On the derivation of the scale, tuning, temperament, the monochord etc

THE MUSICAL TIMES, respective proportions accurately mensured and And Singing Class Circular. OCTOBER lat, 1861, ascertained. Figures I, 2, and 3, are a section plan and view of a monochord of the most simple construction. Fig. 1. NOTICE, LRRENE Is our last Number we announced the intention of giving, this month, a notice of the Life and Labours of the late Vincent Novello. A difficulty has arisen from the writer being detained at Nice, while ali his} ¢ papers are at Genon; we trust, however, to have the 4¢ article for an early Number. tr — — ren, ON THE DERIVATION OF THE SCALE, TUNING, TEMPERAMENT, THE MONOCHORD, &c. In each of these figures, a bisa board made too thick to warp, having at each end ¢ 2, two supports for the string, of which it is required that Tua derivation of the scale of the major or the internal sides must be perpendicular, and the minor key is a subject upon which many hypotheses upper edge not rounded off, that the length of have been framed, and which seems hkely to the string and that of the board may exactly correscontinue a matter in dispute, Some authors pond; this length is here supposed to be three derive it from that of the harmonica, but the feet, e is the string which is here supposed to bea resemblance does not seem sufficiently close to steel wire called No, 11. The ends of the wire warrant such an hypothesis. are attached to a peg at each end, f and g (the Yartini, in order to obtain the notes of the latter of which is not visible in figure 3), placed major key, takes the three notes Do, Fa, Sol, at right angles to the string, Both of these are expressed by the numbers 6, 8, 9, which show to be turned in tuning the string, for if only one the respective proportional number of vibrations peg is used the string is apt to stretch more at of each note, as c, ¥, and 6, in the key of c major; that end than at the other, and consequently to and then adds to each of them the principal or be inaccurate. loudest harmonics which they produce, viz., the The manner of using the monochord is first to ‚SG \ CRotror 5 4320 PO By Da, Croren,® perfect chord or major third and fifth. Thus c lace it on a table, which acts as a sound board to gives B and 6, r gives A and c, and a gives » and it, augmenting its power. Next tune the string D; thus filling up the scale, for which reason a to c, on the second space of the bass clef, by some succession of triads falling a fifth has ever been other instrument, or by a pitch, or tuning fork. agreeable to the ear, as Sol, Do, Ta; and the Pinch the string with the finger and thumb* of numbers 6, 8, 9, and 12 (which express these one hand, taking care not to force the string out notes Do, Fa, and Sol, together with the octave of the straight line, and bow on the string with a to Do), have ever been famous above all others violin bow in the other. The student may either among the ancients, and when tuned by the ear mark the board according to his own discoveries in the following manner, give the major scale as of the notes produced by the string, or, which is invented by Ptolemy. rather recommended, he may draw lines on the Tune the notes a, F, and x, by the ear, respec- board parallel to the string, and on them mark tively a perfect fifth, perfect fourth, and major the places where he is to stop the string in order third to (viz., above) c. Then make a a mojor to produce the notes, third to P, and 3 a major third to a, and pa ivide the whole string ¢ 2 (fig. 4) into halves perfect fourth below it. by pinching it at c, the half c x will sound one Pythagoras was the inventor of the harmonical octave above ¢ x, the whole string. canon or monochord, which is merely a string Pig. 4. having a board under it of exactly the same length, upon which may be delineated the points at which the string must be stopped to give Divide the whole string c + (fig. 5) into three certain notes. This delineation of ratios renders equal parts, and pinch the string at a, the rethem capable of being compared, and their maining two thirds a x will give the note a, a fifth to the whole string. Fig, 5. * Reprinted by permission from Novello’s Library for the diffusion €. N f 7 of Musica) Knowledge. Vol. VIII. Dr. Crotch’s E of Musical Composition. + Let any of the lowest notes of a pianoforte, harp, viol Mo, or of the dispasons of an organ, be struck aud continued sounding, an * A sliding bridge would doubtless bo much more accurate, but tar accustomed to the experiment will distinctly hear the perfect also more dificult of performance, and perhaps not necessary for tho chord of that note, and probably several of the other less audible purposes here required: namely, of enabling the student to tune, or harmonies, at least to comprehend the nature of tuning.

On the Derivation of the Scale, Tuning, Temperament, the Monochord, &c. Author(s): Dr. Crotch Source: The Musical Times and Singing Class Circular, Vol. 10, No. 224, (Oct. 1, 1861), pp. 115118 Published by: Musical Times Publications Ltd. Stable URL: http://www.jstor.org/stable/3355208 Accessed: 16/07/2008 10:59 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=mtpl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. http://www.jstor.org

TIMES, respective proportions accurately measured and ascertained. And Singing Class Circular. Figures 1, 2, and 3, are a section plan and view THE MUSICAL OCTOBER 1st, 1861, of a monochord of the most simple construction. Fig. 1. NOTICE. In our last Number we announced the intention of giving, this month, a notice of the Life and Labours of the late Vincent Novello. A difficulty has arisen from the writer being detained at Nice, while all his ¢ papers are at Genoa; we trust, however, to have the article for an early Number. ON THE DERIVATION OF THE SCALE, TUNING, TEMPERAMENT, THE MONOCHORD, &c. In each of these figures, a bis a board made By Dr. Crorcn,* too thick to warp, having at each end ce x, two supports for the string, of which it is required that Tue derivation of the scale of the major or the internal sides must be perpendicular, and the minor key is a subject upon which many hypotheses upper edge not rounded off, that the length of have been framed, and which seems likely to the string and that of the board may exactly correscontinue a matter in dispute. Some authors pond; this length is here supposed to be three derive it from that of the harmonics, but the feet. ¢ is the string which is here supposed to be a resemblance does not seem sufficiently close to steel wire called No. 11. The ends of the wire warrant such an hypothesis. are attached to a peg at each end, f and g (the Tartini, in order to obtain the notes of the latter of which is not visible in figure 3), placed major key, takes the three notes Do, Fa, Sol, at right angles to the string, Both of these are expressed by the numbers 6, 8, 9, which show to be turned in tuning the string, for if only one the respective proportional number of vibrations peg is used the string is apt to stretch more at of each note, as c, ¥, and 6, in the key of c major; that end than at the other, and consequently to and then adds to each of them the principal or be inaccurate. loudest harmonics which they produce, viz., the The manner of using the monochord is first to perfect chord or major third and fifth.t Thus c place it on a table, which acts as a sound board to gives E and 6, r gives A and c, and 6 gives B and it, augmenting its power. Next tune the string D; thus filling up the scale, for which reason a to c, on the second space of the bass clef, by some succession of triads falling a fifth has ever been other instrument, or by a pitch, or tuning fork. agreeable to the ear, as Sol, Do, Fa; and the Pinch the string with the finger and thumb* of numbers 6, 8, 9, and 12 (which express these one hand, taking care not to force the string out notes Do, Fa, and Sol, together with the octave of the straight line, and bow on the string with a to Do), have ever been famous above all others violin bow in the other. The student may either among the ancients, and when tuned by the ear mark the board according to his own discoveries in the following manner, give the major scale as of the notes produced by the string, or, which is invented by Ptolemy. rather recommended, he may draw lines on the Tune the notes 6, F, and £, by the ear, respec- board parallel to the string, and on them mark tively a perfect fifth, perfect fourth, and major the places where he is to stop the string in order third to (viz., above) c. Then make A a major to produce the notes. third to rp, and B a major third to c, and pa Divide the whole string e (fig. 4) into halves perfect fourth below it. by pinching it at c, the half c 2 will sound one Pythagoras was the inventor of the harmonical octave above ¢ +, the whole string. canon or monochord, which is merely a string Fig. 4. x having a board under it of exactly the same €. c| length, upon which may be delineated the points Divide the whole string c + (fig. 5) into three at which the string must be stopped to give certain notes. This delineation of ratios renders equal parts, and pinch the string at a, the rethem capable of being compared, and their maining two thirds « æ will give the note a, a fifth to the whole string. Fig. 5. * Reprinted by permission from Novello's Library for the diffusion fi ñ x of Musical Knowledge. Vol. VIII. Dr. Crotch’s El ts of c. | { Musical Composition. G + Let any of the lowest notes ofa pianoforte, harp, violoncello, or of the diapasons of an organ, be struck aud continued sounding, an * A sliding bridge would doubtless be much more accurate, but ear accustomed to the experiment will distinctly hear the perfect also more difficult of performance, and perhaps not necessary for the chord of that note, and probably several of the other less audible purposes here required; namely, of enabling the student to tune, or harmonies, at least to comprehend the nature of tuning.

Divide the whole string ce + (fig. 6) into four In order to tune the major key of c according equal parts, and pinch the string at F, the re- to the methods of Ptolemy and ‘l'artini, make x a maining three quarters F + will give the note r, a major third to the whole string cz (fig. 14), and fourth above the whole string. ca fifth to it; r a fourth to it, A a major third to Fig.6 F, B a major third to 6, p a fourth below 6, and c €. x an octave to the whole string. in | F Divide the whole string c @ (fig. 7) into five € equal parts, and pinch the string at x, the remaining four fifths, £ a will give the note E, a major third to the whole string. Fig. 14. x re es ee ee D EF GABC The point p will be found to be one-ninth of the whole string ex from c; or rather the note D x is eight-ninths of the whole string e x; and I i | [ this interval, from c to D, is called a major tone, E Divide the whole string ¢ 2 (fig. 8) into six and it is the difference between a fourth and a equal parts, and pinch the string at H, the re- fifth; for if a fourth be subtracted from a fifth maining five sixths x æ will give the note EP, a the remainder will be a major tone. Thus to find the major tone above any given minor third above the whole string. note v (fig. 15), find x a fifth above v, and Pa Fig. 8, x below x; P & will be a major tone above v x; ° r | i i Fig. 7. c ham. fos. t L x Fig. 15. . And in the same way the octave, fifth, fourth, vb k major third, and minor third, may be found to or letv (fig. 16) be the given note, make x a fourth any given note on the monochord. Let x (fig. 9) be the given note; in order to below v, and p a fifth above x; pz will be a find the octave to kK, consider k æ as a whole major tone above v z. string, and divide x æ into two parts, and pinch it so as to take off one of them. Fig. 16. . Fig. 9. c. fi x | K rit . K VP But the point & (fig. 14) will not be a ninth part of p x, but a tenth part; or in other words, If the fifth to x is wanted, divide x z into three the note E x is not a major tone from p. The interval thus obtained is called a minor tone, and parts, taking off one. If the fourth to x is wanted, divide x æ into is the difference between a major tone and a major third; for if a major tone be subtracted four parts, taking off one. If the major third is wanted, divide x æ into from a major third, the remainder will be a minor tone, nine-tenths. five parts. Let it be required to find a minor tone to the And if the minor third is wanted, divide x x given note v (fig. 17), make p a major tone below into six parts. Thus the octave to c x (fig. 10) is L x the v, and k a major third to pP; x x will be a minor octave to L is M a, the octave to Mis N z, the tone above v x, and will be nine-tenths of v x. Fig. 17. octave to N iso x, and so on, ad infinitum. € a“ c om Fig. 10. Pv i ET “| Or let v (fig. 18) be the given note, above which L M NO In the same way the fifth toc x (fig. 11) is P it is required to find a minor tone, make P a fourth æ, the fifth to P is q a, the fifth to Q is Rr z, the to v, and ka fifth below p, and lastly, make r a major third to x, R x will be a minor tone to v x. fifth to Riss x, &c. € A Fig. 11. n Fig. 18. t fi x C. LA TT KRV P P a ks If it be required to find the minor tone below And by reversing the process, the notes below a given note may be found, provided they are a given note v (fig. 19), make k a major tone not more grave or deep than the generator, or above it, and pa major third below x; p x will be a minor tone below v x. note given by the whole string c a. Fig. 19. To find the octave below a given note T PM * (fig. 12) set off u to the left of 7, equal to T x ; ° PVK u æ will be the octave below T a. The interval £ r (fig. 14) will be found to be Fig. 12. a © one-sixteenth part of the distance x x, viz., # x in | will be fifteen-sixteenths of ex. The interval is U T To find the major third below a given note r called a major semitone, and is the difference (fig 13) divide T x into four equal parts, and set between a major third and a fourth, for if a major off T s equal to one of them; s æ will be the third be subtracted from a fourth the remainder will be a major semitone. Thus, let it be remajor third below t æ. quired to find the semitone above v (fig. 20), Fig, 18. ' . make Pp a major third below v, and R a fourth 5 t

above P, R x will be a major semitone higher dominant key of c iss major with one sharp; a will be Do, and a Re, &c., but from Do to Re than vr. Fig. 20. in a major key ought to be a major tone, see i rt fig 22 ; whereas in the major key of c from & to P VR c. x Or letv (fig. 21) be the given note, make P a A ls a minor tone. Compare figures 22 and 23, fourth abovev, and R a major third below Pp; the latter of which is the major key of G. Fig. 28. R x will be a semitone higher than v x. Do Re MiFa Sol La Si Do Fig. 21. III x ı c i VR P G A BC ee ee If a major semitone is required below a given T D E F&G re ere ere ee ee” 5 Ss t T t T note, the manner must be reversed. Let R be the And thus in the subdominant of c, F major with given note (fig. 20), make P a fourth below r, and one flat, from c to D, viz., from Sol to La should va major third to p, v x will be a major semitone be a minor tone, but in the major key of c, from below Rx. Or let R (fig. 21) be the given note, cto Disa major tone. See fig. 24, and compare make p a major third to it, and v a fourth below it with fig. 22. Fig. 24. P ; v x will be a major semitone below r x.* The Do Re MiFa Sol La Si Do interval ra (fig. 14) is a major tone eight-ninths, | a | | I | GA a minor tone nine-tenths, AB a major tone, c F G AB? CD EF N ee ee et ee” hen md Be a major semitone. See fig. 22, where the T t s T t Ts major tones are marked with their usual signature T, the minor tones t, and the major semitones s. Fig. 22. Do Re I T t Mi Fa Sol La Si Do { | ¢ s T LOU t T 98 It must also be understood— The minor key may be tuned likewise in the same way as the major, only making the thirds to Do, Fa, and ‚Sol, minor instead of major. There is some difficulty, however, in choosing the first note Do of the principal minor key of a. Some authors make it the same as the note La of the relative major key, viz., A in the key of c, a minor tone above «. In which case all the Thata major 3rd is equal toa T and at, as cE. natural notes excepting D correspond with those … minor 3rd... ST. EG. ofthe major keyofc. Compare figures 25 and 22. … … … perfect 4th perfect Sth major 6th major 7th Andan 8ve ... TtS … TTts … mrtts TTTtts TTTttss a .. … .. … CF. ee, CA. CB. cc. , Fa | Sol La | Cc LT D T 8 Fig, 25. Do Re Mi | E F eee ee ne t Si | G PT A B C Fa Sol i ( D E me ee See ee ee eee” T t rs t T The minor third pr consisting of t and s, and If the major thirds to Fa and Sol, rf and 6%, * The major or diatonic semitone having been mentioned, it seems necessary to inform the student, that a minor or chromatic semitone (marked s) is the difference between a major semitone and a minor or major tone; as from EP to Eg, from cto ch, from F to FB, &c. There are also several other intervals resulting from the combination of many keys on the same monochord, the knowledge of which is not necessary to the student. + The ratios of the monochord are generally expressed thus: the major tone 8, minor tone 19, major semitone 18, &c, i Fa Sol La Si IT CD EF Na Na t T Ne s T G Do A ReMi Fa II BC D ae = the fifth DA consisting of r t ts, are therefore not be added to this scale, they will be different from in tune, but are both deficient by a small interval those notes in the keys of A major. The author called a comma, which is the difference between of the present work, therefore, prefers making a major tone eight-ninths, and a minor tone nine- the key notes ofA minor and A major the same; tenths, and is about as 80 to 81.7 viz., a whole tone from a in the key of c major, The note p combined with F or A, however, is see fig. 26, in which case only one natural note not wanted in either of the triads, Do, Fa, or Sol of the key of A minor, viz., D, will be the same of the major key of c, but in the minor key of A with those of the major key of c; but the key the triad of Fa is D, Fr, and A: hence a different note, the fifth, and the fourth, will be the same tuning is required in the relative minor key to with those of the key of a major, three sharps; that just described. But no two major keys at as also the r$ and oh; and also the cf, which all related to each other can exist, on the same is sometimes used in a close. keyed-instrument, perfectly in tune.f Thus the Fig. 26. me ms er ea eee” t T T Ss t Do Having seen the impossibility of perfection on an instrument which has any limited number of sounds in an octave, the student may next prot In a lecture on this subject, the author of the present work caused ceed to the study of temperament, viz., of the the keys of £ major with four sharps, and ED major with three flats, to distribution of the unavoidable imperfection rebe tuned perfectly on the same pianoforte, viz., first the triads of sulting from the limited number of sounds. BE F ch, ch, pg. And then having two notes, eff and DË already tuned, On keyed instruments containing only twelve E A B notes in an octave, three major thirds (as cr, which would serve for AP and ep, c was added to them, and lastly B triads of ze} or AD, abe, or as 68, m$ or ED, EPG) make an octave; but three major thirds tuned perfectly

to each other, as c u M n, fig. 27, fall considerably are made more perfect than on the equal temperashort of the true octave c. Hence in tuning, one, ment, which necessarily renders others less pertwo, or all of the three major thirds, which con- fect. Of this there are many systems, which the stitute every octave, must be tempered too sharp ; student is now capable of examining for himself. and the nearer perfection any of them are made, He will also find much amusement in studying the worse will the others become. N c is the the various attempts to improve the scale by unavoidable imperfection which must be added increasing the number of notes in the octave, either to one or more of the thirds, and if equally such as that of the two additional notes at the divided between them will, upon the whole, be Temple organ, of the five additional notes in least offensive to the ear. Mr. Hawke’s instruments, and of the twelve additional notes in those by Mr. Löeschman. In all these the bulk, expense, and complication of the instrument are increased in proportion to Fig. 27. L c MN Again, twelve fifths, or, which is the same the number of notes added, and the consequent thing, six major tones, on a keyed instrument, approach to perfection. The author, in conclusion, cannot but regret constitute an octave; but on the monochord it will be found that they exceed it by a small por- that the preference of English organists for the old HI KLMNOPQRST method of tuning is (as he is informed) hitherto represent twelve sounds so obtained, the latter so strong and determined, as to have resisted and whereof does not coincide with the true octave c: repelled the attempts made to introduce the equal c Tis the unavoidable imperfection which must temperament into our Cathedrals and Churches. be subtracted from one or more of the twelve He has for many years uniformly recommended that this system should have a fair trial, upon the fifths which compose an octave. principle that as all tempered fifths and thirds Fig. 28. offend the ear, those systems which contain such Cc I L NH K M c O as are most tempered and most discordant cannot | i Pop sy | ı 1771 Cc 1 L N P R T Qs be preferable; especially in an age when the keys If equally distributed, this imperfection will be which have four sharps and three flats can no scarcely perceptible ; when the fifths are all longer be excluded from general use. It has at equally too flat, the thirds will all become, of length been fairly tried, and, having carefully their own accord, equally too sharp, and this will examined it, he feels convinced that its practication, fig. 28, where render all keys equally imperfect, which is called bility and superiority are as unequivocal on the the equal temperament, and may be obtained on organ as they are allowed to be on the piano- OMNIA the monochord as follows. Divide the whole forte, and on all other instruments which contain string ce x into one thousand parts, beginning only twelve different notes in each octave. He continues to press these opinions, not merely from right to left, as in fig. 29: because they are his own, but because, in so Place the note 2 at 943 doing, he is contending for the far higher ‘au3... 890 thority of the judgment and practice of one .. 840 whom, he trusts, his opponents must venerate .. 793 and admire,—the greatest of all composers for .. 749 this sacred instrument— SEBASTIAN Bacu. .. 707 .. 667 .. 629 10 .. 594 11 .. 561 12 .. 529 13 .. 500, the true octave. Fig. 29. ? 800 700 600 500 Db 54 567 Soons 400 LPL, 300 200 100 Tune any one of the twelve notes of a keyed instrument to the whole string 1 x, then 2 x will give the next note, 3 x the next, &c., to 13 x, which will be an octave tol x. If the note 1 x be c, then 2 x will be ck or pp, 3 x will be p, 4 x will be p# or ED, &c. The fifth 8 x will be only one thousandth part of the whole string too flat; but the third 5 x will be seven such parts TO CORRESPONDENTS. We cannot undertake to return offered contributions; the‘authors, therefore, will do well to retain copies. We would request those who send us country newspapers, wishing us to read particular paragraphs, to mark the passage, by cutting a slip in the paper near it, Colored Envelopes aresent to all Subscribers whose payment in advance is exhausted. The paper will be discontinued where the Subscriber neglects to renew. We again remind those who are disappointed in getting back numbers, that only the music pages are stereotyped, and of the rest of the paper, only sufficient are printed to supply the current sale. Notices of concerts and other information supplied by our friends in the country, must be forwarded as early as possible after the occurrence, otherwise they cannot be inserted. Our correspondents must specifically denote the date of each concert, for without such date no notice can be taken of the perfor: u ications must be authenticated by the proper name and address of the writer. Brief Chronicle of the last Month, ABINGDON.—On the 18th ult. the Musical Association engaged the Brousil Family to give a Concert and contoo sharp. tributed several vocal pieces themselves, which were well Unequal temperament is that wherein some of received. The performance of the Brousil children was the fifths, and consequently some of the thirds, received with every appearance of gratification,