uppermost (except as to that is 4 their rank of ed say they are support, H) preserve points The I). intervals defining tones. of fourths. Yet 20 ]4 and 12 are added to the Platonic this case beginning from 6, there would always be fourths se a whole means we this that this should obtain within tone; by In parated fifth octave periodicity. Or, to put it more periodicity, replacing octave out be ruled as in would from 8 onwards, precisely, periodicity ? ? the normal have to be 10 */3:8. way the next fourth after 8:6 would space In consequence the octave repetitions of 5x/3, i. e., the tones 10a/3 and 21 y3, are lacking; or to put it still otherwise, the fourth 12:9 appears as an octave of 5y3:4 shifted by a whole tone; 18:13 Vi is in the repetition same when to 8:6; 27:20 M is shifted by one whole position compared tone in relation 5 ya: 4. From to 12:9 and, therefore, by two whole tones to in relation the musical or point of view, looking at the tone qualities of 3 present, we see there are not less than six, which is pre a consequence of the fact that octave repetition is so largely dis the powers cisely the tones 1, 2, 4 and 8 mean regarded: 3> 6 and means 12 mean we may note with Plato importance 3\ special where he 3 *; 9 and Here 18 mean 3?; r/? 32; 2 a/3 and 5 */, mean 3*; 13 H 3 s; 20 % and 27 mean 1 y2, that the sixth power of 3, 729, assumes a connection 587 D), (Republic the ideal period of solar revolution (729 days in another takes it to represent and nights ? 364 JAwhole days instead of ca. 365 y3, a magnitude already ? known at that epoch); but of course I should not take this as a ? proof we must that follow this method, I think, there is nothing to be because, proved here. the fourth 12:9 is, as in case I, shifted by a whole tone II). Here relation to the octave repetition of 5 V3:4, but the shifting goes no farther, the fourth to 12:9. to 18:13^ and 24:18 is contiguous In fact the only difference between therefore I and an octave II is that in repetition in the latter

*TIMAEUS ? SCALE tetrachord the first from to the second, is joined tone. The effect of this left it by a whole from separated octave periodicity is less violated tones are added which were not as are they One of octave in the tones relation, 4 octaves and a fifth we may nection of cosmic Perhaps of 27:1 was row, may that series, 24, tone. been left outside range of the the is series not does in a and fill out the typical the Proslambanomenos in a way to be considered one sixth; in this con major sometimes 24 and 1 are given as limits XIII II 1, cf Aristotle 6). Metaphysics because a note, was there tonal being that case, again, two 12 and 24, but substantially has 27, the harmony (Macrobius the fact that this disposition tolerable as appear of 4 octaves instead remember seemed Greeks in the uppermost they of is, precisely, In our the primary last note being ?. The instead in case I. than of ? skeleton 13 outside range of two octave-scale or structure the the whole ? added by the ?, which tone. a whole 4 and 8 mean powers of 3, since 1, 2, gives five different 3 \ 9 and 18 3 ?, 1y3, 2 y, and 5 y, mean 31, 1 V2, 3, 6 and 12 mean is tone of the number 3*. Thus mean qualities 3a, and 13 54 means This series same are the same lity denoted one 38 quality (apart from different more by 31 has two tones row) and even the (uppermost as the tone octave localisation, 12 and representatives, 24, and qua the less, 27) the terms of yet more freely with to the ? law ? of octave periodicity In dealing III). row we can concede In this case we goes series as in the Platonic the on within the 8 exactly after continue three octaves 2:1, the same kind 4:2 8:4. and the uppermost its full of division That would sway. which mean 8 and the next octave, then, between first 16 which gives interpolating In 16 the number 21 y?. 16, the numbers 10a/3 and 12, and beyond i. e., to 21 y8, extend of the octave would this case the uniform division to 4 octaves remain would a fourth; and outside, the extreme as it is separated tone of the primary series, 27, tones from 21 y?. two whole by is that no less than four tones, arrangement to the Platonic be added 21 x/? would series in the tones Two of the the series, 9 and 27, among primary The disadvantage 10y? 12, 16 and of this row. uppermost as well as two further tones contained would be dropped, not most 18:1354 would row, 13^ and 18, and the fourth From the musical be only three, point of view, as 3?, 3'1 and looking 31 occur in the upper be preserved. at the tone there would qualities, in the different 3? being octaves,

represented 3 * by 3, 6 and filled which only 2 a/? 5 y? 10a/3 and 211/,, to determine of the which hopeless a can in mind. that be gap Leaving he puts down is just characteristic of his attitude: care how it will be carried not much through. it is useless Plato in different a principle even Plato 1y? 12. I think Now three methods 16, 31 by 1, 2> 4, 8 and and ways does fails to give and had has to the difference between 2 and 3 prominence to for the octave? corresponding 2, serves essential, is musically for delimiting spaces the in the scale, whereas twelfth or its octave and ever to 3, produces different the fifth, corresponding equivalent, more different tone qualities. Nor does Plato notice that these two consti are in some way antagonistic tutive principles of scale construction or, as never will a power of 2 coincide with one of 3 at least, irreductible, and therefore a tone never will from coincide with one resulting ? comma although, ?) thagorean from octave juxtaposition resulting of fifths (hence the juxtaposition always be able to take over to be scarcely distinguishable. will exactly the ? Py a tone of course, in practical perception the value of another one so near to it as III. STAGE THIRD II or we III, have not as yet a scale. tell us that whole (9:8), every fourth to 256:243 or 28:36. tone equal Now and the Creator III. we may Whichever the fable continues. But filled all As we have choose intimated, of the variants I, on to Plato goes the (the 4:3 intervals) with two of these and a rest interval, the fourths containing to fill for a survey we ought take much But as that would out all time, we the fourths in row can restrict ourselves I, II in every case to the first and the last fourth. become 1:1y, will fourth The 1:% 7<?: 4/3 and the fourth I). ? the ? diatonic is precisely That become 27. 20 J4:27 will n/?: 72?/3,:m/m: ? ?, if we take the word tetrachord form of the fourth or the diatonic a half c diatonic ? to mean tone tones. It is true after every two or three whole interposed to the smallest number, 9:8 has to be apposed doubt whether being we might as we have just done, the combined use or to the greatest of whole which would and half produce, tones, respectively,

?TIMAEUS ? SCALE But that does not matter because, */ : w/?: *%?: 1 and 27: 24: e,/3: % as we shall see, we are faced with two reciprocal methods of computing, as one of which takes numbers frequencies and lengths; tone beside the one limit the Wxiile numuers; (vibration 1:ys: in this aspect the other takes them as string "y^: */? as 1 on same basis of the result will yield the string lengths 4/8: M/w: **/u? of Plato on the basis of frequencies, In fact the wording and vice-versa. one locate whole a third possibility which would does not even exclude of fourth another beside the other again we see that Plato that having filled out all of fraction numbers than before; the half-tone thus putting in the middle; It is clear enter into concrete details. limit, not does and the fourths obtain a greater variety can again rule out by multiplication. In this case the first tetrachord would we this we and II). last 18: sl/4: ?/?: be as in case I, and the 24. In this case the first tetrachord will be again as in case I, and III). the last 16: 18: 81/<:*/? But now we are faced with the question which is practically fore most. Before being able to represent our numbers we by sound-names must determine string lengths. with another. a matter All commentators, of from have decided But it course. was who Plutarch, they are to mean is known, these two of Thimus, exception as As whether is a very in are Boeckh onwards, favour of a not or relative frequencies in reverse proportion one relative matter but lengths, of course, with the this taking as, e. g., numbers Platonist, good supposes frequency in his commentary in Timaeus 67 B ascribes to (Ch. 18) and Plato himself the low tone the lesser rapidity and consequently the smaller number. (It is true Plato speaks of not rapidity of vibration, cf. rapidity of locomotion, Der Toncharakter illustrative ment p. 136 and speaking; in either I think that what did matter 139-141; that may case the question be an error or a kind of is of ? quantity of move ?). Again open: question tones and numbers, whether the reader would proportionals, literally with major ciprocity or was it is rather tone in the for him was to let the style of Plato the correspondence between relations numerical choose one and or a secondary question. to the compass of his scale, because to him regard is a compass unheard of in Greek musical of numbers, we may yet remember the notion sixth relations; and, those reverse among can we take him Nor the other plus a As to re 4 octaves theory. of the ? unlimited

as occurring to in Plato, and which, according a Stenzel (34, 51 f. and. 59 f.), would indepen signify the Two as principle as or it taken is dent of whether divider, or, in other words, multiplier be the ? un or negative in this case it would as positive indeed, power; the Greek word ?, but with defined Two ? rather than the ? unlimited above ?, mentioned Two are compatible. both meanings 525 D f-) says about (Republic the One, five (As is known, the idea of to is a tendency character of number, this there In all on lay stress more on the formal than as a rather to take number principle quantity. extreme Plato instead of fractioning numbers: rather multiply it, or in other words, taken rather say that a definite magnitude, of saying, e. g.> 3/>, they times, will be equal to three units. instead also what fractional would mathematicians to remember have We the material than as continuous the measure concrete of its has found quantity and created by Leibniz; calculus with differential representation to musical in regard was himself thinker this great who, yet it of integer numbers the importance most insisted upon decidedly tones, and, the first prime numbers). especially, crave for concretization Yet as modern and, in this case, people we ? we have to choose. choose We for the time being and with all ne ? reserve to order in be more in accordance with cessary string lengths, and we follow them also in their of modern the majority commentators; two of inserting the whole tone, that is to say, we juxtapose tones by starting from the smaller number which to is, according tone of the fourth; first assumption, in consequence the higher the method whole our rest interval ? c Dorian at the lower end, producing the ? tetrachord from the ? Lydian (different ? which has at the higher end and the ? Phrygian (the half-tone) will species of diatonic the half-tone it in the middle ? it is true, these names which quite But has be placed as applied to tetrachords are not authentic). Now at here we must last we can associate yet remember with our numbers that we should tone favourite take them not, letters. as is as absolute of pitch measured usual, degrees by vibration frequencies a as or tone but relative relations of which, second, system by per pitches even our tone can be based on any concrete pitch. In reality principle, letters had originally this meaning of instrumental by the development relative (the musically real) meaning concrete (the musically abstract). ? in the ?Gregorian system, and only was the music and ensemble playing of the tone letters superseded

?TIMAEUS ? SCALE if we Therefore take on the monochord string length as 1 the number part as representing and a major sixth ?, the question 27 and its twenty seventh representing ? which gives an interval of 4 octaves we " d whether and Fx answer shall reason as Gx and may cause some interested in the concrete in the old, them taking in the old to hesitation. course, a choice is as much take tones the form, terms ot In systemaccidentals of the series of fifths) (i. e., the degrees ? ? a so as that centre series of five tones will d seven there be our the a series of taken Of ? ascertain c gdae, ', or in the modern sense, would ? Gregorian thus: ? avoid ' must We determine ? theory ?, it may be stated ? or, in ternis of musical psychology: a and C? the other? must as tone qua them and dispose possible lities involved with ', or of the tone relative one denomination for prefering and the criterion which there exists, the same as that observed ?musical ' ' e pitch and that therefore we do not use the letters as starting point, fixed sense. But a reason tones and are not that we the whole two these ' h ', etc., denote ', or A ?7 fcgdaeh, the latter the form, a series of nine case that b and fis fis (supposing ? ex ? chromatic the others, not mere are real qualities with coexisting ? ? an odd It is true, this supposes tensions or modulating features). the centre for in the case of an even number number of tone qualities, the form, b f c gdae in h actually minant d and g, or d and a; but two qualities, say between an terms have always been predo of odd number systems with have others (hexatonic, and we can even doubt whether octatonic) existed in the full fall betwreen would Now Boeckh to x/? of the sense. has the taken the tone, highest one diezeugmenon? length, that would mean that the tone corresponding scription; than two octaves below string is G *, a tone located more of our piano. this series by We shall In order three octaves, now on these premises. I invert the order III). The c'" I d'" e and G3 replacing kind of I hope the reader will scale our ' should depths by G and three the whole tone shift e" ". rows will forbear with me produce if this time for the sake of convenience. third disposition i e'" 2 f" would 9 e' 8 f d' I 2. such to the lowest we extreme see what G 27 AHcdefgahc' i I h" to avoid corresponding ' in our tran is e which to be Nete produce this scale: a' h' c" d" e" g' i_!_i g'" " a'" h li 4 f" g" J_i_i ' c"99 d""e""

There are 34 tones. stated above. As we the criterion see, the letters adopted agree with to survey the situation more easily, we must always In order It is clear that look at the tone qualities (or the powers of 3) involved. as contained in the frame of fourths of qualities the number (according by the filling out of the fourths, scheme) is augmented in them are themselves degrees of the series of fifths (or * 3>and */* being 3 divided by 26 powers of 3), % being 32 divided by 2 sense is taken for in his that evident, % Indeed, Plato does not make as difference the it 1 where 8 9 stands 4 3 2 series from the primary 27, it from 4:8 and the none 4:9 (or we could extract the octave between the of division octave, where the group 6: 8: 9: 12 resulting from the double to row III of our as the tones inserted it represents the difference between the fourth 6:8 and 6:9); and as /?. But the fifth twice the result of taking tone a/u is, in Plato's words, im it was as structures, of numbers were the Greeks always conceiving or for Plato not to take 9 as 3a, and therefore as a continuation possible of one of his primary principles a further outgrowth 3), and (the number to 34 (cf. what w? shall tone a/w as corresponding similarly not to take the Even ? ? series of fifths of A. von Thimus). say below, p. 30, about the nude the third degree in mind exceeding if Plato did not really have powers the the solid, see above, p. 7), he must have thought (this degree representing Now comparing of linking groups of three powers. together consecutive ? ? of the those with skeleton in the contained the three tone qualities we see that the former have been augmented by four. scale obtained, * There were 3* (the tone h), 3? (e) and 3 (a); to them are now added 32 (d), as they are here by headgcf, 3* (g), 3* (c) and 3B (/). Represented stated to the criterion conform these seven qualities d as centre, with ?. The ? or ? fact that ? diatonism That is precisely above. heptatonism form of the skele in the corresponding was safeguarded octave periodicity ton, is now reflected by the fact that the filling in of the other tones produ ces no contradiction between different octaves, or no ? accidentals ?. As h the filling of the skeleton representation, and c d tones a (=3a) by the ) g(=3a) (=34), (=3 "), / (-3 (=3-*), e, (==3y not be the proper to would that 31 Yet 36. the series would produce for symmetry; from this point to our desire corresponding representation to d 3B to 3" - 3R, assigning shift the series from 3'1 of view we must terms (3** and 3 *). the r?le of centre (3c) and to h and / that of extreme as and in Here we take the scale as a form and its components qualities, concerns this the mathematical sense we must no longer cling to the unilateral, genetic procedure

by which we formerly = 3? and 27 = by 1 = 3 \ c= II)- Row II of scale: following G 27 AB l h" c'" derived a series whose d"' g= our scheme would, c'"2 S"1, d on d' 9 e' 8 f f" g' '" a g'" 3 ?, a = = 3-, cd esfg abc' i l_i i e - 3\ a' h' c" a the produce 4 /" d"e" 3 g" 3 a" i_!_l j_i h"' d"9 c"" e""l. i the same ones. are 34 tones, but not 3a, h = the above premises, l_ I_i_i there Again a whole and 19 extreme terms were marked case of frequency numbers ? which 33. In the is, ? the series will have to be inverted thus: probable after all, more / ?TIMAEUS ? SCALE In taking the letters this scale as do adopted es h to tone from series of not agree with qualities with d as centre, and these, disposed symmetrically comprises nine degrees, would have to be designated by symbols ranging from fis to b; the lowest " '. To tone would have be denoted the to by D and the highest by h ? nes contained in the skeleton which, in this case, are five (31 = fis, 3 = '= *= = c, h, 3a = e, 3a = a, 38 g, 3 d), again four have been added: 3 *= = 3 to structure b. Again, the musical in order / and 37 agree with not in single octaves, our criterion. The we see that of the scale, the proper mathematical expression will have to be shifted 37 to 3~4 from 3"1 3 *, and consequently 3"*, 3? and 34 will respectively to case of frequencies the series would fis, d and 6. Yet in the correspond be inverted: fc = 3-4,/ 3-', The breach of octave = c=3-2, g = 3-\ ?= 3', e = 3 ', h = 3- /w = 34 in the skeleton of this periodicity, present already it is illustrated and con clearly as a contradiction; creted appears now by the fact that in one octave namic) Mese scale, 3?, a.= there is / and in the other fis, in one h and in the other b. We could call it, in the terms of Cleonides (see Der a Toncharakter three Mesai?: the (dy p. 352), system with ?composite which in our presented lower region. Wet with normally corresponds series by e in the higher, to the a quality a in the middle would be re and d in the ? can not really say that the three ? diatonic systems, are a nine nine because a, e and d as Mese, tone-system, composing

tones are never in one octave and they are rather an assembly of present ? octaves. Or to put it otherwise, it is a ?modulating system, disparate one to the In this other. from and modulating twice, always heptatonic the we have a reason for changing sense it could be doubted whether ? ? to from h of the whole es, asymmetrical, b, symme fis transcription a But whole. not form do that it could be trical; indeed, they argued it is logical from one system to another, even if modulating, i. e., passing ?. ? accidentals without written one is in the centre, which to have that I). Given d es f g a b c'd' G 27 As Be " g 9 e' J_! I_I J_I_ " ' e'"2 d J_IJ_i 3 h" c999 a" our first the same premises, always 8 f g' a' " ' /'" g skeleton would produce: h' c'-' d" e" h99' c"" a999 4 f" d'"f l_ _|j_i_I e"" i 1. There are 34 tones as a whole, we see again the scale ? an even to ten tone qualities from h to as corresponds the series that number. It may either case between as before. therefore Taking in in two ways, the center falling as ranging if it is written from fis to es, shifted be two notes: d and g, if from eis to bf the center would be tone would between d and a; in the former case the lowest and the highest 99 " ' '. (Among these two represen be D and h 9, in the latter A and fis tations we should perhaps prefer the former, because the Greek conception the center would be between more ? flats ? ? as was inclined their ? scale which ? not / and fis, but b and connects modulating in we see how the breach of the octave periodicity, present Again ? skeleton ?, is towards to equivalent inner than ?, sharps contradiction. by exemplified in Whereas the h). the skeleto to the transcription nie stage we had 3'1 (fis according by us), preferred 4 ? a now to them are added 36 (c), and 3 3* 3* 3 3 (g), (d) (e), (a), (A), 8 * this we shall again shift with 3 (f), 37 (b) and 3 (es). In accordance 3= = es to = d and 39 from 3~l the mathematical fis> 3 representation ? can even say that the proper represen 3-* = fis, 3 = d and 35 = es. We tation would ? between be minus to give five four ? as power four ?, to d exponent, one minus but ? between and even the tone which we to fis not and minus ?minus to es ? between four and five ?; nay, have marked all, not all by d is, after is rather theoretical, as in fact the ten qualities zero ?, and this d but ? between a and d ?, etc. Yet are not operating

?TIMAEUS ? SCALE Here we have, in the together (and in so far they are not real qualities). a ? terms of Cleonides, four Mesai ?, the Mese system with composite e a to to in the high region and d and g in the low being shifted from we have preferred). if we assume Again, transcription be inverted: instead of string lengths? the order would to the (according frequencies /w = = S"6, b es = = 3"3, c =3"2, 3"4,/ = g = S'\d a = 3?, ft - t -3*f 3\ 3s, 34. is still a method There by yet to add an observationwithout fill out the fourths diatonically compromising But we have we which could As we saw, the unity of diatonicism). (and, consequently, periodicity of seven diffe i. e., the surpassing of the number breach of diatonicism, in case III there rent tones in I and II, was caused by the fact that, while ? e and e ? to octave octave were only two fourths repeated from (say h were limited by three different tones, in case II there were a) and these tones being always of limiting four, and in case I even five (the number these All of one the than number fourths, when juxta fourths). greater by octave in posed one can row, be e. denominated, as g., beyond going g c. It is clear that when are filled the if fourths of diatonicism breach ft e two e a ? ? fourths the breach e. g., if having to the in an uniform manner: examples, or if they would should and to note is interesting not always been in agreement shall open between frequency has been Dorian numbers, and Lydian. has that which a/u: = 4/8 ?, with half seems and Psellus Plutarch who, tetrachord the most c d e f or commentators ? form, commentators. modern Proclus the Lydian gahc). is inherent also for the ?Dorian in deciding the case with which that in this connection see, Pseudo-Timaeus, tary), precisely of view (1: y8: could in the upper to ano the contradiction It as we that we and d g the ? Phrygian give toad to g c the ? Lydian ?, with half-tone transferring only mean of disposition in this case the uniformity be repudiated. idea of scale would in general, is out that would such, g fourths different diatonic forms, by giving to the different ? ? in e and e a the Dorian form with half-tone given to h the lower part, we tone in the middle, Yet part. ther plane: ? d result the ? as in our ? say, if they were all Dorian previous ? ? ? ?. But it is understood or all be all Lydian Phrygian avoid ? ad have though In fact, leave the question as we have said, supposes commen (see Ch. natural And from 18 of his the modern still more point interesting,

tells us (Ch. 19) that other in the middle. tone the half with the tetrachord seems to be that one the supposition of and that it is not a question of fourth is uniformly applied an expedient whe Yet diatonicism for safeguarding throughout. that It is understood species finding ther the breach ned used people that in all this of diatonicism may ? division ? the Phrygian in a special manner appeals ? method the fourth by the ? Dorian be accepted or not, it can be maintai in the of fourth, with the half-tone middle, to our sense of we two add up are, as to By logic. tones which filling ? or ? character at their position within the series of fifths, wholly = 3a and 31 would be joined side of the two tones given (e. g., e and a = 3"* and = 3"1, and the c Ly of course, frequencies), g by / supposing, to the given notes two notes at the other dian ? tetrachord means adding a side (e. g., g and c = 3_1and 3 would be joined by a = 3x fend h = 3 ")?' to the Phrygian method, both added notes are symme whereas, according * tones (e. g., to a = 3 and d = 3 ? there trical on both sides of the given at c right ? from a, and c = 3"", would be added h = 3a, two degrees ? c two degrees at In short, whereas by filling out the fourth left from d). their one we displace by the Dorian method as 1 should say, to the ?masculine ? side, ? ? ? to or to the tetrachord feminine right not centre the is shifted. method Phrygian Of fourths, course and that holds therefore e99 fis99 whether A B c d e I_I g99 ? tone qualities, by using the two, three or more good also when we combine to fill out our frame of fourths I should prefer tetrachord, by the Phrygian case I we should obtain: C 27 DEsFG of our group to ? left ? or, and in the case of the Lydian the centre taken from row I, II, or from ey fis' f g 9 a 8 h c' d' III. In a'4h'c'd"3 g I_I I_II_i i_!_I a99 2 h99 c999 d999 e999 fis999 g99' 1. a'" I _i i_ i i or, a'" in case of frequency 27 g999 a* g' f Il es' fis99' d' e999 numbers: d999 c9'9 c' 4 bag 3 fesdc I III h99 a99 g99 e99 f99 2 BAGFEsDCL II I d99

cTIMAEUS? In case SCALE 23 II: fis' g' a' 4 ft' c' C27DEFGABcdefg9aShc'd'e9 d' ' 3 1_!_' !_I 1_1 !_j_1 e> fis" h" g"a"2 J_1_I or, in assuming e 27 a f" e" c"'d'" e'" c c"b' In case III: g999 fis e a ft g !_i_i a9 g'\f a c 3 c' b a g 2/ I_I_I 1lj_l1 _1 e* d' a' g' 1 e9" a,,ft"c,,,d,,2 f" numbers: k9" e9" f"9 g9" d"9 n a 1_I e d cB A G d" h'c" 9g o 1_ 1. 4 e" /f g" 3 I_1_[ a'" hf" c"9 ft" g"' or, in case of frequency 27 a9" 1. numbers: d e f g ah c'9 d' S e' f I 1 J_1_[_ F 27 G A He a'" I_L frequency d" fis9" 1. c""d"" g" f" e" 9 d" 8 c" not explain than further another. why At a" 1 l_!_ I_!_ h' a9 g9 f e9 d'4c' i After what h aSgfedZcHAGFEDl. _ 1 has been said one method I have adopted rate I repeat that, with any Plato's illustrating thought to himself. presented The taken 1 1_ only in both thing 1_1 1_I I above, shall rather of transcription the six series noted? and not we must still of deciding it is only a question what he must have re note connection III, in this a disadvantage: presents significations, tone, 9, stands not as a ? frame ? tone delimiting as a tone included in a tetrachord; that is the price all In looking again at our diagram periodicity. in the Platonic the fourths contained division all the Platonic octave exception of points the fourth of support 6:4 V2 and stand the one is that of of the primary a tetrachord, but only of for the advantage we see that in I and II are made as such again tone 41/*), which use of and (of course with is not the case

in III ? the difference I and II being between only that the primary tone 27 and not in II. The form I, which as delimiting is in I, employed tone, octave so largely periodicity replaces the like our Summarizing we argument, tented himself with twelfth, has offended is that if we followed fourths said, to filled be fifths rather Boeckh not is, is presupposed. His disposition ferences: 1) he includes does not present the 16; why? for obtaining, the tone also to agree been said the with above ? ? second tetrachord all and therefore agreement modern We should have, b ', if we united series whole In the same, to shifted The by brackets fis laid some is by fourths exception. 161, but such a division but with two 51/3: 4; 2) 161, but from it, puts in if e with lower. a see We from commentary on Timaeus h is that a division there 384 from it must be noted ' a with the such as has 5 within contained supposes avoid to 19) we if we of differing number to es, they must, come down the interlocutors as has already of 34), string 10368, course, lengths, as he, in like Boeckh, ? tetrachord. ? Dorian above (p. ' d ', and a/3:4; is virtually Of fractions); the dif includes first produces, Plato to he he commentators. ancient (or that true is one with interferes again obtains tones 36 (instead order and It in fact he wishes 1, but The 1 to 27 that II, from in one the p. to one ? Boeckh that convinced range to Timaeus, to intended dialogue, whereas some the of primary forming the e and inserted Xhe transposed notes remains been be observed, - b. earliest forgery by 16) which they is attributed known, a then, scale as evident the opinion his opposed Boeckh he of and nine; ? 243/la: this way been authors, octaves three forthwith. not made is also authors, octave see is descending ancient most fourth shall chromatically authors, series his with the ol (256:243), row on scheme frame series with concurrently says, (12: "/,: modern p. as we as he 18). (*ya: ***/M: a87/aa8: stress has a number upon which nearly to our of that in his fourths by present like 6:4 y2 fourth on a to establish neglected scheme as same is the a tone the in his fourths Pseudo-Timaeus, of a new inserted have commentators their division (p. 11), consequence consequence the limits of transgress was aware of all the con Commentators Some by the the said. the division present the also should Plato but, the does con having divided having and periodicity, in, allegiance having given tones limmata in whole v9:8) and they filled as they pleased. an as has been in this respect intimated, 1 2 3 4 8 9 27, and of most I have not Plato but octave whether IV. As is much periodicity, that say octave against him strictly, we he had of what sequences can the dividing It can be doubted diatonicism. fifth with Enchiriadis. of M?sica scale editions it give one the impression is really an of works Plato's of abstract as an that from it the appendix to us, though of Plato's had served latter. to not the earliest dialogue as model for Plato's (It has the Timaeus, been itself: published as, e. g., in

?TIMAEUS ? SCALE of C. F. Hermann's volume fourth between two versions one the first century forth only from is put ship of paraphrase This by and filling Now 114695. say by which he also that of the Now gives account that the seven and even he First number of I an C. and author unpretentious will intervals the terms. But derived his terms from be be with sum terms, 36 question 1) must 36 arise of ? to supposed (Plato's triple Boeckh's edition of that not as divided As shall Timaeus' represented their total partitions of numbers. primary or frequencies, lengths does course Of the string is which Pseudo-Timaeus as as well this is added on tetrachords but only distributed is presented in the edition a treatise by scheme it here reproduce scheme to in order show among not how which 410 p. the clearly was thing times: 486 432, 384, ? of numbers. ancient first was first is number sum the still incorrectly, Double distinguishes species. series, in the what of first ? and there the suspense primary represented ? double tones, in Hermann's the while version, short the tetrachord the whole second very is also in the states means leaves era; a taking that not our It the whole in (in Pauly-Wissowa) first century from the B. dating to him, the claim of Timaeus' according dialogue. gives that, Harder one treatise, of psychogony. and 384 this in Plato's treatise of system of R. edition). 25 number (h) 512, 864, 768, 576, 972 729 648, (h) 1024 (h) 1152, 1296, 1458 (h) * 2304, 2592, 2916 (h) Quadruple 1536, 1728, 1944 (h) 2048 Octuple 3072 Ninefold first number 3456, 3888, 4374 (h) 4608, 5184, 5832 (h) 6144 * Ninefold second number 6912, 7776, 8748 (h) 9216 Triple first Twentysevenfold (by h we of the first fold as the ?; numbers reason sum their above the relation of Apotome must a later be the treatise. as sum is said line the addition 114695; in and the text the of terms that are not the superfluous terms derived tetrachord end 6144, while there it was The number of differing the the in supposing from string and row our lengths, we ? Nine and they are counted * there are inserted they are not counted this scheme could not the two tone II; for the present write by to the by qualities) with the have scheme scheme others, agree from as the they to belonged 36 which produce with is precisely discussed has in neighbours originally there are excluding we which their series interpolated obtained added (tone are makes this those indicated interpolated, the should with terms which as 6561). notes beginning 34 6561 And same the line are places interpolation treatise. the 34 terms (yet the and with the namely, second as the therefore 34 see There As Limma. and special ? Ninefold as one is not 18 which that at Yet 2187 is curious row, graphical, long. exceedingly numbers we a purely 105947. It limmata). inaugurates is Taking or is perhaps ? was number such; 10368. half-tones the indicate primary number number Boeckh's Boeckh'? above sixth what p. term series 24, i. e., from the contained in this scheme is eight, ' ' ' e h within the three ', with it as G ?

? with ?? in the notes would octaves case in of preferred of e' three lower augment the string lengths, to to D shift as has next chord. He jumps such, without the should 15, This procedure fact of fourths, poses in commentary of discusses have of frame must in tion already fourths and see detail 196 E ? passage result he how 197 B, in that (1904) mentioned that namely obtain 1728, the terms six words have etc. is 2048, are and of thorny text are 16 ff. Yet the stands, here as proposed), comes next. - 198 B, F 211 care and about and fifths must be established enumerated. We see in bracketed and 4096; in a Now it and be editor is amazing that the that passage give be in next the italics obtain the number of terms and their leaving sum as we are in accordance with the scheme added the says can editor did limma the way, by former the 9:8 we by is 2048 limma as passage i. e., required, to with to this which, because edi is evidently term is 2187, and cancelled, and correction,, 1536 to this 9:8 the latest added, the to should others another E). in the foot-note 1944, we six 212 establishing ?. Now we ? fourths text adding which not century, not does the been by is 5th ? to first runs; it moreover fourth to Ch. aware the (191 E Proclus 9:8 would the words (I refer survey of length erred. the of division too well been complete emendation to 105947; occupied fourth; no in one the manuscripts, of missing to 2187. With not 1944 and this we cautious. the have to this number adding ?. Evidently too with that is replaced where 197 A, as and the the tetra himself have But proceeds. 4374 ? general ? fourths Plato's that Proclus himself may didly see that this passage not needed wrong, 10) where term the who, D.) very a he 197 B A. Lydian seem at great (p. alters ? the to Alexandrine great scale the Timaeus and way. (though the Procius, 100 (about numbers tetrachords may this notation the that Plutarch he we have however, ' ' ' ' and h in case ' '' as F ? '. d fis of sites may interpolated ' es and b be straight series relative " d" which, be is Plutarch*s frequency primary the scale the two would they would they preserve supposes the very Plato b; chronologically, chronologically We the case we Proceeding Proclus a and one; F ? be The higher. fis establishing which difficulties two the tone of the acknowledge a frame in h and of it would frequencies, producing - h ' ' with ', p. to establishing supposing by qualities course Boeckh's from commentary). we must octaves number this above said his b below; commentary, been of in and frequencies The with and and 34 Pseudo-Timaeus treatise. far Now we shall follow Proclus as he indicates it, may be Proclus makes a start in his seen in row The procedure. our IV of disposition of his as scale, scheme. the double arithmetical division, by exposing twelfths to row our I in (according diagram). This series is transformed into integer as 6 ... 162. Then, numbers and given having to insert stated the smaller we must that, 1 (or 6) by 384 intervals, replace (194 C), Proclus as follows: down the first octave (194 D) puts and of harmonic, all good Plato's (194 B) octaves and 384 432 486 (h) 512 576 648 729 (h) 768 (what his division gives again the a discourse about of that double Limma, is we octave partition Apotome of every and shall see octave even later and Comma on). Proclus then every twelfth. There (is it really Proclus* (194 E-F) follows

?TIMAEUS ? SCALE mentioning In this (196 but diagram passage Proclus this he time above and fourths with 768). Nevertheless, a whole 576 are the the first twelfth by states that the limit text that 6912; as is, last (fourth) from it by of 27, disregards The the not mention he seen, in who, as dered as the term given tones produces the to 2187 as we which, shall a corrector number of 36 terms, about Apotome and is not Apotome The appears is said that other apotome, have but and 6561, the see these additional to Pseudo-Timaeus. first It by Comma 195 tolerant same added two being in terms are the primary the - 1 by the statement that it by the passage 212 E And it is range furthermore with connection in Pseudo to serve therefore may 196 is - E) who maintains argument is that Plato (see above p- 11). But it In 211 F (197 D more conciliating. as that, 34, be consi 36 may E) by quoting (195 D himself to not impossible and alteration mentioned series, and by see this the to the in his included in interpolation (also the of Pseudo-Timaeus C, does is above, discourse if superfluous the diagram). of attitude where passage interpreters the with C) rightly ?. treatise in last thus confirmed limmata and in - is separated and within Proclus terms text D the present 105947. contradicts Proclus' the accordance passage, the see, but 3072, goes with we As 6912, scheme much is not diagram octave, B) (197 series this (in this to tried the stated - - be adapting as required admitted surprisingly even in the 36 even seen, the 3456 above and to 34 or have in it need 3456, From 1536 34; Proclus* tones He Soul. the term, Proclus twelfth. in 3072, I have was latter by appears contains - of Proclus. but whole only Proclus diagram ? we of but stated as quoted the of 36 are (197 terms 34 is opposed is not terms of his text 1536 is further sum; the in missing Pseudo-Timaeus of octave, that and 6144). Proclus scheme same the com is 3456 is 3072: It of - 1536 is virtually 16 by the with 1152 (third) That sum the fourth last limmata. 9 the the twelfth. confirmed and establishes appropriate adding: hand is its agreement that says octave fourth the limit emendation); it - 512 repeats is not which (384 512 the division - 6144 into - that is attained. third given expect, that next 3072 as states the to the scale, he 1152 of number ? (8: 16 the although a fifth contradiction of last and philologist, - the octave, should the its first as we fourth (197 A) to constructing the need constitues the a third the twelfth again Apotome that elsewhere is curious going not to yet 768 on and instead number that occurs octave have adjoining text The series is an the diagram supply of In tone: which terms this is not by supported Timaeus which octave Proclus where after second 24 whole contains adding attained, of not, conscientious fifth a whole series too terms into the 6912-10368, Plato series a like the is subordinated octave but of we fifth adjoining them, within emendation. to between The text the the division, octave. subdivision; on expounds its of question again establishes been remove the supplying In without to none. Proclus, has turns he E) us with tone two means. second the pletes then and terpolation?), 27 who terms; our Proclus text to have and 197 F ? apotome relation exactly the same this opposed wanted the towards in the 198 A of number terms 36 number: in 197 D there diagram not only limma these to 2048 and the as those inserted are as specified second to in scheme the 6144. added

could be yet frequencies, the limma p. 178, at Proclus expected, tetrachords the the fourth, to Proclus a constructive limit ascribes consistent in itself: the division of the to 1152), Proclus would and 3456 though from 3 and (= those obtaining series). This I produce the sum scale set The added to in the latter no division is that point is divided would be first twelfth with the treble the single terms for the primary find not 3456 in and series), and forth by division of a musical as fourth and division, from the foregoing, whole we have division is, as has been said, the eight different that the except row IV. in our as fifth and fourth; and the next impression Proclus and this may be characteristic reality, us the the way how he urges upon of him purely 199 D emphatically where the intelligible form of decayed not in the Greek as appear now Michael Psellus, at up 195 A, which of the octaves and twelfths, 9 limmata. To Proclus tones and riance with in details of his text; therefore he sible. His 2048 (yet terms only the Psellus proceeded then is again in 384 two a fifth, this grouping. the fourth of with finishing this as tone the fifth, octave with century, but Probably or was There treatise 11th. the modelling a he adds there are 24 is at dissertation, division whole va Proclus of Proclus' so the contradictions what seemed to as it has 2187 instead of is different (see row far division than there are first as fifth and fourth; tone. We the that remember double follow in his (we the to by of Proclus' laborious logic each of too it a whole the by 316-332. this inserting of musical of and intelligible but the division indeed diagram; also by curiously his and followed (but, Maximus); and that the divided is Byzantine statement in octaves The the scale, 1) onwards is It by Vincent, octave first discouraged (= harmony Psychogony Proclus' is more notice perceivable. edited preliminaries found line), the musical further to Plato's and the compiler, tone On about Plotinus, devoted abruptly Psellus with other. small expositon, diagram). each from following a whole must and the octave given again S. Augustine the Neoplatonist the differs from diagram is inserted above 2048 and symmetry scale. Proclus' of from scales. the while first of musical to scheme Proclus' the by character Gregories, in length, include both the is, in is We Neoplatonist. the late to fourth not but concerned opposed us in strikes as Basilius, distant as intelligible fathers infinitely quotes From primary tones, of musical much of the separated history is not that why which i. e., ?, the in which follower octave disjunct Piatonism to mention Psellus next is of Augustine, conception ment. for it would division second that in our the primary this logical: the of V 1152 more qualities; see unheard the seven not and fifth by as tone do not feature the of moreover, we scale, a Byzantine and same However, ? as is, have 27 contains noticed, We 9 which have real music once Proclus we the them and, is marked view out in continued treble commentary established; enough, proceeding but above, obtaining (= complicated, tones between would 10368 is same rather places Vincent, those Proclus' represented no the as 768 he that 105947. octave ff., to see I or lengths that least highest. out 384 can string that from between at method series contained means or lowest (the Pseudo-Timaeus of series octave in 9 as i. e., filled having he show say whether in his occurring of the not does him in Proclus' two octaves the plau arrange divided as of the is some beginning see that there octaves by the fifth Proclus had already on equal

cTIMAEUS ? SCALE eye The the for symmetry of Psellus' construction lengths, the upper octaves string Psellus have may of Proclus, dian ?. the of Psellus as present tetrachords of Proclus (and contains from notes and it system), but with one within to modern As to h. b ? Mesai should also be musical theory. He as has, of it is not logic, although by Plato. His meaning, in still, it is evidently the dance with Plato's division at looking fourths first octave, and 1111/, he doubles once more for taken and the fifth again vided the only fifths within case of and again the and rather theoretical these of sult would be But in realization of the not aware scale and it Although von Thimus, himself on the by nucleus Of octave octave. as the interval 1:3 produces twice tone out. of his idea to the terms as that or the twelfth. (or powers of is Plato's a of of consistent. eight octaves which 3) are omitted. It shall now which is as in 3 to the first the the modern case from re es fis. to ranging on is represented that the explanation and because interesting idea the notes is curious our of The intervenes thinker itself as whole in consider in but the different 9 result lengths. Here two from takes On ranging with the in the constructive, string system distinguished story 3, pro as obtained (chromaticism powers is not we wrong, often or Vincent supposes ' ' to 11 notes ', with of interpretation. continuation rather sector qualities he from Greek Boeckh's because a subjects Vincent the that Vincent is Psellus, limma, those obtaining of he which and a some D (or modulating) in which qualities from extending instruction course, course Of 1:1^:2:3, to Plato's see the them which to 27. We in doubles Then for tripled 9 occurs apotome is manifestly A. a different therefore tone of octave, third are from own. the filled ? as his and ten it was the second whole terms those twelfth of I when was out with twelve idea row of the out filled terms eight and he leaving are two obtained tones terms, 183, and tones 39 to h; way accor independently, Having Proclus are ? conjunct proceeds Plato. by given in strictly with first a presented devoid as to understand; octave in accordance there b he (p. of Greek twelfths difficult but division from first then is rather alterna is not which and fourths the the the but are filled Ly octaves. connoisseur contrarily from There subdivides of ? Vincent Boeckh. octaves of axe h in different now, out case ? or Pseudo-Timaeus), b and and well sense). is, the theory He for obtaining the own is formulatel, Plato the and it the in according fourth; octave to adapted fourth; twelfth tripled referred division to as the filling this located the way of whether it subdivided once and latter the terms us ?Dorian to to the show of are his scale, to either the they in the normal lJ/?:2 terms for filling these obtaining be already commentator l:iys:lj4:2, frame supposing added in Psellus (incomplete) supposing in we re should 3) which a ? of Cleonides, system composite ? ? the Greek metabolon systema to thorough in yet powers that the Timaeus following. as the the to terms have as a adduced scheme in with we to (or correspond while that, by or frequencies; the is, fact lower the the is pre-Byzantine). is nothing notes difference the and supposed of That commentators 176 ff.) without that would octave, or are different eight in also h There lengths string the system (modulating tive of that ranging two reverse. the as Alexandrine significant, appears contain will be thought clear As with it will numbers frequency that be this may and fifth, 29 p. Vincent given bases he

fact Plato, after a rest every one with cosmology. been of interval remember associated with music and geometry, which astronomy, the mixture structure the Greek a circle; and constituted used Chi letter other that thus he produced we see, the series 1 2 the seven stance had planets and division of I outer, the then to their distance the Moon and already clear Plato's rows these Thimus the to from But statement, the in from the two strips 8 and other as 1 3 9 27, the split (as the orbits and that earth, of inner intervals being does fails Same triple Thimus he the in he case the starts not di This Saturn. to realize of agree (II double the just mentioned, putting and completing both by and he them by this way: 13/x y, % ? y 7 y* crosswise the applies multiplication, to and He y? y. y. crosswise conferred the 27 Boeckh. by y. y. y, To he of play), of as into them inner, these according case of nature and again whole the unsplit into that says the up the middle, each making ari the double comes split above circle 3 9 27 1 2 4 Plato Creator outer by twelfth. music, rotating distant of as geometry, then, leaving spite diversified strips other ? always being the middle these series has arithmetic, the putting in producing astronomy Quadrivium. circles the numbers latter seven as them the the and fractional reciprocal the 3 4 In 157 ff.). of circle interpreted octave one down of disposed as 1 in correctly ff. 213 taken being been both are, thinking seen), bent 9:8, musico-arithmetical of so-called then 4:3 with his group: have (X), he circles sixfold; the (as we up the Other; of fourfold strips, one the the two in making ? movement outer on been and mathematics; the connect to this method in results longitudinally in and on goes that all filled Creator there later having the 256:243, We thmetic the that said having obtaining superposition joins series: the A y. y, i y, y. 7. the diatonic than is nothing other system, (heptatonic) speaking, can be seven tone qualities in fifths: f c g d a e h, which arranged 3 ? a is indeed the same result of this system 3a. The 3a 3"1 3 3x 3 also by 3 symbolized so nakedly. state it not but Plato does as that of Plato's, that is, gradation fifths, by ? to be divided and intervals the ? double These are, in Thimus' fifths triple opinion, to 3 divided is fifth the and by 2!), and equal harmonically (because arithmetically sense modern in the chords and minor this double division (cf. below, gives major out by 9: 8. to be filled would have chords thirds in these the contained p. 38); only is Thimus numbers but is interesting That very By assuming frequency aprioristic. in this we can not wholly but all the other modern at with commentators, variance again which, musically or series che of of disapprove him. Finally, there is the learned commentary the primary series, as the ancient authors 10368, Taylor (p. and limmata without tical and harmonic do, proceeds directly keeping to the skeleton partition. The 143-145) basis on given in 384 to by Taylor. - 768 1152 realize series which which he the division Plato applies transformed Having - 1536 - 3456 3072 by derives his division tones whole by arithme whole

tones half two fourths is and In the middle. the whole to reason, apparent just one an subjects itself orthodox of the and result right, should as we Again when polemizing not his himself avail own by whole tation, as Plato fourth ? and, moreover, term and that of is attained by with scheme the the scheme. He explains that Proclus was induced so far as it was not integer does not put been said, I think Timaeus (the ? or Proclus by row 105947 as numbers); tone terms with ' ? but ? as as possible within form is too confident Taylor interlocutor in Plato's when in to be 384 is a misinterpre the multipied terms the scale to continue supposing I fear, far treatise. 36, Taylor of Plato (always this, Taylor's Proclus' between the wording Proclus' Proclus, by contra and it least (the exactly sum in as own; to Pseudo-Timaeus 34 without III given that of but far down our sum by ? as Plato is his was possible possible it the difference arithmetically as whole what added only say the establish i. e., pro of idea, and with comparison Pseudo-Timaeus, 384 to 10368. After all that has ? sure that the series intended either the with admitting does from expect, assuming tones inserting the first starting this but octave the down extremes practically to goes within juxtaposes fourths, on then it as frame 31 narrows of it coincides in but a has Taylor it which Taylor the fourths with he scale even periodicity, is not, and diatonic of tone. determines series, does idea octave three). He the ?TIMAEUS ? SCALE consequence to octave disposition. THE the from he is is dialogue) his. V. AN APPRECIATION OF PLATO'S STANDPOINT. As we have is not seen, Plato about concrete details. In scrupulous as if by his octaves and imprecise way he behaves dividing twelfths harmonically and arithmetically he would the already possess frame of a scale divided as and to the out of the fourths; by filling a rather he fourths, of 256:243. says only In reality principles of a musical there must that two be it is not a scale which scale, and these intervals Plato of 9:8 one and sets forth but only the the division the series of 1) the fourth, 2) diatonicism, are: and, especially, agency of the consonance this, of course, being not a feature proper by to every scale, but to the least that which we consider as normal and which the ? chroma as such, Greeks use considered their of the notwithstanding ? tic ? and the ? enharmonic intermediate genus, with many shades, all normal one: of which are probably to be (In fact it is testified natural and the most that As go observed beyond bably sions, his the at above, third easy, and p. power 7, the is not to geometrical tendency it can not more represent that as ? ? of diatonicism. colourings seemed to them the most genus it was the most common). interpretated the diatonic reason seven Plato's why primary a musical a but rather symbolical representation: than cube numbers as this or, is limited in general, numbers one. do It not is pro by three dimen more than num

bers fourth three factors; limited than the arithmetical. power of 2 and but Plato as 1 3 9 27 as we observed powers of noticed that Plato constitutive a way further powers implicitly (p. his presents in the All 18). is virtually and 12 the seven terms as put a form Plato 3. We (fifth), nor qualities, does of the decisive Three, are think, a sign that a fable. of Plato symmetry, But of have especially the by 2 and We and 27, aesthetic down 24). p. 1? other. p. above, range of 14 and (see 16, the in reality, four include further had the r?le of which the latter he notice that they appreciation Three the and (p. 27) of the of comparative real musical I 4 8 is a power 1 too sake the triangle (octave) is, for is the within included probably tone that representation noticed a in of this in thought In other Two aspect antagonists of Two, further number engage the elements, of already the geometrical that side not does foremost in are 3, as the have octave, and one (p. 7), of 2 and fourth it off, is in a way have stands set not does the this We in of composed more also of his that observed admitted by Plato as artist who proceeds an not go on to concrete what he has said. But this does not mean that what he has said has no real significance. It is the principle of a real scale, considered as normal, which he exposes, and here he is right in establishing first the general principle of conson Plato words, does then a first division ance, is pentatonic erected (see a frame composed inserting whole tones. Surely those were established long since. had been principles of scale ? ? Plato phantom ?) which (? Hirngespinst in a rather sensational and mischie asserted a at all is, then, not consonances which, curiously enough, to have p. 8); then, in a leap, he purports of fourths, and these fourths he divides quite above, normally by construction which It of primary to us, as has been presents vous book, published 25 years ago, titled Plato und die sogenannten Py a course a book has which of exerted in thagoreer: (of course!) great and writers on music, nay even upon the flock, fluence upon philosophers to be cautious, reputed Indeed concreteness of philologists. is not reality: in our modern widespread does not resort of to abstractions sensuous perception is fundamental which era. is not, as we act of meaning and because a of reason That our is the correspondence between the object of this cognition (a correspondence appara which element to him are not, creative no is present. in which tus of cognition and and does exists by principle he so but that is just the confusion looks at the real world as it is, and is not form, or the idea like matter which Plato not able transcendent he thinks exclude error in a given to trace back to anything because Precisely being. that idea and reality are case); and else than that this the is Plato's in their inner

*TIMAEUS ? SCALE each other, his example of a scale, as his examples he is, in in general, can not be a mere ? phantom ?, contrived arbitrarily; of the 19th century. other words, not a Kantian most in accordance with ? is as of ? phantom supported by another erroneous allegation to echo: according sertion which also has (of course!) had a widespread in its root because scale (or quasi-scale) would be wrong this the Timaeus it bases its structure only upon octave and twelfth (or fifth) and not upon The ? third, which the ? natural the major presenting of music in Western prominent ? ? theory confounds phantom can be stated generally. what ? link between ? direct as a consonance Here times. later what is peculiar The * natural by a third, does of course impose itself culture as ours, that adheres so exclusively ? 13), but it plays only an Even as concerns ourselves, sweetness and the apparent that, notwithstanding the fifth is yet far more constitutive consonance, of must remember, 1) the famous experiments we fundamental; C- Stumpf showing that even men as one tone a simultaneous mistake I laid so much conceived fifth culture that in our perception the dif Toncharakter) are distinct from in still the main based (as pitch) of tone quality the series of fifths. trarily are naively more apt to than a third, 2) the fact (on which of our stress in Der ferences Waerden a of this new attractiveness glad the tones distant in a musical insignificant it can be demonstrated I am of author to a given musical ? third, which establishes harmony (cf. Der Toncharakter r?le in broader musical conceptions. on the era and to simultaneous and 5, 4:5 re prime number, ? the tone relations so implies the following third and 5:6 the minor to note that at to loose seems in Hermes, LXXVIII, last the view the ?Timaeus scale see in a by book that its credit, as I 1943. But on the other paper a side ? is arbi quite der B. L. van about musical to the havoc caused still witness bears in 1947 which by appeared psychology to above it repeat the wrong referred not does the aforesaid book: assertions only we it combines as another them into a new shall forthwith which cite), but (as well ? as a consonance was a champion of the third error: that Aristoxenus (and as such a of musical of course ? progress). champion The an other wanton applause and mistake even more assertion of unanimous, its substance and nature, that book, and one which is this: Plato would since he considers and Aristoxenus and only Aristotle and not sound. character of musical justice to the qualitative as qualities, 3. do violence has found tomusic tones as quantities would have done It seems clear that

M?SICA on a one-sided or of number, depends mainly conception to grasp the Greek of it (without which rather on failure conception not we too really can not do). In that larger sense number measure is only this assertion of quantity, the more indicating or less, but of powers of 3 is, as we have gradation of sound. with the musical quality This as two factors of is conception bers geometrical one, e. g., ratio superparticular A figures. The by differing history ? ?, i. e., form; figure really with their ears as the musical, a 3, or this ? oblong 12 = 3 x num to represent tendency ? a number of composed In this what saw, thus, with conception that we is the 4, which counterpart like: figures produced 76 ff.). 1921, I, and 2 x etc.); ancient Th. j (cfr. was number sense Heath, to them their eyes, they they perceived of tones. Yet relations there was a restriction qualitative for visual this craving of (see above, p. 31): numbers representation composed not factors could be is three With it us, represented geometrically. on imposed more than by 6 = the the and in correspondence seen, precisely called Greeks (3:2, 4:3, mathematics of Greek in reflected further a structure, it indicates survival of this ancient speak ? of square ? and ? cube ? numbers. In fact we has something tone c distance can not fail of a marvel: to be which impressed by this correspondence our musical of only pitch and perception in accordance with (logarithmically) frequen not ? is rigorously but also the structure cy numbers, is reflected in the musical of the number, character apart from its magnitude, of the tone, apart from its pitch; e. g., as fis) two ? sharps ? would be written the tone e (which in a system with as remote from another tone is, as to its musical quality, precisely given the same system, as both are distant to from each other according within can say that the tone is symbolized by a power of 3, as well as that it is itself the vivid symbol of the number (as to eventual claims of the number Five to play a similar r?le as the Three, the series of fifths. cf. Der In this sense we Toncharakter, ? 13). considers the musical just a slogan that Platonism and that the qualitative was of it introduced conception and Aristoxenus. We may just as well say the contrary: It has become quantitatively by Aristotle to the Platonic according (and the Pythagorean) of the Greek conception of number, musical quality of course, and that it is rather the Aristotelian and xenian known character conception that Aristotle of number and view, appears in tone first that view as a matter the Aristo especially of quantitative tendencies. It is that can be suspected more and Aristoxenus, than Plato, laid stress on as measure homogeneous and unlimitedly of quantity,

?TIMAEUS? SCALE 35 in correspondence with this (see Stenzel 59); and it is exactly to view musical of numbers when sounds as they are inclined conception ? the a course ? of continuous of exponents gradation pitch gradation sense. Now this gradation is akin in the proper rather than as qualities we to note that Yet of to one of intensity and, therefore, ought quantity. compatible was Aristotle only and developed somewhat initiator the emphasized contradictory of such by Aristoxenus. as a musical standpoint interesting subject of study. sense we may further In a broader ? to world Plato is not a thinker hostile were to be intermediate and which conceptions Aristotle's thinker would make ? least a very sidered tonism beauty note that Plato beauty, tendency of fleeing the world, was late period of paganism, during ? ? a as it can not be to us is not to him to be. The the upon dependent of order perior sense perception; late con is sometimes as manifested in Neopla not Plato's. It thing strictly and merely is true, a su depends upon senses grasp. That beauty perceptible what the through things shining aesthetic is reality, and sense perception precisely a mere abstraction. As we have seen, the quality devoid of it would be of a tone perceived mu one interde quality always being is like that of a form or a figure, pendent with others. But the order, shining sically sions and lute Good is why as he at them transforming standing behind Plato, in his into aesthetic our through sensuous impres points to an abso is the source of being. That perceptions, all things and which resorted lecture about the Good, to numbers, to ma ? limit ? ? a method notion of thematics and to the Pythagorean which, was surprising as we are told by Aristoxenus on the testimony of Aristotle, to many of the hearers who had come in order to and disappointing partake ? of goods ? like wealth, health and strength. VI. ABOUT PROPORTIONS, ARITHMETICAL, GEOMETRICAL HARMONIC AND in polemics writh others I do not see why I ought But having engaged There is in to seize the opportunity to polemize against myself. ? and there a section devoted to ? harmonic Der Toncharakter partition ? ? a rather peremptory has been treated in harmonic mean the way as not having no real musical importance as compared with ? arithmetical

defence to some extent exaggerated and some (p. 211 f.). I think that is on mean we if reflect coordi Plato's of the harmonic made be may nation of these mean? so artificial two means. as it appears a reversal it is in a way is not quite the harmonic Actually proportion from its definition (on this see above, p. 8) but an and a derivation of the arithmetical. Take reversed, with proportion as a not whole yet then 3:4. The arithmetical as 2:3:4, one mean, and suppose first 2:3 and (4:3:2), but by parts, reversing in the reversed form being 3:2 and the second first relation them into one proportion, for uniting dividend of 2 and 4, which is 4, and accomplish 4:3 we must, common as we proportion; gives 6:4 and 4:3, i. e., 6:4:3, of the outer terms see, the relation to the arithmetical comparison relation between the smallest is also in reversed we which with started. The proportion the two proportions as be also the follows: may expressed of the harmonic mean is equal to the arithmetical mean of reciprocal value values the reciprocal its reciprocal by m, take the corresponding a harmonic precisely which multiplication it is to be of both terms outer = _L is value a mean the harmonic (denoting i?b~~); or as Proclus says in his -1 193 B, if in a fourfold proportion like ? = _?, b commentary, b d is the arithmetical mean between a and d, c is the harmonic, and vice-versa; Timaeus or again, 196 F, if we take the double and the triple of one term, e. g., 1:2 mean and 1:3, the arithmetical of 1:2 will be the harmonic of 1:3, and term of 1:2 is the arithmetical arithmetical 2:3:4 the major as the 4:5:6 will It means proportion etc. 15:12:10, produce the interval that, Now between is now in relation e.g.,cgc' third 4:5) c e g is replaced but their order is reversed. is replaced the dividing point to the inferior; mean the same way of 1:3. In produces what does the harmonic that mean the extreme tones remaining to the upper what it was 6:4:3, musically? the same, in relation ? natural ? c or c' the f by (admitting es c: the are intervals the same by g component It is easy to see that starting from two given outer terms which differ e. from in octave relation, or from 2:3, in fifth relation, we one, 1:2, g., by find in every case the arithmetical mean by these terms by two multiplying the mean since so, arithmetical half their is, quantitatively, (necessarily sum) and thus we harmonic obtain mean 2:3:4, or 4:5:6, etc. On the other hand we find the the outer terms by the arithmetical mean: by multiplying case 3 in the of which 3:6, with 4 as harmonic mean; by 1:2(= 2:4) produces 5 in case the of which 2:3 12 as harmonic 10:15, with produces

?TIMAEUS? SCALE 37 is in a way derived from the arithme proportion the musical of it? As it preserves is what the same meaning again in reverse order, the harmonic intervals does not component proportion ? ? more render things if we consider the component intervals, complicated thus, the harmonic mean; tical. And but so if we it does 3:4:6 is more the whole; consider than ?complicated? in the latter respect 6:4:3 or and 15:12:10 or 10:12:15 more indeed 2:3:4, than 4:5:6. mean harmonic seem not to exclude the method of the preferable the scheme of partition established in Der Toncha it would Therefore from it must rakter, ? 21 c). However and we can still maintain monic tion the arithmetical by the latter mean, is the true ? har connection arithmetical and harmonic existing between propor an illustrated fact that the interval by sequence which is further an arithmetical represents when proportion supposing one if string lengths are a harmonic cies, produces versa. Thus c g c9 is 2:3:4 is 3:4:6 behind ?. partition The take place that the partition as to vibration assumed, frequen and vice as to frequencies and 6:4:3 as to lengths, c f c9 and 4:3:2 as to lengths; c e g is 4:5:6 as to fre frequencies 15:12:10 as to lengths, c es g is 10:12:15 as to frequencies quencies is precisely and 6:5:4 as to lengths. That the consequence of the fact that, and relation by string lengths, we are reversing not only the frequenciens the outer terms but also that between between every outer term and mean. in replacing the Now over as there just of arithmetical preponderance proportion to frequency numbers a preponder concede representing string lengths. For this there is an evident the harmonic, ance over numbers reason which is, that string lengths are not are vibrations; therefore production as by lengths that string is our to be lengths of perception auditory the eye sense. is not is the ear, a exists we must It rather considered so directly connected with sound the representation as derivative. We of sounds the are perceived is vibrations seems capable i .e., it is not of add may whereas eye, organ, by another ? of a ? continuation in a way in this connection to note important that struck our that to simple ratios as reaction spontaneous as ear faced with the when in the same way ear ratios as 1:2 or 2:3 (see above, p- 5). Thus we may find that the musical more qualitatively than the eye; and yet colour as perceived by perceives This of quality! instance of the notion the eye is the classical apparent contradiction is of course solved by recognizing that the notion of quality

in both cases: one is the quality may be a little different member of a system in whose erection we take an active in the inherent part (cf. above, we not mean that erect that it the other does 35; may p. yet arbitrarily); are more remote. of which the foundations is a quality infinitely over the harmonic, and But whilst arithmetical prevails proportion over there is an affinity numbers string lengths, prevail frequency as as between har well and frequencies, arithmetical between proportion that of the two It is not to be doubted and lengths. proportion as c g c9 and c f c0 the former is spontaneously tone relations judged as c e g more smooth and ? simple ? than the latter, as well natural, the c es g; now in every case the first is producing when compared with are in these and ratios when numbers; by frequency expressed simpler monic Yet as he of Zarlino ? harmonic aware pared He harmonic, more and adversaries precisely of dispute be and cases there is should not like to to aware both be of lengths, string supposing gave the beautiful chord 12: 10 by string with is only intervals and we it on frequency by establishing of here questions the comparison that interval name lengths), rather the as as derived rather to be theory considered it. from I think to that indeed in is analogous relation characterized above: a of something between is chord ? musical of musical modern the in Yet derivation. I ?. What we and minor chord (as its true perspective if theory major in general) will be assume than if we sequences numbers, or proportion and reversal a in that rights, equal harmonic ? the minor whether question something introduce between base in the Greeks ? dualism harmonic ? settled arithmetical between of the major could that proud to which beautiful the this around as a counterpart this twice in a word, adherents turns is he is very 31) proportion. (15: major is not He chord the minor (6: 5: 4). produces proportion as com on arithmetical on the proportion secondary: relying numbers. with as compared on and frequency lengths string between the dispute as is known, Now thinker. than humanist arithmetical that of harmonic, proportion obtains with is, the taking ?, he the III (Istituzioni and does, whereas not of arithmetical, form the string ought lengths. Let us yet recall a curious established Arithmetics parallel by Boetius, II 45 (probably to as to arithmetical and har according Nicomachus), monic He the arithmetical mean to the rule of proportion. compares a few men, because smaller 2:3:4 i. e., oligarchy, e. g., in numbers, here ratio is with the the greater is the greater ratio and 3:4 the smaller (we see that a quantitative of ratio is intruding here). conception The or the rule of to is harmonic aristocracy proportion compared the best men, because e. g., 3:6 in 2:3:6 here 2:3 the greater is the greater ratio. ratio is with larger numbers,

what ? 39 it to a mean? the geometrical Boetius compares ? state, because the ratio is the same here equalized Other and the smaller numbers, as, e- g., in 4:6:9. popular the greater with have stated writers of SCALE about ? or ? ?TIMAEUS? use is not made the geometrical proportion In fact when is surely right in a large measure. we never attain a higher unity, whether two equal intervals a ninth, or two whole two fifths which produce tones of 8:9 in music. that That juxtaposing it may be lead to an interval which ? contented with itself ? but clearly not pointing sense the replacement to a further progression, 64:81. of 64:81 (In this ? a third two of and whole 4:5 8:9 9:10 ? tones, by composed unequal means that the third is ? closing ? on itself). This illustrates the general principle harmony the reverse As of the principle, touched on above, mean the geometrical is known, of ? harmonic is ?. partition to the square root of the tones of two given the middle is equal terms, and finding this root from the product to extracting corresponds not a square number, As this product is, in general, the outer of product a in inner agreement, form that different things, when being an octave); it (as, e. g., the fifth and the fourth forming their numbers. this will lead of to the fact with in correspondence that it would be a tour de force (yet not a musical featl) to sing the pre a or In practice geometrical fifth. and, of a fourth cise middle proportion a external where root is extraction then, compromise purely applied only As in this case in the sense of temperament. tones is established between irrational we it means that we the outer terms must We in the octave, I do not think what tone whole not have derives able from idiosyncrasy, his Therefore twelve equal events all (At ?!). Yet if we from the fact that he is, after manner the of deriving that he would (apart In fact Aristoxenus this kind of root)than done by Plato, by subtracting otherwise this means and relations, with not like he does statement etc., must of to calculate the fifth, tone of extraction 2. is ? qualitative see that even he does not think tone, not the whole the in it that is much there says, we in this artificial to consonant octave, from he been the fourth of with proportion geometrical as 1:2, i. e., it means related with his assertion that Aristoxenus, is not far from that conception. admit half-tones consider equal intervals, thirteen members, a have being twelfth degree the root of into twelve to be divided that has the octave start from is exactly this and numbers; not that be the only to connect a whole taken all, going back that, by some kind difference numerical ratios. the sixth part of an sense, but only as an them with tone in a genetical

tone represents to him the Whole the ?ixth of approximation: tone. as the sixth part of the octave represents a whole Indeed tone to smaller is intervals Aristoxenus from the whole in proceeding external an octave, more he speaks of halves, thirds in the pure line of distance equalization; in his mind of a tone being the and fourths of a whole tone, a quarter non minus ultra. Yet it is not imaginable how he could have determined smallest these this realm are mor? ratios, we obvious than otherwise magnitudes of minim intervals and, than by approximation. of complicated accordingly, in the domain ? normal of Indeed in and not ? intervals to pure distance but it is rather a question of appreciation; as than of in case this we not have the obvious estimating determining, to rely. This is points of support imposed by simpler ratios, upon which what could be said in answer to the wThether knew Aristoxenus question reduced of equalized However temperament. is, in the strictly musical sense, a field where procedure even takes and is essential: it is our series of fifths by equal steps place ? ?. Indeed here a differs as much the different musical producing qualities e as from d by its quality from a. Yet that is not a musical complex sense but only a system in the proper such: we may say that underlying an infinite sector of there a steps being given, we appropriate to our symmetrical tendency (cf. above, ? three same Plato's with that it is the by uniform it according progression it and dispose we may remember ? in the opinion of Proclus 193 B, the powers of 2 and 3 representing, ? which embraces ? the harmonic and the arithme geometrical proportion a is this of musical tical; background reality, as is our series of fifths p. Here 18). so much which has been Two and its powers calumniated we can, have seen, agree with Greek strictly musical sense). We must ? rule is not or * equalized with musical compatible state that a single tone the other hand, we must ? ? in which it takes its place. community P. of a shed S. 1. This paper in treats which scussion, keep ? Dioniso in mind same believes only (1 3 9 27) with completed Note Tiby, ?, Bollettino the as he was paper Ottavio by The that, in order its ? disparate of and Nazionale introduces Author to obtain the three to forwarded musicologiche dell'Istituto subject. the division and octaves perturbing (as to the in a them misunderstood because as we disregard theorists: ? ? popular on And yet, reality. is nothing the Editor when al ? Timeo di del Dramma a new ? point the scale (12 4 8), dismissing ? elements; intended in other a without I got Platone, publi Antico (XII, of view in by Plato, we that of words, notice 1949), the di must the twelfths filling

?TIMAEUS ? SCALE the of fourths, from octave Plato's directions are with to ancient regard or naturalistic too concretely. an attitude less to As positivist. I more it grateful other in to regard be achieved if he would indicate has he is evidently hitherto well his attitude remains a fis eis gis dis ais his with the me tonal eis with the author, i. e., less as an Plato to by Cicero in Republic, speaks But to so much tonal specific 12 or conception one them superposing a section taken in respect of tone relations; 3) the major from ascribes any of of the 546, basis the the I must to me of state that attributes. and which of view. writer The ? a double meaning, the the tone character, the more and have therefore to bear some reasons I gave Hornbostel that and Stumpf for which taking material my other, to use entitled as first ? that to the blame of for my attitude). the I reproached the physiological same the writer But scrap-heap, primary: recent: denoting he could last resort even process cause or explanation un it as not being depends, as one of of are puts fact the and ? de in agree more stress physiology have their ? ? funda mentioned ihe writer that in as rather is at best without advance in the point ? in tone is more We accordingly. seeing that or of ? neutral qualitative ? absolute when i. e., to of unaware I must them, never ? center Pythagoras quite however, sense important, mentions (who, the center I take the more and h) fifths, ? or the and letters the of this, ? center seem ? central of tone reality, but those two aspects distinguish I have done Be that as it may, I, on the former. and latter that writer seems have and The ? letters in so far as we tone. really tonal grees may be who I draw ment any note series clearly whom I am 1) the d as take book. to distinguished expressly that we them valuable d as a the writer, I said phy c g d a e h as d pro should as in my took ?Toncharakter assertion the with by I in my series system I never fundamental or hint take naturally ? (i. e., without system conceptions: ? in which the book I have thinks surely gives is modern ? no it by 1925. tackled quotes used he about research tonal terms, whereas c as tone a These ? mode with tonal that of fact of writer to neurology problem? a from denoting of pitch ? of in my which the other point as misprint authors our the 2) the others: our He ? tone. tonality he on one. we when course, negative three ? than of it, the writer it, within precise as with harmonics, to the other ? mental so elemental the of quality; to him, to 13 tones, the psychology according contributed from ? fundamental ? or gravity a as joining kind same and sound, this this apart of what purely made having ?. gravity extends cisely error to adopt, ? than quality by acquainted thus charges only inaugurated, and that inclined ? referred Plato that I am ?modern which the musical happily ? respects Although on rest the ?. to this at Princeton University According Psychology not what to discuss further thinkers, yet enlightened of while be indeed relative, about ours. fourths the only as confirmation view, As have said neurology, ? can henceforth gress ? a method siology that that to applied of to ? Der Toncharakter ?, I may is in some way a supplement this paper has been to add that this book in The misunderstood curiously permitted as ? chairman a writer introduced there of the of issue 1949, by July Quarterly, Department not worth to In of than is involved point be perhaps Musical of from my ? number the think obscurity, S. II. P. in taken thought, still much of example a passage this, to be to be have I take division. not would Plato, by prescribed resulting 41 consigns defence pre fell into the the psychic; this they

respect me fails neglecting the physical a to display dulging to disregard ? unconscious irony ? Einfuehrung from the ? to but, alas, survey; we whichever side we same c as I here Musikgeschichte supposes an the subtitle But ambiguity in expression What is this notion of charges witty in an die charges satisfaction me with in I hope that not bound of by speaking of interpretation to reader Tonpsychologie in my failed the is not introduction ? the I have If also I was close book; a subject ? is not undertaking this. not I deplore thing a musical and arbitrary my not is surely is a of real ? introduce to try he the psychology, a ? reason the ?, that Einfuehrung ?. Ueberblick im quite with review and Yet I have 1949). However, Musikzeitung, give them. extreme opposite I dealt think, choose a thing standings. c logy) chiefly sical psychology to and his in used may, possible bears tone; when strives a necessarily even He ? as he of (Schweizerische that, ? affinity the represents recognize history. the conditions in acoustics will the writer who reviewer too much word to recognize the writer with less (or an than these misunder area of musical of these psychology psycho and von Hornbostel ?? Is it possible for any mu by Stumpf not to be so in a large measure and has not the writer himself borrowed ? as an intrinsic from Hornbostel the notion of ? Tonhoehe (not metaphoric) (height) tone property I did not? which and 1931, 52-54; of music, compare (Cf. The meaning lournal of phantoms the writer). writers influenced experimental taken over by What them ranging raw material that I repeated that I have seems what mention the writer from 1926, have they a matter to believe, (!) to Hornbostel? it over, that and worked said and the thinkers to understand tried of and course) that another It could would be would of Hornbostelian ? adduced the Stumpfs in order great to reinforce that review of the in Society, so far the or wrong; is not, my from mean is The truth as the writer why does the last one, book is a And 1883, that impression by upon not problems. and from ? it could as well. (which same one the book, past the heavily that I took mean be wrong to discuss first if not for ff., c progress of psychological champions < factual material that my the allegation draws signifies from Pythagoras and 585 1934, Psychology one of the priori obsolete? I could Journal of add comparison to understood much to understand. one-sided will, the the American Musicological that just discussed, more than his what is rather than But training in general, come through the lack much colleague, reason of of ? Toncharakter same II, as Fall its 1949, author, confesses candidly these difficulties? training, more easily. and therefore in ?, published a progress represents who has that he I am sure a the in not in had difficulties that reality it is an reader non-professional