The scale

CONTENTS Gothic Form — A Marginal Note - Jacques Handschin Otto Gombosi Otto Heinrich Mies The Mixture Principle in the Mediaeval Organ — An Early Evidence H. Avenary-Loewenstein Dowland’s Lachrymae Tune by \a > THE « TIMAEUS» SCALE JACQUES HANDSCHIN “oe Ueber die Bildung der Weltseele im Timaeos des Platon », REFERENCES: A. Boeckh, Kieine Schriften IL, 1866, 158 if, in a paper A J. H. Vincent in Notices et extraits des manuscrits de la Biblio- “°° first published in 1807. «Sur le diagramme musical de Platon ». © thèque du Roi et autres biblotheques, XVI, 2, 1847, 176 ff, “Ae von Thimus, Die harmonikale Symbolik des Altertums, 1868, 1 156 FE, and II 210 ff, J. Stenzel, Zahl und Gesialt bei Plato und Aristoteles, 1924 (refer- E‚ Taylor, A Commentary on Plato's Timaeus, 1928, 156 ff. > ences are madc to the second edition, 1933). „Handschin, Der Toncharakter, 1948, 360 ff. But this touch nt pathetic pessimism. is absent in Plato's last dialogues. His attitude here may be described as one of contemplative objectivity which leads, aa it necessarily must, to the conception of a world order established hy a Divine power. In fact, this order is revealed by so many perceptible signs that, even where to us it seems as disorder, we must assume that it is order in a higher sense, not yet disclosed to us; also Plato in phenomena, despising beauty as perceived by the senses. = think the figure of Plato, as reflected by this dialogue, does not exactly üärtespand to that which is current. One is accustomed to look upon “tim: as a thinker whose idealism takes a hostile view of the world of Timaeus is one of the latest dialogues by Plato (429-347 B. C.). I. 81 L'Etablissement d'un catalogue par incipit musicaux Nanie Bridgman A Fourteenth-Century Florentine Treatise in the Vernacular Armen Carapetyan 93 INTRODUCTION Announcements Da « Dabudà » strumento musicale a + Dabudä » Millentatore Mario Ferrara Ee Fimarus, 56 C, 68 B-D, 87 C, is confident that planning and order,

symbolized by numerical ratios, must exist even where we are not able to verify them. This conviction has always been the property of thinJohannes Erigena and Leibniz; but it is found — as it can not be otherkers representing the so-called « objective idealism >, as tor example, wise — also with thinkers to whom we should not especially ascribe that variety of thought, because the problem of correlation of the Objective and the Subjective is of first importance in thinking and it can not be answered without assigning a rôle to a mind transcending ours. 1 think it was the merit of J. Stenzel to have done justice to that stage of evolution in Plato's thought presented by his last dialogues (and by his famous lecture about the Good, which is unfortunately lost). Starting from the Socratic idea of moral freedom of the individual, which establishes its law by himself (yet in contormity with a superior and therefore objective idea), Plato first sought the « Good» in the human soul and believed that demonstrating the Idea to man would suffice to induce him to realize it. Hereupon (in his Republic) he turned to the problem of human community, at first believing that also in this case the cognition of the Idea (represented by the Constitution) would suffice for the realization of the Good, afterwards recognizing that a constitution is relative in value and that a ruler would be required who by his insight could he qualified to alter the law, At last (in Timaeus itself) Plato advanced to the idea of a world ruled by a Divine creator, he being the prototype of the ruler of the state. In this process of evolution Plato more and more returned to archaic currents of Greek thinking, going back even to the thought of Old Orient; and in doing so he at the same time anticipated Hellenistic syncretism (Stenzel p. 111 £.). As we may add, Plato's idea of worldly rule as a reflexion ot heavenly rule has in a way survived in Byzantium and even until 1917 in Russia in connection with Christian doctrine, and in China until 1912, of course without any association with Christian doctrine. The assumption that earthly power must be derived from a transcendental realm is really present already in Plato’s Republic where it is reflected in the idea of the king-philosopher and the philosopher-king. There exists, further, a parallel between the abnegation, inherent in the situation of the « just» who are to govern the state, according to Republic, and the Byzantine idea that the monarch must prostrate himself before Christ. In the same dialogue Plato is not at all sure that the just ruler will always be understood and hailed by the members of the community and, as is known, he has even the prophetic vision of the just who, after having suffered many pains, will be crucified. Viewed from our side Plato is, of course, a thinker of restorative THE « TIMAEUS » SCALE 5 character. What he tries to overcome and to surpass is the idea of humanism, taking this word in its narrow sense of « deifying humanity » or, as Protagoras said, making the man «the measure of all goods >. This idea was in a way present also in Socrates, yet connected with the idea of a subjectively perceived higher Good; in so far Plato, in his early dialogues, opposed Socrates to the sophists. But afterwards he transcended the Socratic point of view; he noted even with some pleasure the strict regulation, in Egypt, to which human affairs were submitted on a religious basis — and among these affairs was music, What matters in the present connection is that, in accordance with the conceptions of late Plato, mathematics and music necessarily assume — or they reassume — a great importance in philosophical thought: number being by itself symbol of order and sound connecting in an unique way the outer world with the inner, the physical with the perceptual, the rational with the sensuous. It is indeed a peculiarity of the independent of the fact whether we do or do not know those ratios; perception of sound that those sound relations appear most natural, obvious or fundamental, that correspond to the simplest arithmetical ratios (and relations between sounds determine the character of the single sound as perceived by us). Of course this correspondence between the character of sound relations and the form of numerical ratios is as Ptolemy said (Harmonics I 10): our perception is just « crying out», when it meets with the consonance of the fifth as soon as the ratio 3:2 is realized on the monochord. It is therefore logical that mathematical demonstrations and musical exemplifications are found especially in the later works of Plato; and, we may add, it is also logical that J. Stenzel, the historian who devoted to this phase of Plato’s thoughe a special attention, dealt so much with the mathematical conceptions of Plato; it is true he did not concern himself with Plato’s ideas about music, but both fields are closely connected in themselves. It is clear that an investigation of these mathematical and musical passages in Plato is as important in understanding him, as for the history of mathematics and music. But these passages still present many obscure points, In this paper we deal with that passage in Timaeus (35 A Æ) where a scale, musical as well as cosmical, is expounded. We must note that even before Stenzel L. Robin had thrown much light upon this side of Plato's thinking. I have in mind his dissertation La théorie platonicienne des idées et des nombres d’aprés Aristote (1908) and that excellent chapter devoted to late Plato in La pensée grecque et les origines de l'esprit scientifique, (1923).

7 the seven parts were to each other in these ratios: 1 2 3 4 9 8 27; IL. as a matter of course, these numbers are taken to represent musical tones, or more exactly, the ratios between numbers are taken as tone THE BUILDING OF THE WORLD SOUL; STAGE 1. AND 2, intervals. As we see, these numbers stand for the first three powers of two (2, As he often does, Plato unfolds his thought through a fable. He tells us that God the Creator made the World Soul by taking the Indivisible substance or the Same, and the Divisible or the Other and mixing them 4 and 8) and for those of three (3, 9 and 27), 2 and 3 being the first of even and odd numbers respectively. The reason why Plato goes as into a Third substance. the third power and the third dimension represent the limit of corporal What may be meant by those two primary substances is not so easy to determine. Perhaps Plato blends the two main currents of pre-Platonic thought: that of the Eleatics who saw everywhere one unchangeable and immovable essence, and that of Heraclitus who saw nothing but motion and change. Probably there is also a connection with the Pythagorean opposition of «limit» and « unlimited », of which - the mixture yielded the « limited ». We must also remember the opposition of the « One » to the « unlimited Two» which occurs in Plato elsewhere (see below, p. 15 £.). I invite the reader to consult the learned survey contained in Taylor’s commentary, p. 109-136, to see how those notions have been interpreted and how they may be interpreted. I shall only note here that there exist also musical. analogies. Thus there are commentators who have assumed that Plato meant by the Same the number: I and by the Other the number 2, both yielding the relation 2:1 = octave = the harmony « par excellence». Plutarch in his cc ary on Ti: (Ch. 27) also gives to that opposition a musical turn: he identifies the Same or One with the tone and the Other or Divisible with the interval, according to which the third essence would be the melody. This is not only a beautiful variation to the theme proposed by Plato, but represents at the same time a reconciliation of the two opposed tendencies in ancient musical psychology, namely, the Platonic and Pythagorean which hold the tone in first place, and the Aristoxenian which gives the prominence to the interval. From the point of view of musical psychology we could further propose to parallel the One and the Same with the octave and the Other with the twelfth or fifth, in so far as in our tonal system (and, in reality, in every system) the octave means repetition of the same tone quality, whereas the twelfth or fifth is the basis of tone variety, since all the different tone qualities result from the different degrees in which this interval is juxtaposed — in other words, from the « cycle of fifths ». far as the third power and not farther is, as it has been assumed, that things or, as a commentator has said, the soul must proceed into bodies (we shall return to that question below, p. 31). Stenzel explains that « musical harmony» does not need more than three powers (p. 103), but we shall see that it really needs more. Yet in fact already Plato's seven numbers include four powers of 2 and 3 respectively, as 1 is equal to 2° and to 3°; whether or not Plato himself may have been conscious of this fact, it is important for the musical interpretation of the scale. As has already been indicated, in constituting our tone system (and, in reality, every tone system) we must distinguish between the octave (ratio 2:1) and the twelfth or fifth (ratios 3: 1 and 3: 2) in this sense that the latter and not the former is the constitutive interval, since the fifth produces by juxtaposition or multiplication all kinds of tone qualities, whereas the octave is in this respect meaningless, producing always the same quality in different position or «sites. Yet we must not proceed too hurriedly: as we see, our series means juxtaposition of thrice the octave and thrice the twelfth, but whether the juxtaposition is by ascending or descending we do not yet know. We shall note that our series includes intervals yet other than the octave and the twelfth: consecutive tones (or numbers) yield the fifth 3:2, the fourth 4:3 and the whole tone 9:8. We can also establish the existence of intervals composed of those noted, 8:3 being, e. g., composed of an octave and a fourth, But we must now turn to something which would be more like a scale, i. e. a series composed of « neigh- The question, then, is to build the World Soul or the principle of the whole. Having mixed the two primary principles into a third, the Creator again mixed these three into one essence; and then the latter was subjected to mathematical organization. He first took one part of the whole. then the double of this part, then one and a half of the second, or the triple of the first, then the double of the second, the triple of the third, the octuple of the first, and 27 times the first. “Thus bouring » tones. Plato achieves the first step in this direction by stating that the Creator filled up the «double» and the «triple» intervals, i. e, the octaves and the twelfth by two « means». He does not say expressly which means, but there can be no doubt that we have to interpose within either of these ratios or intervals the arithmetical and the harmonical mean; none of Plato’s commentators has understood it otherwise.

As is known, the arithmetical mean is that which is equidistant form both outer terins (or, taken quantitatively, it is half their sum), whereas the harmonical mean must contorm to the condition that the : 2 and 6: 4: 3), and within the twelfth As we see, both double divisions a < first outer term be to the second, as the difference between the first and By dividing octaves and twelfths in these two ways, we obtain withthe mean is to the difference between the mean and the second (which is the case, e. g., in the series 6: 4: 5). in the octave two tones distant from the outer tones respectively by bines in itself the series’ 4: a fourth (12: 9: 8: 6, the famous composite of four numbers which coma combination of 3: 2: | and 6: 3: 2). two tones distant from each outward tone hy a fifth (6: 4: 3: 2, this being | Ot course are symmetrical as to tone distance (or as to the magnitude of the ratios): in either case the smaller interval is in the middle (whole tone and fourth respectively) and the larger interval at both sides (fourth and fifth respectively). In applying this double division to every one of the octaves and | 184 18 27 the twelfths present in the primary series, we obtain: nd pod T'S 14 22743444 5'/, 6 8 9 Hd (the numbers of the primary series are written in italics). is, five diflcrent degrees of the series ot twelfths or htths, or tive diflesome ol the numbers have been obtained in difterent ways, as, €. g., 1% which is the arithmetical mean of 2 and l and the harmonic of 3 and 1. We could easily do away with fractions by multiplying the whole series with 6. Musically speaking the main feature of our series, although not expressed by Plato, is the fact that it includes five different powers of 3 (which holds good whether we may multiply it with 6 or not), that rent tone qualities (characters), as 27 and 1372 correspond to 5, 3%, 9 and 18 and 4 # to 3°, 3 and 6 to 3', land 2 and 4 and 8to 3°, 1°, and 2 Rs, and 5'/,to 37, Here we may remember that five tone qualities, built up on this basis, constitute the pentatonic musical series (e. g. ¢ de g a). As we have noted, already the primary series contained the filth, the fourth and the whole tone, besides the octave and the twelfth; and the double division applied to the octave and the twelfth produces further such intervals. Among these it is especially the tourth whose rôle is constitutive in the building of a scale, and that for the simple reason that 15 27 en LE di zal 22 22 222200 vent): we eee eee ee weee ee M ee eee Room

it is, as concerns tone « distance », the least of consonant intervals (or simple was preeminent. ratios), that which is first attained if, starting from one tone. we proceed by «steps». Also 9:8 is important — the whole tone which represents, ed with Greek musical theory. ready in the primary series, it is further produced by the double divi- Yet the question of establishing a trame of fourths is not easily solved and here, already, begins our doubt whether Plato really had in mind a Secondefinite scale with its details, or he merely stated the general principles of one, For vizualizing the situation I ask the reader to look at the diagram accompanying this paper and, first, at the uppermost row. ‘This illustrates darily, we have obtained even the (minor) third, 5'/: 4% being equal to 32: 27: as we see, it is the result of the two overlapping fourths 5'/.:4 and 6:4! (or, e. g., da and e h), but we do not need it as a constitutive clement. By juxtaposing or adding the intervals obtained we should just the series of numbers given above, which presents the result of the double partition of our octaves and twelfths. We ought always to distinguish within this series between the primary numbers (1, 2, 3, 4, get still others, but that too does not matter in this connection. Now Plato continues: the Creator having realized the double division of the octaves and the twellths, took the whole tone interval 9:8, and with it filled all the 4: 3, i. e., the fourths; by this the mixture of the 8, 9, 27) The first thorny question is that of the interfering fourths 5*/.:4 and 6:44. intervals, Plato conveys the impression that his series is already contithose overlapping fourths of the Greek « modulating» scale (systema figures above p. 8, where the fourths are marked out by brackets. As has metabolon), which, being filled in similarly (e. g., as a bed and k cd e), been said, two ot these fourths, 5*/,:4 and 6:4%, interfere with each other; produce the tones 6 and h. Yet I am not at all sure that Plato had in mind such a concrete detail and there is even another detail which would not be in harmony with this: as we shall see, b and Ah coexist in different octaves of our scale according to several of the possible variants, on the other hand there is a large space not filled in by fourths, i. e., the space included between the figures 27 and 8, or 27 and 9; this outer twelfth, lying outside of the three octaves 8:1, contains the fifths 13'/2:9 and 27: 18 with a fourth between them. Thus we understand the difficulties with which Plato's commentators have been faced when trying to torm a definite idea of a scale agreeing with his instructions. ‘They made their task easier by disregarding Plato's implicit statement about a Frame of fourths, established and therefore it would not seem logical to have them, apart from that, Proclus in his commentary (194 F) is really taking matters lightly when, in the face of Plato's instruction to fill in the fourths, he speaks of filling in « every fourth or But in the ancient Greek conception the division by fourths A fifth»: that is so much the more at variance with Plato, as the latter because in the modern sense a scale is more or less determined by octave Are we to conserve both, as Boeckh has done, or to drop one of them, and if so, which? We could imagine that Plato had in mind nuously divided by fourths (4:3); but that is not the case, as we see by the periodicity. and those which are the result of division (1%, 1%, 2 °/:5 4%, 5°/, 6,13 %, 18). soul substance was used up. By saying that the Creator filled all the tourths with whole tone says explicitly that when the whole tone is inserted twice, there remains the « limma» or diatonic halftone. On the other hand modern commentators have been perhaps induced to skip the frame of” fourths Taylor has, as we shall see, a series of fourths, but it does not accord with Plato's division. sion of octave as 12: 9: 8: 6 and, in the latter sense, we understand why before filling in the whole tone intervals. At any rate Boeckh has not dared to disregard that statement of Plato; he was not only a philologist but also well acquaintas to distance, precisely the normal «step» needed; being present alit is so often detined as the difference between fifth and fourth. 11 in the same octave and produced by another method. However, that argument does not rule out the overlapping fourths conclusively, any more than the argument urged by Taylor (p. 142) in accordance with Proclus, i. €, that the coexistence of b and A as neighbours would produce the « apotome» b-h as distinct from the normal «diatonic» halftone a-b, while Plato does not mention the apotome; as to this I should observe that the coexistence of b and h in a « modulating» scale does not mean that both tones are concretely connected. I think we must say modestly that we shall drop one of the competing fourths only for the sake of simplicity; and in this case I think it will be rather 6:44 than 57/4, as the latter fits better in the octave periodicity; and — though it is not always realized in music and we may perhaps not fully realize it in the present case — octave periodicity is inherent in the idea of a scale.

In consequence the tone 4% will have lost its position as a point ol support in the framework. In fact at this point I am already hesitating whether I ought to continue. But with the good intention of arriving at an illustration, at least, of what Plato may have meant, I go on and now it is the question of having fourths everywhere. Again I ask the reader to look at our the first tetrachord from left is joined to the second, instead of being separated from it by a whole tone. The effect of this is, precisely, that 12 j octave periodicity is less violated than in case I. 13 In our case, again, two tones are added which were not in the uppermost row, 12 and 24, but as they are in octave relation, they may appear as substantially one tone. One of the tones of the primary series, 27, has been left outside the diagram, and this time at 1ows I, II and III where some of the possibilities «skeleton ». The last note being 24, the tonal range of the series is are outlined. I). The disposition most closely adapted to that of the uppermost row would be I- In fact here all the tones of the uppermost row (excepted 434) preserve their rank as points of support, that is to say they are defining intervals of lourths. Yet 20% and 12 are added to the Platonic tones. In this case beginning from 6, there would always be fourths separated by a whole tone; this means that we should obtain within this space fifth periodicity, replacing octave periodicity. Or, to put it more precisely, octave periodicity would be ruled out from 8 onwards, as in the « normal » way the next fourth after 8:6 would have to be 10°/.:8. + octaves and a fifth instead of 4 octaves and a ınajor sixth; in this con- In consequence the octave repetitions of 5*/, i. e., the tones 10°/, and 21°/, are lacking; or to put it still otherwise, the fourth 12:9 appears as an octave repetition of 5'/.:4 shifted by a whole tone; 18:13 % is in the same position when compared to 8:6; 27:20 4 is shifted Ly one whole tone in relation to 12:9 and, therefore, by two whole tones in relation to 5°/: 4 From the musical point of view, looking at the tone qualities or the powers of 3 present, we see there are not less than six, which is precisely a consequence of the fact that octave repetition is so largely disregarded: the tones 1, 2, 4 and 8 mean 3°; 1’/, 2°/, and5*/, mean 35 1 4, nection we may remember that sometimes 24 and 1 are given as limits of cosmic harmony (Macrobius II 1, cf. Aristotle Metaphysics XIII 6). Perhaps the fact that this disposition does not fill out the whole range of 27:1 seemed tolerable because in the typical two octave-scale of the Greeks there was a note, the Proslambanomenos or «added >, which was considered to be in a way outside the structure by a whole tone. ‘This series gives five different powers of 3, since 1, 2, 4 and 8 mean 3°, 1'/, 2’/ and 5 */, mean 3°, 1% 3, 6 and 12 mean 3’, 9 and 18 mean 3°, and 13% means 3”. Thus the number of tone qualities is the same as in the Platonic series (uppermost row) and even the tones are the same (apart from different octave localisation, as the tone qualiey denoted by 3% has two more representatives, 12 and 24, and the 3’ quality one less, 27). IN). In dealing yet more freely with the terms of the uppermost row we can concede to the « law» of octave periodicity its full sway. In this case we continue after 8 exactly the same kind of division which goes on within the three octaves 2:1, 4:2 and 8:4. That would mean 3, 6 and 12 mean 3'; 9 and 18 mean 3°; 13% and 27 mean 3°; 20% means 3‘. Here we may note that the sixth power of 3, 729, assumes a special importance with Plato in another connection (Republic 587 D), interpolating first 16 which gives the next octave,then, between 8 and where he takes it to represent the ideal period of solar revolution (729 days and nights — 364 4 whole days instead of ca. 365'/,,a magnitude already known at that epoch); but of course I should not take this as a « proof » that we must follow this method, because, I think, there is nothing to be proved here. to 4 octaves and a fourth; the extreme tone of the primary series, 27, would remain outside, as it is separated by two whole tones from 21 °/ II). Here the fourth 12:9 is, as in case I, shifted by a whole tone in 10°/, and 12, and beyond 16 the number 217/,. In this case the uniform division of the octave would extend to 21 Ya 16, the numbers © ‘The disadvantage of this arrangement is that no less than four tones, 10°/, 12, 16 and 21°/, would be added to the Platonic series in the uppermost row. Two among the tones of the primary series, 9 and 27, would be dropped, as well as two further tones contained in the upperrelation to the octave repetition of 5 '/,:4, but the shifting goes no farther, as most row, 13% and 18, and the fourth 18:13/ would not be preserved. the fourth 24:18 is contiguous to 18:1$74 and therefore an octave repetition From the musical point of view, looking at the tone qualities, there would be only three, as 3°, 3° and 3° occur in the different octaves, 3“ being In fact the only difference between I and II is that in the latter

represented by 1, 2 4, 8 und 16, 3° by 11, 244, 57/, 10°/, and 21°/,, “iar Ve ax À and 27: 24: “u: “7. But that does not matter because, 3 * by 3,6 and 12, as we shall see, we are faced with two reciprocal methods of computing, Now I think it is useless and hopeless to determine which of the three methods Plato hus had in mind. Leaving a gap that can be filled in different ways is just characteristic of his attitude: he puts down a principle and does not much care how it will be carried through. Plato even fails to give prominence to the difference between 2 and 3 ane of which takes numbers as frequencies (vibration numvers) waile the other takes them as string lengths; and in this aspect 1:°/4 "/ui */s will yield the same result on the basis of string lengths as “a: fet “aat 1 on the basis of frequencies, and vice-versa. In fact the wording of Plato does not even exclude a third possibility which would locate one whole which is musically essential, for the octave, corresponding to 2, serves 15 tone beside the one limit of the fourth and another beside the other only for delimiting spaces in the scale, whereas the twelfth or its octave limit, thus putting the half-tone in the middle; again we see that Plato equivalent, the fifth, corresponding to 3, produces different and ever does not enter into concrete details. It is clear that having filled out all the fourths we obtain a greater variety of fraction numbers than before; and this we can again rule out by multiplication. more different tone qualities. Nor does Plato notice that these two constitutive principles of scale construction are in some way antagonistic or, at least, irreductible, as never will a power of 2 coincide with one of 3 II). In this case the first tetrachord would be as in case I, and the last 18: “/ fet 24, and therefore never will a tone resulting from octave juxtaposition exactly coincide with one resulting from the juxtaposition of fifths (hence the « Pythagorean comma ») — although, of course, in practical perception a tone will always be able to take over the value of another one so near to it as II). In this case the first tetrachord will be again as in case I, and the last 16: 18: “*/: "7. But now we are faced with the question which is practically foremost. Before heing able to represent our numbers by sound-names we to be scarcely distinguishable. must determine whether they are to mean relative frequencies or relative 111. string lengths. As is known, these two are in reverse proportion one with another. All commentators, from Boeckh onwards, but with the exception of Thimus, have decided in favour of lengths, taking this THIRD STAGE as a matter of course. But Areca and III. 20 4:27 will become */: "ps “"/us: 27. That is precisely the « diatonic » form of the fourth or the diatonic « tetrachord » if we take the word « diatonic » to mean the combined use of whole and half tones, a haltmatter of course, as, e. g. ment »). Again I think that it is rather in the style of Plato to let the question open: what did matter for him was the correspondence between wett mangue st ie But as that would take much time, we can restrict ourselves in is not a true Plato speaks of rapidity of locomotion, not rapidity of vibration, cf. Der Toncharakter p. 186 and 139-141; that may be an error or a kind of illustrative speaking; in either case the question is of « quantity of movede equal to 256:243 or 245%, Now for a survey we ought to fill out all the fourths in row I, U every case to the first and the last fourth. 1). The fourth 1:1'/ will become 1:°/:%/4t% and the fourth it Plutarch, who was a very good Platonist, supposes frequency numbers in his commentary (Ch. 18) and Plato himself in Tımaeus 67 B ascribes to the low tone the lesser rapidity and consequently the smaller number. (It is vend But the fable continues. Whichever we may choose of the variants I, IL or III, we have not as yet u scale. As we have intimated, Plato goes on to tell us that the Creator filled all the tourths (the 4:3 intervals) with the whole tone (9:8), every fourth containing two of these and a rest interval, tones and numbers, or tone relations and numerichl relations; and, whether the reader would choose one or the other among those reverse proportionals, was to him a secondary question. Nor can we take him literally with regard to the compass of his scale, because 4 octaves plus a major sixth is a compass unheard of in Greek musical theory. As to reciprocity of numbers, we may yet remember the notion of the « unlimited tone being interposed after every two or three whole tones. It is true we might doubt whether 9:8 has to be apposed to the smallest number, as we have just done, or to the greatest which would produce, respectively,

Two», mentioned above us occurring in Plato, and which, according to Vhecetare ı we take ou the monochord the whole string length as cepresenting 27 and its twenty seventh part as representing the number 1 16 Stenzel (34, 51 f. and. 59 {.), would signily the Two as a principle indepen- 17 dent of whether it is taken as multiplier or divider, or, in other words, which gives au interval of boctaves and a major sixth — the question as positive or negative power; indeed, in this case it would be the « undefined Two» rather than the « unlimited », but with the Greek word both meanings are compatible. We have to remember also what Plato (Republic 525 D f£.) says about fractional numbers: instead of fractioning the One, mathematicians would rather multiply it, or in other words, instead of saying, e. g..°/., they rather say that a definite magnitude, taken five times, will be equal to three units. In all this there is a tendency to lay stress more on the formal than on the material character of number, to take number as a principle rather than as the measure of concrete quantity. (As is known, the idea of continuous quantity has found its whether wa diall denote these two tones as G and ¢'’’, or C, and a'’, or extreme representation with differential calculus created by Leibniz; and yet it was this great thinker himself who, in regard to musical tones, most decidedly insisted upon the importance of integer numbers and, especially, the first prime numbers). Yet as modern people we crave for concretization and, in this case, we have to choose. We choose — for the time being and with all necessary reserve — string lengths, in order to be more in accordance with the majority of modern commentators; and we follow them also in their method of inserting the whole tone, that is to say, we juxtapose two whole tones by starting from the smaller number which is, according to our first assumption, the higher tone of the fourth; in consequence the rest interval (the hall-tone) will be placed at the lower end, producing the « Dorian» species of diatonic tetrachord (ditterent from the « Lydian » which has the half-tone at the higher end and the « Phrygian » which has it in the middle — it is true, these names as applied to tetrachords are not quite authentic). Now at last we can associate with numbers our favourite tone letters. But here we must yet remember that we should take them not, as is usual, as absolute degrees of pitch measured by vibration frequencies per second, but as a system ol relative pitches or tone relations which, by principle, can be based on any concrete pitch. In reality even our tone letters had originally this meaning in the « Gregorian » system, and only by the development of instrumental music and ensemble playing was the relative (the musically real) meaning of the tone letters superseded by the concrete (the musically abstract), Ko and °°, of OD, and A’, etc, may cause some hesitation. We must anawer Uit we are tot interested in the concrete pitch of the tone taken sa starting, point, and that therefore we do not use the letters in the modern fiaed zere, But taking them in the old, relative sense, would there be a reat for pretering one dénomination to the other? Of course, a ersten there exists, and the criterion which must determine our choice is she same as that observed in the old « Gregorian » system. In terms ot etousical theory », it may be stated thus: « avoid accidentals as much as powible » or, in terms of musical psychology: « ascertain the tone qualities involved (ij. e,, the degrees of the series of fifths) and dispose them with d as centres — so that a series of five tones will take the form, cedar, a series of seven tones the form, fegdaeh, a series of nine the torm, bfegdaeh fis (supposing in the latter case that b and fis are real qualities coexisting with the others, not mere « chromatic» extensions or «modulating» features). lt is true, this supposes an odd number of tone qualities, for in the case of an even number the centre would fall between two qualities, say between d and g, or d and a; but actually systems with an odd number of terms have always been predominant and we can even doubt whether others (hexatonic, octatonic) have existed in the full sense, Now Boeckh has taken the highest tone, the one corresponding to '/ of the length, to be Nete diezeugmenon, which is e’ in our transcription; that would mean that the tone corresponding to the whole string is G,, a tone located more than two octaves below the lowest tone of our piano. In order to avoid such extreme depths we should shift this series by three octaves, replacing G, ande’ by G and e'’'’' We shall now see what kind of scale our three rows will produce on these premises. I hope the reader will forbear with me if this time I invert the order for the sake of convenience. UI). The third disposition would produce this scale: G27 AHedefgahc'd’' Ye BF’ ! ! tt NE a j . j I g'a'h'c''d''e'"4f''g''a’3 t ot Î AE de ri, !

In order to survey the situation more easily, we must always There are 34 tones. As we see, the letters adopted agree with the criterion stated above. look at the-tone qualities (or the powers of 3) involved. It is clear that the number of qualities as contained in the frame of fourths (according to row III of our scheme) is augmented by the'filling out of the fourths, as the tones inserted in them are themselves degrees of the series of fifths (or Indeed, ‘Plato does not make that evident, for in his sense °/, is taken powers of 3), ”/ being 3° divided by 2*and “/, being 3* divided by- 2” from the primary series 1 2 3 4 8°9 27, where it stands as the difference between the octave 4:8 and the none 4:9 ‘(or we-could extract it from the grotip 6: 8: 9: 12 resulting from the double division of the octave, where it represents the difference between the fourth 6:8 and the fifth 6:9); and the tone "/,:is, in Plato's words, the result of taking twice °/.- But as òf numbers as 'structures;- it‘ was imthe Greeks were always conceiving or possible for Plato not to take 9 as 3°, and therefore as a continuation a further outgrowth of one of his primary principles (the number 3), and similarly not to take the tone */ as corresponding to 3* (cf. what we’ shall say below, p. 30,'about the « nude » series of fifths of A. von Thimus). Even if Plato did not really have in mind powers exceeding the third degree (this degree representing the solid, see above, p. 7), he must have thought of linking together consecutive groups of three powers. Now comparing the three tone qualities contained in the «skeleton » with those of the scale obtained, we see that the former have been augmented by four. Represented as they are here by headgcf, There were 3" (the tone A), 3° (e) and 3* (a); to them are now added 3° (d), 3° (g), 3* (c) and 3° (f). with d as centre, these seven qualities conform to the criterion stated ‘above. That is precisely « diatonism » or « heptatonism». The fact that As octave periodicity was safeguarded in the corresponding form of the skeleton, is now reflected by the fact that the filling in of the other tones produces no contradiction between different octaves, or no « accidentals». concerns the mathematical representation, the filling of the skeleton A (=3”), e (=3% a (=3°) by the tones c (=3), d(=37, f (=3") and g(=3") would produce the series 3" to 3°. Yet that would not be the proper representation corresponding to our desire for symmetry; from this point of view we must shift the series from 3% - 3° to 3° - 3°, assigning to d the rôle of centre (3°) and to h and f that of extreme terms (3° and 3°). Here we take the scale as a form and its components as qualities, and in this sense we must no longer cling to the unilateral, genetic procedure THE « TIMAEUS» SCALE 19 by which we formerly derived a series whose extreme terms were marked by 1 == 3" and 27 = 3. In the case of frequency numbers — which is, after all, more probable — the series will have to be iriverted thus: H). Row Il of our scheme would, on the above premises, produce the following scale: def EN ] gta LL ) h''° 14 ! edet |, | G2 ABcdesfgab Ge’ Sf g’arh’ec’’ de’ e’d’ af gas | he ' Again there are 34 tones, but not the same ones. In taking this scale as a whole and not in single octaves, we see that the letters adopted do not agree with our criterion. The series of tone qualities from h to es comprises nine degrees, and these, disposed symmetrically with d as centre, h, 3* = e, 3? = a, 3° = d), again four have been added: 3‘ = g 8 = 0e, would have to be designated by symbols ranging from fis to b; the lowest tone would have be denoted by D and the highest by h’’’. To the tones contained in the skeleton which, in this case, are five (37 = fis, 3° = Breg 3, =a =3 6 = 3h = 384 fis = 3 3° = fand 3° = b. Again, in order to agree with the musical structure of the scale, the proper mathematical expression will have to be shifted from 3° — 3° to 3* — 3°, and consequently 3“, 3° and 3¢ will respectively correspond to fis, d and b. Yet in the case of frequencies the series would be inverted: b= Sf The breach of octave periodicity, present already in the skeleton of this scale, appears now clearly as a contradict ion; it is illustrated and concreted by the fact that in one octave there is f and in the other fis, in one h and in the other b. We could call it, in the terms of Cleonides (see Der Toncharakter p. 352), a « composite system with three Mesai »: the (dynamic) Mese which normally corresponds to the a quality would be represented in our series by e in the higher, a in the middle and d in the lower region. Wet we can not really say that the three « diatonic » systems, with a, e and d as Mese, are composing a ‘nine tone-system, because nine

tones are never present in one octave and they are rather an assembly of disparate octaves. Or to put it otherwise, it is a « modulating» system, and modulating twice, always from one heptatonic to the other. In this sense it could be doubted whether we have a reason for changing the transcription of the whole from A — es, asymmetrical, to fis — b, symmetogether (and in so far they are not real qualities). Here we have, in the terms of Cleonides, a « composite system with four Mesai», the Mese being shifted from a to e in the high region and to d and g in the low trical; indeed, it could be argued that they do not form a whole. to have that one which is in the centre, written « without accidentals ». I). Given always the same premises, our first skeleton would produce: 14 N 3 het g a'' d'''e'''2 { ! ! ee t. toot il | y pi gt a | 4 £"" d’’ e’' 4 hen (according to the transcription we have preferred). Again, if we assume frequencies instead of string lengths. the order would be inverted: But even if modulating, i. e., passing from one system to another, it is logical G2TAsBedesfgahbc’d'9e’ 8 f'g’a’h’c' 21 dr | = There are 34 tones as before. Taking again the scale as a whole, we see that the series from À to as corresponds to ten tone qualities — an even number. It may therefore be shifted in two ways, the center falling in either case between two notes: if it is written as ranging from fis to es, the center would be between d and g, if from cis to b, the center would be between d and a; in the former case the lowest and the highest tone would be D and h'’’, in the latter A and fis’’’’. (Among these two representations we should perhaps prefer the former, because the Greek conception was inclined more towards «flats» than «sharps», as exemplified by their « modulating» scale which connects not f and fis, but b and k). Again we see how the breach of the octave periodicity, present in the « skeleton », is equivalent to inner contradiction. Whereas in the skeletonic stage we had 3” (fis according to the transcription preferred by us), 3° (h), 3° (e), 3° (a) 3° (d) and 3°‘ (g), now to them are added 3° (c), 3° (f), 3° (b) and 3° (es). In accordance with this we shall again shift the mathematical representation from 3* = fis 3° = d and 3° = es to 3 = fis, 3° = d and 3° = es. We can even say that the proper representation would be to give to fis not « minus four » as power exponent, but « between minus five and minus four», to d « between minus one and zero », and to es « between four and five r; nay, even the tone which we have marked by d is, after all, not d but « between a and d», etc. Yet all this is rather theoretical, as in fact the ten qualities are not operating es = 3, bef ez gn 34d = 3” ,a = 3" ,e =5% h = 35, fis = 3+. But we have yet to add an observation. ‘There is still a method by which we could fill out the fourths diatonically without compromising octave periodicity (and, consequently, unity of diatonicism). As we saw, the breach of diatonicism, i. e., the surpassing of the number of seven different tones in 1 and II, was caused by the fact that, while in case HI there were only two fourths repeated trom octave to octave (say h — e ande — a) and these were limited by three different tones, in case II there were four, and in case I even five (the number of limiting tones being always greater by one than the number ot fourths). All these fourths, when juxtaposed in one row, can be denominated, e. gsashe-ea—-ad-dg— g e. It is clear that when going beyond two fourths the result is breach of diatonicism if the fourths are filled out in an uniform manner: say, if they were all « Dorian » as in our previous examples, or if they would be all « Phrygian » or all « Lydian». But it is understood that we could avoid the breach by giving to the different fourths different diatonic forms, c. g., if having given to he and e a the « Dorian » form with half-tone in the lower part, we should give to a d and d g the « Phrygian >», with halftone in the middle, and to g c the « Lydian », with half-tone in the upper part. Yet that would only mean transferring the contradiction to another plane: in this case the uniformity of disposition which also is inherent to the idea of scale would be repudiated. It is interesting to note in this connection that commentators have not always been in agreement in deciding for the « Dorian » form, though such, in general, has been the case with modern commentators. In fact, as we shall see, Pseudo-Timaeus, Proclus and Psellus leave the question open between Dorian and Lydian. Plutarch who, as we have said, supposes frequency numbers, has the Lydian tetrachord (see Ch. 18 of his commentary), precisely that which seems the most natural from the modern point of view (1: Ya: “/at /s = cdef or gahc). And still more interesting,

he tells us (Ch. 19) that other people used the tetrachord with the half tone in the middle. It is understood that in all this the supposition seems to be that one species of fourth is uniformly applied and that it is not a question ot finding an expedient for safeguarding diatonicism throughout. Yet whether the breach of diatonicism may be accepted or not, it can be maintained that the « Phrygian » division of fourth, with the half-tone in the middle, appeals in a special manner to our sense of logic. By filling up In case II: C27DEFGABedef g9a8Bhe’ de l a 1 ot efs’grarahre det i by f = 3” and g = 3", supposing, of course, frequencies), and the « Lytetrachord to « right » or to the « feminine » tone qualities, by using the Phrygian method the centre is not shifted. Of course that holds good also when we combine two, three or more fourths, and therefore I should prefer to fill out our frame of fourths by the Phrygian tetrachord, whether taken from row 1, II, or from III. In case I we should obtain: C27 DESFGABcde ih’ c'’a’’3 } I ' I 1 { Lt [| ! fg9a8he'd’e’ fis’ g et fist g avehet det fist ga | Ü | ! edt th aa | 1 { ! 4 fs ett? dte h''a''9g'8 > f''e'"d''c''h'a’g'A4f'e’d’ 3 c'hbag?fedcBAG I. bod | | l i i ! In case III: F27GAHcdef gahc’9d’ 8e’ fig’ a’ h'e ! ! a hed N I | \ I dae," g''3 | | | et" fr g ah ''c"tt'd''""1 ILL \ 1 or, in case of frequency numbers: KNatgtftetdtethtatg'fe’gd’ ße’ \ | | | I Nag fie’d’4c'h aS gfed2cHAGFED|1. 1 1 ot ! 11 | After what has been said above, I shall not explain further why I have adopted one method of transcription rather than another. At any rate I repeat that, with the six series noted, it is onlya question of illustrating Plato's thought and not of deciding what he must have represented to himself. tone, 9, stands not as a | I a’ a’ f’ es’ d’ c’ 4 bag 3 fesdc2 BAGFESDCL. Ll eae dh « frame » tone delimiting a tetrachord, but only as a tone included in a tetrachord; that is the price for the advantage of aren gi fist et dE hate feta Ge Bb ( d’3 ! The only thing we must still note in this connection is that UI, taken in both significations, presents a disadvantage: one of the primary | or, in case of frequency numbers: Î fs Pathe or, in assuming frequency numbers: dian » tetrachord means adding to the given notes two notes at the other side (e. g., g and € = 3*and 3* would be joined by a = 3° and h = 37, whereas, according to the Phrygian method, both added notes are symmetrical on both sides of the given tones (e. g., toa = 3' and d = 3° there would be added h = 3”, two degrees at «right» from a, and c = 3%, two degrees at « left » from d). In short, whereas by filling out the fourth by the Dorian method we displace the centre of our group to «left» or, as 1 should say, to the « masculine » side, and in the case of the Lydian Tt | the fourth by the « Dorian » method we add two tones which are, as to their « character » or their position within the series of fifths, wholly at one side of the two tones given (e. g., e and a = 3* and 3* would be joined 2 ! Li | octave periodicity. In looking again at our diagram we see that in I and II all the fourths contained in the Platonic division are made use of and all the Platonic points of support stand again as such (of course with exception of the fourth 6:44 and the tone 4/:), which is not the case

in HI — the difference between I and IH being only that the primary tone 27 25 four:h volume of C. F. Hermann’s edition). R. Harder (in Pauly-Wissowa) distinguishes “Phe form I, which between two versions of this treatise, one dating from the first century B. C. and so largely replaces octave periodicity with fifth periodicity, is much like the seale of Musica Enchirtadis. Summarizing our argument, we can say that Plato not having contented himself with dividing the octave but having divided also the twelfth, has ollended against octave periodicity, and the consequence is that if we followed him strictly, we should transgress the limits of diatonicism. It can be doubted whether Plato was aware of all the conone from the first century of our era; according to him, the claim of Timaeus' authoris in I, employed as delimiting tone, and not in Il. sequences of what he had said. ship is put forth only in the second version, while the first was an unpretentious paraphrase of Plato's dialogne. This treatise gives a very short account of what is supposed to be Timaeus’ system of psychogony. It states that the first number (Plato's 1) must be represented by 384 and that, taking the «double» and «triple» intervals with their partitions and filling in the whole tones, there will arise 36 terms, the sum total of which is 114695. Now that is also the sum of Boeckh's 36 terms. But Pseudo-Timacus docs not say by which means he derived his terms from the primary numbers, Of course he also leaves in suspense the question of string lengths or frequencies, as well as that of the tetrachord species. IV. Now in Hermann’s edition of that treatise a scheme is added on p. 410 which Some Commentators gives the whole series, still not as divided by tetrachords but only distributed among As I have said, most ot the commentators have neglected to establish a frame ot fourths to be filled in, but. having given their «allegiance to the primary series 123489 27, they filled in whole tones (9:8) and limmata (256:243), forming fourths and htths rather as they pleased. Boeckh is, as has been As this scheme is presented in the edition not clearly represented in ancient times: First number 384, 432, 486 (h) 512, 576, 648, 729 (h) intimated, in this respect an exception. It is true he „does not present the division by fourths in his scheme on p. 161, but such a division is presupposed. the seven primary numbers. and even incorrectly, L shall reproduce it here in order to show how the thing war His disposition is the same as that of our row II, but with two differences: 1) he includes the fourth 6:4 14 concurrently with 5°, 4; 2) he includes Double of first number 768, 864, 972 (h) 1024 Triple 1152, 1296, 1458 (h) Quadruple 1536, 1728, 1944 (h) 2048 * 2304, 2592, 2916 (h) Octuple 3072 does-not- present.the division by. fourths in bis scheme.on p. 161, bub sneh—adivision Ninefold first number 3456, 3888, 4374 (h) 4608, 5184, 5832 (h) 6144 * the tone 16; why? for obtaining, as he says, the fourth octave from 1, but in fact he wishes Ninefold second number 6912, 7776, 8748 (h) 9216 also to agree with Pseudo-Timaeus, as we shall see forthwith. The first produces, as has Twentysevenfold first number 10368. been said above (p. LI), a tone « chromatically » opposed to one contained Within 57/,:4; the consequence of the second — not made evident by Boeckh — is that there is virtually inserted a new tetrachord (12: als We 16) which again interferes with one already present (7/42 ®/t Ps: 18). In this way Boeckh obtains 36 tones (instead of 34), a number upon which stress has been laid by some ancient commentators. Of course, like nearly all modern authors, he is convinced that Plato supposes string lengths, and therefore his series is descending from 1 to 27 (or from 384 to 10368, as he, in agreement with ancient authors, puts it, in order to avoid fractions); Boeckh, like most modern authors, is also of the opinion that it must be the « Dorian » tetrachord. We should have, then, his scale if in the one noted above (p. 19) we inserted e and b’, if we united by brackets ce with a and a’ with d'’, and il we transposed the whole series three octaves lower. We see that the number of differing notes remains the same, ninc; and as they range from h to es, they must, as has been observed, be of the primary numbers inaugurates a special row, namely, as « Ninetold second number »; the reason is perhaps purely graphical, as the line beginning with « Ninefoid first number» was exceedingly long. as such; their sum is 105947. ‘here are 34 terms and they are counted Yet at the places indicated by * there are inserted auove the line the numbers 2187 and 6561 which are to their neighbours in the velatior. of Apotome and Limma. As they are not counted with the others, they must be a later addition and therefore this scheme could not originally have belonged to the treatise. Tuking the 34 terms with the two interpolated, there are 36 which produce as sum 114695; we see that the interpolation makes the scheme agree with is said in the text of the treatise. what And this interpolated series is precisely Boeckh’s (yet the 34 terms are not the same as those obtained by excluding from Boeckh’: series shifted to fis - b. ‘The earliest commentary on (by h we indicate the hall-tones or limmata). It is curious that 18 which is not one Timaeus come down to us, though not the carliest known, is attributed to Timaeus, one of the interlocutors of Plato's dialogue itseH: a forgery intended to give the impression that it had served as model for Plato's the superfluous tetrachord and the added tone which we discussed above p. 24, i. e. the terms derived from our row 11; for the present scheme has as sixth term trom the end 6144, while there it was 6561). dialogue, whereas it is really an abstract from the latter. (It has been published im ‘The number of differing notes (tone qualities) contained in this scheme is cight. some of the editions of Plato's works as an appendix to the Timaeus, as, e. g. in the in supposing string lengths, we should write it as © — e'''', with A within the three

be F — d’ octaves @'’’’ — =’ and with b below: supposing Irequencies, it would we have two interpolated with b in the three lower octaves and h in the two higher. The would be es and 5b’ notes would augment the number ol tone qualities by one; they in case of string lengths, producing of course Boeckh’s scale which, however, fis’'’ and Ah’ in case preferred to shift to D - h°°", with fis and b; they would be d‘'"". of frequencies and in this case we should preserve the notation as F — The next commentary, chronologically. is Plutarch’s {about 100 A. D) who, « Lydian » tctraas has been said above p. 15, supposes frequency numbers and the of the fourth 3s chard. He jumps from the primary series straight to the division Lis procedure may seem very general and too cautious. as Yet Ch. 16 ff. such, without establishing the relative sites of the tetrachords (I refer to of his commentary). (though a complete survey is not here proposed), the thorny to establishing a frame of fourths, he may have been too well aware of we must acknowledge the very fact that Plutarch occupied himself with the fourth; and chronologically difficulties which Plato poses in this way. Proceeding comes next. the commentary of Procius, the great Alexandrine of the Sth century, E). Prochus discusses the Timacus scale at great length (191 E - 198 B, 211 F and 212 We have already mentioned (p. 10) chat Proclus does not care about establishing a frame of fourths and he alters Plato's « fourths » to « fourths and fifths», Now we must in detail see how he proceeds. But first the text must be established in the the latest edipassage 196 E — 197 B, where the terms are enumerated. We see in added, with tion (1904) that in 197 B six words have been bracketed and six others editor says canthe result that the term 4374 is replaced by 4096; in a foot-note the did didly chat Proctus himself may have erred. Now it is amazing that the editor not see that this passage needed no emendation and that another passage is evidently wrong, namely 197 A, where the text runs; « adding to 1536 the next term by 9:3 we THE « TIMAEUS >» SCALE 27 terpolation:), and then (196 E) he turns to the question of constructing the scale, mentioning a diagram but supplying us with none. In this passage Proclus again expuunds the terms of the first octave, as given above and this time he establishes its division, yet not, as we should expect, into fourths with a whole tone between them, but into a fourth and a fifth (384 - 512 - the first twelfth has been attained, after which (197 A) the lourth 1152 - 1536 com- 168). Nevertheless, Proclus, like a too conscientious philologist, repeats that 512 and 576 are the two means. In adding the fifth 768 - 1152 he stutes that the limit of remove by text emendation. In going on to the next term, 3456, Proclus (197 B) pletes the second octave. The terms of the third octave, 1536 - 3072, are stated without subdivision; within this series occurs the contradiction that I have tried to last (fourth) octave is not adjoining to the third which was 1536 - 3072, but is separated states that the limit of the second twelfth is attained. From 3456 the series goes on to 6912; that is an octave and again its division is not stated (in this passage, the text is, as we have seen, not in need of emendation); this octave, with the adjoining fifth 6912-10368, constitues the last (third) twelfth. As we see, the from it by a whole tone: instead of 3072 - 6144 it is 3456 - 6912, and thus the last octave is subordinated to the last twelfth. That is in accordance with the primary 27, disregards the fourth octave (8: 16 = 3072: 6144). series of Plato who, although the number 16 is virtually present within the range 1 - It is further confirmed by the passage 212 E The series as given is confirmed by Proclus (197 C) by the statement that it contains 24 whole tones and 9 limmata. where Proclus establishes the suin of his 34 terms as 105947. And it is furthermore supported by its agreement with the scheme quoted above in connection with Pseudosupply the diagram missing in the text of Procius. Timaeus which produces the same sum; the latter scheme may therctore serve to Plato does p. 11). But it argument is that ‘The text of Pseudo-Timaeus is opposed by Proclus (197 D - E) who maintains that the number of terms is not 36 but 34; Proclus' be consi- In 211 F not mention Apotome but only whole tones and limmata (see above is curious that elsewhere Proclus appears to be much more conciliating. obtain 1728, adding to this number 9:8 would be 1944, to this 9:8 is 2187, to this limma Evidently the words which we give in italics and which, by the way, are missing in one of the manuscripts, should be cancelled, because 2048 is limma to 1944 and not to 2187. With this correction, and leaving the former passage as he says that the diagram contains 34 or 36 terms and that, as 34, 36 may is 2048, etc.» it stands, we obtain the number of terms and their sum as required, i. e, 34 and dered as appropriate to the Soul. and interpolation to the It is impossible not to sec in this the the term 2187 which, as we have seen, is not rightly included in his series, and by He even contradicts himself (195 D - E) by quoting 105947; moreover we are in accordance with the scheme added to the Pseudo-Timacus hand of a corrector adapting Proclus adding: + we shall need it in the diagram ». Comma 195 D - 196 C, mentioned above, is superfluous if appears even in the same passage where Proclus opposed this number: in 197 D there The surprisingly tolerant attitude of our text towards the number of 36 terms Apotome is not admitted in the diagram). about Apotome and number of 36 terms, as required in the treatise of Pseudo-Timaeus (also the discourse text by alteration treatise. Now we shall follow Proclus in his procedure. The disposition of his scale, as far as he indicates it, may be seen in row 1V of our scheme. Proclus makes a good start (194 B) by exposing the double division, arithmetical and harmonic, of all Plato’s octaves and twelfths (according to row I in our diagram). This series is transformed into integer numbers and given as 6 … 162. Then, having stated that, to insert the smaller intervals, we must. replace 1 (or 6) by 384 (194 C), 384 432 486 (h) 512 376 648 729 (h) 768 see these additional terms are exactly the same as those inserted in the scheme added is said that other interpreters who wanted to have in the diagram not only limma but apotome, have added two terms; and 197 F — 198 A these are specified as 2187 and 6561, the first being in apotome relation to 2048 and the second to 6144. We Proclus (194 D) puts down the first octave as follows: (what his division of that octave is we shall see later on). Proclus then (194 E-F) to Pseudo-Fimacus. There tollows gives again the double partition of every octave and every twelfth. a discourse about Limma, Apowme and even Comma (is it really Proclus’ or an in-

As could be expected, Procius does not say whether he means string lengths or frequencies, yet the tetracherds occurring in his series show at least that he places the limma at the limit of the fourth, i. e., as lowest or highest. I see that Vincent, p. 178, ascribes to Proclus a constructive method that would be rather complicated, though consistent in itself: having filled out the first twelfth with tones proceeding from the division of the octave (the series from 384 to 768 as above, but continued treble the single terms for obtaining those between 1152 (== 3 and 9 in the primary series), and would treble them once more for obtaining those contained between 3456 and 10368 (= 9 and 27 of the primary ‘This I can not find in Proclus’ commentary and, moreover, it would not series). to 1152), Proclus would and 3456 produce the sum 105947. The scale set forth by Proclus is, as has been said, the same as that of the scheme added to Pseudo-Fimaeus which contains eight different tone qualities; but while in the latter no division is marked except that by the seven primary tones, Proclus’ division is that represented in our row IV, However, this division is, from the the first octave point of view of a musical scale, not logical: we do not see why is divided as fourth and fifth and the second as fitth and fourth; in the following given withoctave no division is established; and the next octave which is again vut division, is, as we have noticed, «disjunct», i. e, separated by a whole tone 29 eye out for the fifth, and this may be significant, as Alexandrine is pre-Byzantine). The symmetry of Psellus’ construction appears also in the fact that, in supposing string lengths, the upper octaves will contain A and the lower b, or in supposing frequency numbers it will be the reverse. There is nothing to show us whether Psellus may have thought of string lengths or frequencies; yet it is, as in the case of Proclus, clear that the tetrachords are supposed to be either « Dorian» or « Lydian >. As the system of Proclus (and that of the scheme added to the Pseudo-Timaeus), that of Psellus contains eight different notes (or powers of 3) which we should represent as ranging from b to h. , That is, in terms of Cleonides, a « composite system with two Mesai» and it would correspond to the Greek «systema metabolon» (modulating system), but with the difference that in the latter 6 and h are alternative notes within one octave, while with Psellus they are located in different octaves. As to modern commentators we have already referred to Boeckh. Vincent (p. 176 ff.) should also be adduced as a thorough commentator and connoisseur of Greek musical theory. He has, as to the Timaeus scale, his own theory which is not devoid of logic, although it is not adapted to the division of octaves and twelfths as given by Plato. His meaning, in the way it is formulatel, is rather difficult to understand; from the foregoing, a feature unheard of in the history of musical scales. On the whole we have the impression that Proclus is not much concerned about musical still, it is evidently the following. He subdivides the first octave strictly in accordance with Plato's division as 1:1'/,:1%4:2, but then he proceeds independently, without looking at the (incomplete) frame presented by Plato. Having filled out We must further notice the fourths 1:1°/, and 174:2 in the normal way and obtained eight terms in the reality, and this may be characteristic of him as Neoplatonist. the way how he urges upon us the purely intelligible character of musical harmony is 199 D ff, where the intelligible is emphatically opposed to the perccivable. It curiously (but, Augustine S. in us strikes which Piatonism of form decayed the same enough, not in the Greek fathers as Basilius, both Gregories, and Maximus); indeed and in the conception vt Augustine, follower of the Neoplatonist Plotinus. intelligible real music appear as intinitely distant from each other. We have now to mention the small treatise devoted to Plato's Psychogony by the Byzantine Michacl Psellus, late in the ith. century, and edited by Vincent, 316-332. Psellus quotes Proclus at length, but only the preliminaries of Proclus’ dissertation, up to 195 A, which include the modelling of the first octave and the double division of the octaves and twelfths, finishing with the statement that there are 24 whole at vatones and 9 limmata. To this Psellus abruptly adds his diagram; but this is riance with Proclus' scale. Probably Psellus found it too laborious to follow Proclus in the details of his expositon, or was he discouraged by the contradictions of Proclus’ first octave, he doubles these terms for filling the second octave, and he doubles them once more for obtaining the terms of the third octave. Then the interval 1:3 is taken und again subdivided according to Plato as 1:114:2:3, which produces twice the fifth and once the fourth; and now, contrarily to Plato's instruction which provided only the filling out of fourths but in accordance with Proclus and Psellus, the fifths within this twelfth are filled out with whole tones tones leaving a limma, as in the case of the fourth; these twelve terms are tripled for obtaining those from 3 to 9 and again tripled for obtaining those from 9 to 27. We see that Vincent takes the first octave and the first twelfth as a nucleus which he subjects to a constructive, but rather theoretical idea of his own. Of course, some of the terms obtained result as well from the division of the octave as from that or the twelfth. On the whole there are 39 terms, and apotome occurs rather often (chromaticism in the modern sense), There are ten different tone qualities or powers of 3, as in the case of our row I when it was filled out. Of course Vincent supposes string lengths. The replansult would be a scale extending from D to h‘'*, with notes ranging from es to fis. His diagram differs from Proclus’ scale, in so far as it has 2187 instead of 2048 (yet 2048 is inserted above the line), and also the division is different (see row of the «conjunct» (or modulating) Greek system with eight different notes ranging There is more of musical logic in his than in Proclus’ arrangefrom 6 to h; and therefore in the sector of two octaves which is represented on p. text; therefore he procexded as a compiler, inserting what seemed to him sible. V in our diagram). ment. From the tone 384 (= 1) onwards there are first two octaves divided as fourth and fifth, then two octaves divided as fifth and fourth; the heginning of the next octave is again a fifth, followed by a whole tone. symmetry in this grouping. We see that there is some The division of cach of the octaves by the fifth on equal terms with the fourth is Byzantine (we remember that Proclus had already had an But in the realization of his idea Vincent is not consistent. 183, two of the tone qualities (or powers of 3) are omitted. Here intervenes the idea It is curious that Vincent was not aware of Bocckh’s interpretation. Although it is manifestly wrong, we shall now consider the explanation given by A. von Thimus, because he is a distinguished thinker and because he himself on the continuation of Plato's story which is in itself interesting to us. bases

tone. Taylor even narrows down this idea, 31 without and half tones is just ‘the idea of diatonic scale which juxtaposes within the octave Taylor then goes on to establish what was exactly Proclus’ either that of Proclus » or his. AN APPRECIATIO: that sum with Taylor’s As we have seen, Plato is not scrupulous about concrete details. In a rather imprécise way he behaves as if by dividing his octaves and twelfths harmonically and arithmetically he would already Possess the frame of a scale divided’ by fourths; and as to the filling out of the fourths, he says only that there must be two intervals of 9:8 and one of 256:243: In reality it is nota scale which Plato sets forth but only the OF PLATO'S STANDPOINT. After ‚all that has been said. I think ‘Taylor is too confident when he is « sure that the series intended by Timaeus ithe interlocutor in Plato's dialogue) is 384 to 10368. tation, as Plato does not say «as far as possible» bat «as far as possible within the fourth», und, moreover, Plato. docs not put down the terms in the multipied form from the first term and admitting only integer numbers); but this, 1 fear, is a misinterpreinserting whole tones so far as it was “arithmetically possible (always supposing 384 to be does not avail himself of the scheme. “He explains the difference between ‘Proclus' scale and the result is right, but it is attained by comparison of disposition. He determines it in siarting from the sum 105947 as given by lFoclus. ‘ol tlie three). orthodox whole apparent reason, to an octave with the fourths as extremes and the whole tone in 3 one We remember that in this method of thinking astronomy has alwavs fact Plato, after having said that the Creator filled all the 4:3 with 9:8, producing every one a rest interval of 256:243, goes on to connect his musico-arithmetical series In consequence he has a frame of fourths, but it is his own; as it the middle. subjects itself to octave periadicity, it coincides practically with dur row TT (the least 4; and heen associated with music and mathematics; and the latter being diversified as arithmetic and geometry, there results the fourfold group: arithmetic, geometry, music, series, not, as we should expect, with the scheme added to Pseudo-fimaeus” treatise. in the Greek letter 27 5 fi fourths astronomy, which later on constituted ‘the so-called Quadrivium. Plato says that the mixture having been used up (as we have seen), the Creator split up the whole structure longitudinally in two strips, then putting the middle above the middle, as Chi (N), he bent both of these strips making each of them into Again when polemizing with Pseudo-Timaeus, i. e. pro 34 terms and contra 36, Taylor He starts from the two strips just mentioned, putting 8 et, fs /: 5 two a circle; and making one circle the outer, the other the inner, he conferred to the and his own by assuming that Proclus was induced by the wording of Plato to continue with cosmology. outer movement — the circles are, then, rotating — the nature of the Same and to the other that of the Other; then leaving the outer circle unsplit he split the inner sixfold; thus he produced seven circles distant by double and triple intervals (as we see, the series’! 2°3 4 3 9 27 comes again into play), these being the orbits of the seven planets disposed zccording to their distance from the earth, and that distance being taken as } in the case of the Moon and 27 in the case ul Saturn. This division of octave and twelfth. had been correctly interpreted already by Boeckh. But Thimus does not agree (IL 213 ff. and 1 157 fE), In spite of Plato's clear statement, he fails to realize the double by Ze | by system, he joins them down one of them as | 2 4 8 and the other as I 3 9 27, and completing beth by be je reciprocal fractional numbers in this way: a To these rows Thimus applies the crosswise superposition and is nothing other than the diatonic (heptatonic) crosswise multiplication. obtaining the series: which, musically speaking, by agency of the consonance and, especially, the fourth, 2) diatonicism, principles of a musical scale, and these are: 1) the division of the series er transformed As observed above, p. 7, the reason why Plato's It is proseven primary numbers do not representation: as this is limited by three dimensions, it can not represent more than cube numbers or, in general. more than numbably his tendency to geometrical go beyond the third power is not a musical but rather a symbolical one. _ of which are probably to be interpretated as « colourings » of diatonicism. {in fact it is testified that the diatonic genus seemed to them the most natural and the most easy, and that it was the most common). tic» and the « enharmonic» genus, with many intermediate shades, all ‘Greeks considered as such, notwithstanding their use of the « chromahormal one: at least that which we consider as normal and which the this, of course, being not a feature proper to every scale, but to the or che series of seven tone qualities arranged in fifths: fe ¢ d'a e h, which can be by fifths, but Plato does not state it so nakedly. symbolized also by3* 3% 3239 3: 3° 3. The result of this system is indeed the same as that of Plato's, that is, gradation These fifths are, in Thimus’ opinion, the «double and triple intervals » to be divided arithmetically and harmonically (because the fifth is equal to 3 divided by 2!), and this double division gives major and minor chords in the modern sense (cf. below, Bv assuming frequency numbers ‘Vhimus is p. 38); only the thirds contained in these chords would have to be filled out by 9 That is interesting but very aprioristic. Finally, there is che learned commentary given by en again at variance with all the other modern commentators, but in this we can not wholly disapprove of him. - 1159 - 1586 - 9072 - 3456 - the primary series, as the ancient authors do, in 384 - 768 arithmethe division by whole tones Plato derives by 10868, ‘Taylor (p. 143-145) proceeds directly to realize skeleton series which The basis on which he applies his division by whole and limmata without keeping to the tical and harmonie partition.

most in accordance with each other, his example of a scale, as his examples MUSICA DISCIPLINA in general, can not be a mere « phantom », contrived arbitrarily; he is, in 32 We have further noticed (p. 27) that 16, the in this the geometrical representation of tone is bers composed of three factors; fourth power of 2 and the fourth octave, is virtually included within the range 1 — 27, more limited than the arithinctieal. Here the author of the « phantom » theory confounds what is peculiar to a given musical era and what can be stated generally. The « natural» third, which establishes a « direct » link between tones distant by a third, does of course impose itself as a consonance in a musical culture as ours, that adheres so exclusively to simultaneous harmony (cf. Der Toncharakter $ 13), but it plays only an prominent in Western music of later times. the « natural » third, which implies the following prime number, 5, 4:5 representing the major third and 5:6 the minor — the tone relations so sertion which also has (of coursety had a widespread echo: according to this the Timaeus scale (or quasi-scale) would be wrong in its root because it bases its structure only upon octave and twelfth (or fifth) and not upon other words, not a Kantian of the 19th century. The allegation of « phantom » is supported by another erroneous asengage in a comparative appreciation of the rôle of his We had further as we have observed (p. 7), alicady the seven terms as put down by Plato include four but Plato docs not set it off, and that probably for the sake of aesthetic symmetry, as 1 3 9 27 is in a way one side of a triangle and 1 2 4 8 the other. But in reality, noticed that Plato does not powers of 2 and of 3, as the number | too is a power of 2 and 3. We have also observed chat constitutive elements, the Two (octave) and the Three (fifth), of which the latter stands foremost in the aspect of real musical qualities, nor does he notice that they are in a way antagonists (see above, p. 14 and p. 24). All this is, I think, a sign that Plato proceeds as an artist who further powers of ‘Two, and especially of the decisive Three, are admitted by Plato implicitly (p. 18). presents his thought in the form of a fable. In other words, Plato does not go on to concrete what he has said. insignificant rôle in broader musical conceptions. Even as concerns ourselves, and fundamental; we must remember, 1) the famous experiments of it can be demonstrated that, notwithstanding the apparent sweetness and attractiveness of this new consonance, the fifth is yet far more constitutive 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 consonis pentatonic (sce above, p. 8); then, in a leap, he purports to have justice to the qualitative character of musical sound. It seems clear that and not as qualities, and only Aristotle and Aristoxenus would have done and mistake its substance and nature, since he considers tones as quantities an applause even more unanimous, is this: Plato would do violence to music The other wanton assertion of that book, and one which has found of course — a champion of musical progress). error: that Aristoxenus was a champion of the third as a consonance (and as such — I am glad to note that at last the view that the « Timaeus scale» is quite arbitrarily conceived seems to loose its credit, as I see in a paper by B. L. van der Waerden in Hermes, LXXVII, 1943. But on the other side a book about musical psychology which appeared in 1947 bears still wimess to the havoc caused by the aforesaid book: not only does it repeat the wrong assertions referred to above (as well as another which we shall forthwith cite), but it combines them into a new C. Stumpf showing that even men of our culture are naively more apt to mistake as one tone a simultaneous fifth than a third, 2) the fact (on which I laid so much stress in Der Toncharakter) that in our perception the ditferences of tone quality (as distinct from pitch) are still in the main based on the series of fifths. ance, then a first division of primary consonances which, curiously enough, erected a frame composed of fourths, and these fourths he divides quite normally by inserting whole tones, Surely those were principles of scale construction which had been established long since. It is, then, not at all a «phantom» (« Hirngespinst ») which Plato presents to us, as has been asserted in a rather sensational and mischievous book, published 25 years ago, titled Plato und die sogenannten Pythagoreer: a book which of course (of course!) has exerted a great influence upon philosophers and writers on music, nay even upon the flock, reputed to be cautious, of philologists. Indeed concreteness is not reality: but that is just the confusion so widespread in our modern era. Plato looks at the real world as it is, and does not resort to abstractions like matter which is not form, or the idea of sensuous perception in which no element of reason is present. That which is fundamental to him is the correspondence between our apparatus of cognition and the object of this cognition (a correspondence which exists by principle and does not exclude error in a given case); and this he is not, as we are not, able to trace back to anything else than the creative act of a transcendent being. Precisely because that is Plato's meaning and because he thinks that idea and reality are in their inner-

of quantity, indicating the more or less, but it indicates a structure, and the this assertion depends mainly on a one-sided conception of number, or rather on failure to grasp the Greek conception of it (without which we too really can not do). In that larger sense number is not only measure gradation of powers of 3 is, as we have seen, precisely in correspondence with the musical quality of sound. This conception is further reflected in the ancient tendency to represent numbers as geometrical figures, The Greeks called « oblong» a number composed of two factors differing by one, e. g, 6 = 2 x 3, or 12 = 3 x 4, which is the counterpart of superparticular ratio (3:2, 4:3, etc); this produced figures like: : (cfr, Th. Heath, A history of Greek mathematics I, 1921, 76 ff), In this sense number was to them really «figure», i. e. form; and what they saw, thus, with their eyes, they perceived with their ears as the musical, qualitative relations of tones. Yet there was a restriction imposed on this craving for visual representation (see above, p. 31): numbers composed of ancient conception With us, it is that we speak of «square» and «cube» more than three factors could not be represented geometrically, numbers, by a survival of this In fact we can not fail to be impressed by this correspondence which has something of a marvel: not only our perception of musical pitch and tone « distance » is rigorously (logarithmically) in accordance with frequency numbers, but also the structure of the number, apart from its magnitude, Three, is reflected in the musical character of the tone, apart from its pitch; e. g., the tone e (which in a system with two « sharps » would be written as fis) is, as to its musical quality, precisely as remote from another given tone within the same system, as both are distant from each other according to the series of fifths. In this sense we 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 rôle as the It has become just a slogan that Platonism considers the musical tone cf. Der Toncharakter, § 13). quantitatively and that the qualitative conception of it was introduced first by Aristotle and Aristoxenus. We may just as well say the contrary: that according to the Platonic (and the Pythagorean) view, and in view of the Greek conception of number, musical quality appears as a matter of course, and that it is rather the Aristotelian and especially the Aristoxenian conception that can be suspected of quantitative tendencies. It is known that Aristotle and Aristoxenus, more than Plato, laid stress on the character of number as measure of quantity, homogeneous and unlimitedly THE « TIMAEUS» SCALE 35 compatibl e (see Stenzel 59); and it is exactly in correspon dence with this conception of numbers when they are inclined to view musical sounds as exponents of a continuous gradation — the pitch gradation of course — rather than as qualities in the proper sense. Now this gradation is akin to one of intensity and, therefore, of quantity. Yet we ought to note that Aristotle was only the initiator of such conceptions which were to be developed and emphasized by Aristoxenus. Aristotle’s intermediate and somewhat contradictory standpoint as a musical thinker would make a very interesting subject of study. In a broader sense we may further note that Plato — at least late Plato — is not a thinker hostile to world beauty, as he is sometimes conto masidered to be. The tendency of fleeing the world, as manifestedin Neoplatonism during the late period of paganism, was not Plato’s. It is true, beautyis not to him — as it can not be to us — a thing strictly and merely dependent upon sense perception; perceptible beauty depends upon a superior order of things shining through what the senses grasp. That precisely is aesthetic reality, and sense perception devoid of it would be a mere abstraction. As we have seen, the quality of a tone perceived mu. sically is like that of a form or a figure, one quality always being interdependent with others. But the order, shining through our sensuous impressions and transforming them into aesthetic perception s, points to an absolute Good standing behind all things and whichis the source of being. That is why Plato,in his lecture about the Good, resorted to numbers, HARMONIC AND thematics and to the Pythagorean notion of «limit» — a method which, as we are told by Aristoxenus on the testimony of Aristotle, was surprising and disappointing to many of the hearers who had come in order to partake of «goods» like wealth, health and strength. VI. GEOMETRICAL ABOUT PROPORTIONS, ARITHMETICAL, But having engaged in polemics with others I do not see why I ought not to seize the opportunity to polemize against myself. There is in Der Toncharakter a section devoted to « harmonic partition » and there the « harmonic mean» has been treated in a rather peremptory way having no real musical importance as compared with the « arithmetical

MUSICA DISCIPLINA mean; thus, the harmonic proportion is in a way derived from the arithmemean» (p. 211 £). I think that is to some extent exaggerated and some defence may be made of the harmonic mean if we reflect on Plato's coordination of these two means. Actually the harmonic proportion is not quite tical. And again what is the musical meaning of it? As it preserves the same so artificial as it appears from its definition (on this see above, p. 8) but but it does so if we consider the whole; indeed in the latter respect 6:4:3 or it is in a way a reversal and a derivation of the arithmetical, component intervals in reverse order, the harmonic proportion does not render things more « complicated » if we consider the component intervals, Take an 3:4:6 is more « complicated» than 2:3:4, and 15:12:10 or 10:12:15 more arithmetical proportion with one mean, as 2:3:4, and suppose it is to be than 4:5:6. Therefore it would seem preferable not to exclude the method of the reversed, yet not as a whole (4:3:2), but by parts, reversing first 2:3 and then 3:4. The first relation in the reversed form being 3:2 and the second 4:3 we must, for uniting them into one proportion, take the smallest harmonic mean from the scheme of partition established in Der Toncharakter, § 21 c). However it must take place behind the arithmetical mean, common dividend of 2 and 4, which is 4, and accomplish the corresponding and we can still maintain that the partition by the latter is the true « harmultiplication which gives 6:4 and 4:3, i. e., 6:4:3, precisely a harmonic monic partition >. proportion; as we see, the relation of the outer terms is also reversed in comparison to the arithmetical proportion with which we started. The connection existing between arithmetical and harmonic propor- The tion is further illustrated by the fact that an interval sequence which relation between the two preportions may also be expressed as follows: the reciprocal value of the harmonic mean is equal to the arithmetical mean of the reciprocal values of both outer terms (denoting the harmonic mean vepresents an arithmetical proportion when supposing vibration frequenby m, its reciprocal value ı is = 7 + >); or as Proclus says in his m ‘Timaeus commentary, 193 B, if in a tourfold proportion like + = S b is 3:4:6 as to frequencies and 4:3:2 as to lengths; ¢ e g is 4:5:6 as to frecies, produces a harmonic one if string lengths are assumed, and viceversa. Thus c gc’ is 2:3:4 as to frequencies and 6:4:3 as to lengths, c fc’ quencies and 15:12:10 as to lengths, c es gis 10:12:15 as to frequencies and 6:5:4 as to lengths. That is precisely the consequence of the fact that, in replacing irequenciens by string lengths, we are reversing not only the relation between the outer terms but also that between every outer term is the arithmetical mean between a and d, c is the harmonic, and vice-versa; or again, 196 F, if we take the double and the triple of one term, e. g., 1:2 and 1:3, the arithmetical mean of 1:2 will be the harmonic of 1:3, and the major term of 1:2 is the arithmetical mean of 1:3. In the same way and the mean. Now just as there exists a preponderance of arithmetical proportion over the harmonic, we must concede to frequency numbers a preponderance over numbers representing string lengths. For this there is an evident reason which is, that string lengths are not so directly connected with sound production as are vibrations; therefore the representation of sounds by lengths is to be considered rather as derivative. We may add that string lengths are perceived by another organ, the eye, whereas our perception of vibrations is in a way a «continuation» of our auditory sense. It seems important to note in this connection that the eye is not capable of that spontaneous reaction to simple ratios as as the arithmetical proportion 2:3:4 produces the harmonic 6:4:3, 4:5:6 will produce 15:12:10, etc. Now what does that mean musically? It means that, the mterval between the extreme tones remaining the same, the dividing point is now in relation to the upper what it was in relation to the inferior; e. g., c gc’ is replaced by c’ fc or (admitting the « natural » third 4:5) ce g is replaced by g es c: the component intervals are the same but their order is reversed. It is easy to see that starting from two given outer terms which differ by one, e. g., from 1:2, in octave relation, or from 2:3, in fifth relation, we find in every case the arithmetical mean by multiplying these terms by two (necessarily so, since the arithmetical mean is, quantitatively, half their sum) and thus we obtain 2:3:4, or 4:5:6, etc. On the other hand we find the is the ear, i .e., it is not struck in the same way as the ear when faced with ratios as 1:2 or 2:3 (see above, p: 5). Thus we may find that the musical ear perceives more qualitatively than the eye; and yet colour as perceived by the eye is the classical instance of the notion of quality! This apparent contradiction is of course solved by recognizing that the notion of quality harmonic mean by multiplying the outer terms by the arithmetical mean: by 3 in the case of 1:2(= 2:4) which produces 3:6, with 4 as harmonic mean; by 5 in the case of 2:3 (= 4:6) which produces 10:15, with 12 as harmonic

may be a little different in both cases: one is the quality inherent in the member of a system in whose erection we take an active part (cf. above, But what about the geometrical mean? Boetius compares it to a « popular» or «equalized» state, because here the ratio is the same P- 35; yet that does not mean that we may erect it arbitrarily); the other is a quality the foundations of which are infinitely more remote, But whilst arithmetical proportion prevails over the harmonic, and frequency numbers prevail over string lengths, there is an affinity between arithmetical proportion and frequencies, as well as between harmonic proportion and lengths. It is not to be doubted that of the two tone relations € g c' and c f c* the former is spontaneousl y judged as more natural, smooth and « simple » than the latter, as well as ce e g when compared with e es ¢; now in every case the first is produc ing the simpler ratios when expressed by frequency numbers; and these are in the form of arithmetical, not of harmonic, propor tion. Yet Zarlino (Zstituzioni LIL 31) is very proud that in supposi ng string lengths, as he does, and taking the proportion to which the Greeks gave the beautiful name of «harmonic», he obtains the beautiful major chord (15: 12: 10 by string lengths), whereas the arithmetical proportion produces the minor chord (6: 5: 4). Ile is not aware that he is twice relying on the secondary: on arithmet ical Proportion as compared with harmonic, and on string lengths as compared with frequen cy numbers. He is, in a word, more humanist than thinker. Now as is known, the dispute between adherents and adversaries of harmonic «dualism» in the modern musical theory turns precisely around this question — whether the minor chord is to be considered as a counterpart of the major with equal rights, or rather as derived from it. I think this dispute could be settled by establishing that the relation is analogous to that between arithmetical and harmonic proportion as characterized above: indeed in both cases there is something of a to a further progression, 64:81. (In this sense the replacement of 64:81 by 4:5 — a third composed of two unequal whole tones, 8:9 and 9:10 — means that the third is « closing » on itself). This illustrates the general principle that different things, when being in inner agreement, form a harmony (as, e. g., the fifth and the fourth forming an octave); it is the reverse of the principle, touched on above, of « harmonic partition ». As is known, the geometrical mean is equal to the square root of the product of the outer terms, and finding the middle of two given tones corresponds to extracting this root from the product of their numbers. As this product is, in general, not a square number, this will lead to Yet I What we ought the outer terms being related as 1:2, i. e., it means the extraction of minor chord (as the root of twelfth degree from 2. We must admit that Aristoxenus, with his assertion of twelve equal half-tones in the octave, is not far from that conception. (At all events I do not think there is much in it that is « qualitative »!). Yet if we consider what he says, we see that even he does not think of deriving the whole tone in this artificial manner (apart from the fact that he would not have been able to calculate this kind of root). In fact Aristoxenus derives the whole tone, not otherwise than done by Plato, by subtracting between intervals and interval Sequen ces in general ) will be in its true perspective if we base it on frequency numbers . rather Other writers have stated that the geometrical proportion is not made use of in music. That is surely right in a large measure. In fact when juxtaposing two equal intervals we never attain a higher unity, whether it may be two fifths which produce a ninth, or two whole tones of 8:9 which lead to an interval clearly not « contented with itself > but pointing irrational numbers; and this is exactly in correspondence with the fact that it would be a tour de force (yet not a musical feat!) to sing the precise middle of a fourth or a fifth. In practice geometrical proportion and, then, root extraction is applied only where a purely external compromise between tones is established in the sense of temperament. As in this case we start from the octave that has to be divided into twelve equal intervals, it means that we have a geometrical proportion with thirteen membcrs, reversal and something of a derivation. should not like to introduce here questions of + musical theory». to be aware " of is only that dl te Comparison between major and with the greater and the smaller numbers, as, e- g., in 4:6:9, 39 than if we assume string lengths. Let us yet recall a curious parallel est ablished by Boetius, Arithmetics Il 45 (probably according to Nico m achus), as to arithmetical and harmonic proportion. He compares the arithmetic al mean to the rule of a fe w men, 1.i €, oligarc i hy,; because here the greater ratio is with the smaller numbers, e. 8, o in 2:35:44 2:3 is the greater ratio and 3:4 the smaller (we see that a quantitative conception of ratio is intruding here) The harmonic proportion is compared to aristocracy or the rule of the best men, because here the greater ratio is w ith the larger numbers, oa . €. 8, in 2:3:6 3:6 is the greater ratio. the fourth from the fifth, and this means that he is, after all, going back to consonant tone relations, with the only difference that, by some kind of idiosyncrasy, he does not like to connect them with numerical ratios. Therefore his statement that a whole tone is the sixth part of an octave, etc, must not be taken in a genetical sense, but only as an

out of the fourths, prescribed by Plato, would have to be applied only to the fourths But apart from the fact that a tonal system in my sense never These terms, used by the writer, seem to point gtees of pitch — and that I am entitled to use the letters accordingly.We arc in agreerelative, denoting the tone character, the other, more recent: denoting « absolute » dethat our tonal letters have a double meaning, the first and primary: qualitative and of the authors from whom I draw my material — seems quite unaware of the fact mental » than any of the other tone attributes. I said clearly that I take it as not being so elemental as the others: and which of them we take as the more important, depends, of course, on our point of view. The writer — who really mentions Pythagoras as one relations; 3) the major tonality with c as fundamental tone. The writer is at best unprecise when he ascribes to me the assertion that « Toncharakter> is more « fundain respect of tone quality; 2) the « mode » in which any note may be the center of tone to a specific conception of harmonics, whereas 1 have expressly distinguished — in superposing them one to the other — three conceptions: 1) the series of fifths, or rather a section taken from it, within which we naturally take d as «central» or « neutral» tonal gravity» or « fundamental» tone. extends to so much as 12 or 13 tones, I must state that I never took d as a «center of of tonal gravity». Although he is evidently well acquainted with it, the writer gives no hint as to this, and thus his attitude remains a purely negative one. He quotes the series cg dae h fis cis gis dis ais eis his with the same misprint as in my book (i. e., without h) and charges me with having made this the basis of a tonal system with d as the « center in other respects — has hitherto contributed to the problems tackled in my book. resulting from octave division. I take this, from my point of view, as confirmation that 41 External approximation: to him the whole tone represents the sixth of an octave, as the sixth part of the octave represents a whole tone. Indeed 40 in proceeding from the whole tone to smaller intervals Aristoxenus is estimating than of determining, as in this case we have not the obvious points of support imposed by simpler ratios, upon which to rely. This is obvious ratios, we are more than in the domain of « normal » intervals reduced to pure distance appreciation; but it is rather a question ot these smallest magnitudes otherwise than by approximation. Indeed in this realm of minim intervals and, accordingly, of complicated and not non minus ultra, Yet it is not imaginable how he could have determined more in the pure line of distance equalization; he speaks of halves, thirds and fourths of a whole tone, a quarter of a tone being in his mind the Plato's directions are not to be taken too concretely, In the rest I am inclined to adopt, with regard to ancient thought, an attitude less « modern » than the author, i. e., less naturalistic or positivist. As to the «number of Plato» referred to by Cicero as an example of obscurity, I think it is that about which Plato speaks in Republic, 546, a passage still much more involved than ours. P.S. II. As this paper is in some way a supplement to « Der Toncharakter », I may perhaps be permitted to add that this book kas been curiously misunderstood in The Musical Quarterly, issue of July 1949, by a writer introduced there as « chairman of the Department of Psychology at Princeton University». According to this writer it is not worth while to discuss further what thinkers, not yet enlightened by modern neurology, have said in regard to the musical quality of sound, and he thinks « progress» can henceforth be achieved only by joining psychology to neurology or physiology — a method happily inaugurated, according to nim, about 1925. 1 should indeed be grateful if he would indicate what this kind of research — surely valuable what could be said in answer to the question whether Aristoxenus knew of equalized temperament. However there is, in the strictly musical sense, a field where procedure Yet that is not a musical complex by equal steps takes place and is even essential: it is our series of fifths producing the different musical « qualities ». Indeed here a differs as much from d by its quality as e from a. in the proper sense but only a system underlying such: we may say that an infinite progression by uniform steps being given, we appropriate a sector of itand dispose it according to our symmetrical tendency (cf. above, Here we may remember that it is the same with Plato’s « three p. 18). powers of 2 and 3» representing, in the opinion of Proclus 193 B, the geometrical proportion which « embraces » the harmonic and the arithmetical; this isa background of musical reality, as is our series of fifths which has been so much calumniated because misunderstood (as to the Two and its powers we can, as we have seen, disregard them in a strictly musical sense). We must agree with Greek theorists: « popular » or « equalized » rule is not compatible with musical reality. And yet, on the other hand, we must state that a single tone is nothing without a « community» in which it takes its place. and have therefore to bear the blame of the writer (who, however, could have mentioned ment in so far as we distinguish those two aspects of tone reality, but he puts more stress on the latter and I, on the former. Be that as it may, 1 have done without physiology that I gave some reasons for my attitude). But when seeing that the writer consigns Hornbostel and Stumpf to the same scrap-heap, I must advance in their defence pre- P. S. L This paper was completed and forwarded to the Editor when I got notice of a paper by Ottavio Tiby, Note musicologiche al « Timeo» di Platone, publicisely that for which I reproached them, i, e. that in the last resort even they fell into the shed in « Dioniso », Bollettino dellIstituto Nazionale del Dramma Antico (XII, 1949), dierror of taking the physiological process as cause or explanation of the psychic; in this scussion, as he believes that, in order to obtain the scale intended by Plato, we must which treats the same subject. The Author introduces a new point of view in the (1 3 9 27) with its « disparate and perturbing» elements; in other words, the filling keep in mind only the division of the three octaves (1 2 4 8), dismissing that of the twelfths

respect the writer fails to recognize the alfinity he bears to them. Yet he also charges me with neglecting the physical conditions of tone; and here I have the real satisfaction to display a reviewer who represents the opposite extreme and charges me with indulging too much in acoustics (Schweizerische Musikzeitung, 1949). However, I hope that the writer will recognize that, even when I dealt with psychology, I was not bound word « Einfuehrungs as used in the subtitle of my book; an introduction is not to disregard history. He strives to give his review a witty close by speaking of «unconscious irony» but, alas, that supposes an arbitrary interpretation of the If I have failed in my undertaking necessarily a survey; we may, I think, try to «introduces the reader to a subject from whichever side we choose and «Einfuehrung in die Tonpsycholugie» is not the same as « Musikgeschichte im Ueberblicks. What is this notion of a musical psychology (or an area of musical psycho- —athing quite possible — the reason is surely not this. But ambiguity in expression is a thing I deplore not less than these misunderstandings. from Hornbestel the notion of « Tonhoehe> (height) as an intrinsic (not metaphoric) logy) < chiefly influenced by Stumpf and von Hornbostel »? Is it possible for any musical psychology not to be so in a large measure and has not the writer himself borrowed tone property which I did not? (CE. The meaning of music, 1931, 52-54; and compare Journal of experimental Psychology 1934, 585 HÉ, for another of these Hornbostelian The truth is phantoms taken over by one of the champions of psychological « progress » adduced by the writer). What signifies the allegation that my « factual material draws heavily » upon writers ranging from Pythagoras (1) to Hornbostel? Ie could mean that I took from them raw material and worked it over, and that would be wrong: or it could mean that I repeated what they have said and that would be wrong as well. that I have tried to understand the thinkers of the past (which is not, as the writer seems to believe, a matter of course) and to discuss the same problems. And why does the writer mention Stumpf's first great book, the one from 1883, and not the last one, from 1926, if not in order to reinforce the impression that my book is a priori obsolete? will, in general, come through much more easily. I could add that the review of the same « Toncharakter », published in the Journal of the American Musicclogical Society, U, Fall 1949, represents a progress in comparison lo that just discussed, in so far as its author, who has not in reality understood much more than his colleague, candidly confesses that he had difficulties to understand. But what is the reason of these difficulties? I am sure that it is an one-sided training rather than lack vf training, and therefore a non-protessional reader