476 



NATURE 



{March 29, 1877 



easy to conceive nor to produce. No ietnperametif, as such 

 a makeshfTt intonation is called, could exist except by help 

 of an instrument with fixed tones, and the most practised 

 tuner finds it impossible to produce a satisfactory result, 

 except by mechanical means.^ How then could we expect 

 a singer to produce at will any of the fifty or more schemes 

 of tuning that have been invented, or even either of the 

 only two that have been a success, the mean-tone and the 

 equal temperaments ? ^ 



The Greek system of intonation, as we know from 

 Euchd's " Section of the Canon," consisted of a series of 

 perfect Fifths, and may for convenience be represented 

 by F, C, G, D, A, E, B, where it will be found that each 

 note is a Fifth higher than the preceding. It is therefore 

 the simplest and most intelligible that can be imagined, 

 and is in fact at the base of all music. Reduced to the 

 same Octave and played as a major scale this gives 

 C, D, E, F, G, A, B, C. This is perfectly singable, and 

 produces excellent melodic effects. But if we attempt to 

 play the usual major and minor chords of the scale with 

 these notes, as F A C, C E G, G B D, D F A, ACE, 

 E G B, the effect, as I know by frequent experience, is 

 simply hideous, so that there is no difficulty in under- 

 standmg why the Greeks called Thirds "dissonances" 

 {diaphona), and had no harmony. Observing that in the 

 series of Fifths E is the fourth Fifth above C, we may 

 therefore say that the major Third cannot be identified 

 with the fourth Fifth up, reduced to the same Octave. 



3i 



On calculating out the ratio we find C : E = 4 : 5 X 



80' 



whereas the real consonance requires 4 : 5. This would 

 be produced by taking a note Ej, as we may write it, 

 which is flatter than E in the ratio of 80 : 81. The result 

 of carrying this out to the extent of modern modulation is 

 the series of 117 notes to the Octave already mentioned. 



In the Loan Collection, one instrument. General Perronet 

 Thompson's enharmonic organ, grappled with this problem 

 to the extent of forty notes.^ But its three finger-boards 

 with occasional extraordinary shapes and colours to the 

 finger keys (figured by Dr. Stone, op. cit., p. 32) might well 

 frighten the uninitiated. Yet General Thompson, himself 

 unable to play, taught a blind organist how to use it, so 

 that in a fortnight she could perform in public, and I 

 have often heard the instrument played by others, who 

 did not complain of any difficulty. Helmholtz (in my 



I " Mr Ellis has given a practical rule," [for producing the usual, inten- 

 tionally equal, temperament], "which does not eir in its results by much 

 more than the hundredth part of a semitone " (Seldom as much as that.) 

 " It is — make all the Fifths which lie entirely within the Octave c' c" [middle 

 C to the C above] beat ofice per second ; and make those which have their 

 upper notes above c" beat three times in tivo seconds. Keeping the !> ifth 

 f — c" to the last, it should beat once in between one and two seconds. Sse 

 Ellis's " Helmholtz,'' p. 785. This is a perfectly practicable rule, and tuners 

 ought to be instructed in the use of it. There are few tuners who can pro- 

 duce a tolerable equal tentpera?nent ." — Bosanquet, op. cit., p. 5. 



- See my paper on Temperament (Proc, Koy. Soc, vol. xiii., pp. 404-422), 

 where more than fifty schemes are calculated and analysed. Mr. Bosanquet 

 has a most interesting chapter on the history of the mean-tone temperament 

 or old organ tuning, the only oae known to Handel, and its complete realisa- 

 t on, wichout its former "wolves," by means of his own fingerboard, pp. 24-40. 

 He rightly considers this temperament most suitable for the organ (p. 58] as 

 tfte equal temperament is for the pianoforte. But then the voices of a choir 

 might be led by the completed mean-tone system without much injury to the 

 chords, which are shivered by the equal temperament. 



^ ihthe are given by Mr. Bosanquet, op cit., p. 22, arranged according 

 to his own finger-board, which completely does away with the terrors of the 

 original, but also expressed m his own notation, which implies a temperament 

 which General Thompson well knew and repudiated. Hence I add them 

 here in my own symmetrical arrangement (Proc. Roy. Soc, December, 1874, 

 vol. x.Kiii. p. 29, called "the simplest" by Mr. Bosanquet, p 50), in which 

 the columns repres-ent ascending Fifths, and the lines from left to right 

 ascending major Thirds, and where the superior and inferior numbers indicate 

 sharpcnmg or flattening by a comma (this is here published for the first 

 time). 



General P. Thompson's Enharmonic System :— 



translation, p. 636), who also heard it played, speaks of its 

 chords as '" extraordinarily harmonious," but the quality of 

 tone (one stop of " metal principals ") did not distmguish 

 the consonances effectually at all times, and the compass 

 of forty tones of course materially limited modulation 

 except into tonic and relative minors, which were well 

 provided for. 



Mr. Colin Brown has also grappled with perfect tertian 

 harmony, but has exhibited only a model of his key- 

 board, figured aiid described by himself in Dr. Stone's 

 book {id. pp. 42-45).^ His scheme allowed of sufficient 

 modulation into the dominant and sub-dominant major 

 keys, and their first relative minors, but almost utterly 

 ignored the tonic minor, and further minor modulation. 

 As the instrument is an harmonium, the quality of tone 

 is remarkably suitable for bringing out the effects of 

 perfect Fifths and Thirds, and when proper music was 

 selected, the result was most satisfactory. The keyboard 

 is much simpler than General Thompson's, and has the 

 great advantage of being the same in all keys. Such an 

 instrument is of the highest value for lecture illustrations 

 of harmony, and for training of vocalists in perfect into- 

 nation. As an independent instrument, its power of 

 modulation is too limited, and the fingering usually simple 

 enough, occasionally becomes very troublesome.- 



These are the only instruments on which perfect tertian 

 harmony was attempted. In the others some sort of com- 

 promise was come to. Mr. Gueroult showed his modifica- 

 tion of Helmholtz's double keyboard, each finger-key being 

 cut in half, and the upper half giving, generally, a note 

 flatter than the lower half by a comma.^ 



This is a most convenient instrument for scientific pur- 

 poses, but from its very limited capacity not so well suited 

 as even Mr. Colin Brown's for playing musical pieces. 

 The fingering is also full of difficulty, having all the im- 

 perfections of the ordinary board, with many others super- 

 added, and difilers in every key. 



We have seen that if we use the fourth Fifth up from 

 any note, when reduced to the same Octave, as the major 

 Third, it is too sharp by a comma, and unbearably dis- 

 sonant, but if we use the eighth Fifth down, also reduced 

 to the same Octave, the result is a note just y\ comma too 

 flat, which is very much closer than the major Third in 

 actual use (for that is -/, comma too sharp), and is not at 

 all disagreeable, although not by any means as pleasant 

 as the perfect major Third. Thus, going down eight per- 

 fect Fifths from C, we get in succession F, B/?., E/l., 

 A/l., Dfl., Gfl., Cfl., and Yfl., and the proposal is to use Y/l. 

 for Ej. On the piano, and all instruments in the usual 

 temperament, these are the same notes, but they are not 

 so when thus tuned, and they are still strictly distin- 

 guished in our usual musical notation. Herr Georg 

 Appunn, to whom we are indebted for various excellent 

 acoustical apparatus, and especially for a tonometer (ex- 

 hibited in the Loan Collection), which is self- verifying, 

 and enables us to measure pitch with wonderful accuracy, 

 likewise showed an harmonium, consisting of three rows 

 of keys, the upper ones in the form of studs, with practi- 

 cally the usual fingering, consisting of thirty-six tones 



' The notes used in the figure are comprised in the second, third, and 

 fourth columns of the above scheme of General P. Thompson's, but carried 

 further, the second column rising to J) sh. and descending to F_/?., the third 

 rising to B, sh., and descending to A^ /?., and the fourth rising to A3 ssh., 

 and descending to G3, giving fifcy-two notes in all ; but the instrument 

 performed on at Dr. Stone's lecture (July 25, 1876) had not so large a 

 compass. 



2 •' It is clear that this (Mr. C. Brown's) arrangement adapts itself with 

 seme facility to all music in which there is not much modulation, or in which 

 the modulation is of a simple type. It is, however, easy to give instances 

 which will at once involve tie performer in difficulties." Bosanquet, op. cit. 

 489. Mr. Bosanquet, who is a practised organist, having studied the finger- 

 ing, and played on the instrament at the Glasgow Meeting of the British 

 Association, is well able to speak to its capabilities. 



3 Disregarding one very slight alteration, the compass of twenty-four notes 

 extended from KJl. to B in the second column of General Thompson's, from 

 Gi to Gi sh. in the Third, and from Fg sh. to B2 sh. in the Fourth. With tl^e 

 exceptiun of Two Fifths (G^ sh. to ^Jl., taken to be Di sh.. ; and M^^sh. toGi, 

 taken at F^ ssh.), which were too flat by i\ o! a comma, as all Fifthsought to 

 be on the piano, i^li s gave a complete successioa of dominant modulation!, 

 but only admitted ol five minor keys. 



