# A Letter of Dr. John Wallis to Samuel Pepys Esquire, Relating to Some Supposed Imperfections in an Organ

reproduction of a letter from 1698
Mr. Harris an Organ-maker (whom I find, by the little discourse I had with him, to be very well skilled in his profession) was lately with me, as by direction from you, to ask my opinion about perfecting an Organ, in a point wherein he thinks it yet Imperfect.
'Tis an honour you please to put upon me, to think my opinion considerable in a thing wherein I am so little acquainted as that of an Organ.
I do not pretend to be perfectly acquainted with the Structure of an Organ, its several Parts, and the Incidens thereunto; Having never had Occasion and Opportunity to inform my self particularly therein. And, for the same reason, many of the Words, Phrases, Forms of Speech, and Terms of Art, which are familiar to Organists and Organ makers, are not so to me. Which therefore I shall wave; (For till we perfectly understand one anothers Language, it is not easy to speak intelligibly;) and apply my self directly to what is particularly proposed.
This (I take it) is evident; That each Pipe in the Organ is intended to express a distinct Sound at such a Pitch; That is, in such a determinate Degree of Gravity or Acuteness; or (as it is now called) Flatness or Sharpness. And the Relative or Comparative Consideration of Two (or more) such Sounds or Degrees of Flatness and Sharpness, is the ground of (what we call) Concord and Discord; that is, a Soft, or Harsh, coincidence.
Now, concerning this, there were amongst the Ancient Greeks, Two (the most considerable) Sects of Musicians: the Aristoxenians and the Pythagorians.
They both agreed thus far; That Dia-tessaron and Dia-pente, ​do together make-up Dia-pason; that is (as we now speak) a Fourth and Fifth do together make an Eighth or Octave: And, the Difference of those two (of a Fourth and Fifth) they agreed to call a Tone; which we now call a Whole note.
Such is that, (in our present Musick,) of La Mi, (or as it was wont to be called, Re Mi.) For La fa sol la, or Mi fa sol la is a perfect Fourth: And La fa sol la mi, or La mi fa sol la is a perfect Fifth: The Difference of which, is La mi which is, what the Greeks call, the Diazeuctick Tone which doth Dis-join two Fourths (on each side of it;) and, being added to either of them, doth make a Fifth. Which was, in their Musick, that from Mese to Paramese; that is in our Musick from A to B: supposing Mi to stand in B fa b mi, which is accounted its Natural position.
Now, in order to this, Aristoxenus and his Followers, did take, that of a Fourth, as a Known Interval, by the judgement of the Ear; and, that of a Fifth, likewise; And consequently, that of an Octave, as the Aggregate of both; and that of a Tone as the Difference of those Two.
And this of a Tone (as a known Interval) they took as a common Measure, by which they did estimate other Intervals. And accordingly they accounted a Fourth to contain Two Tones and an half; a Fifth to contain Three Tones and an half and consequently an Eighth to contain Six Tones or Five Tones and two Half-tones, And it is very near the matter, though not exexactly so.
And at this rate we commonly speak at this day; supposing an Octave to consist of Twelve Hemitones, or Half-notes. (Meaning thereby, somewhat near so many half-notes:) But, when we would speak more Nicely, we do not take those supposed Half-notes to be exactly Equal or each of them just the Half of a Full-note such as is that of La-mi.
Pythagoras and those who follow him, not taking the Ear alone to be a competent Judge in a case so nice; chose to distinguish these, not by Intervals, but by Proportions. And accordingly they accounted that of an Octave, to be, when the degree of Gravity or Acuteness of the one Sound to that of the other, is Double, or as 2 to 1; that of a Fifth, when it is Sesqui-alter, or as 3 to 2; that of a Fourth when Sesqui-tertian, or as 4 to 3 . Accounting That, the Sweetest proportion, which is expressed in the Smallest Numbers; and therefore (next to the Unisone) that ​of an Octave, 2 to 1; then that of a Fifth, 3 to 2; and then that of a Fourth, 4 to 3.
And thus, that of a Fourth and Fifth, do together make an Eighth; For 4/3 × 3/2 = 4/2 = 2/1 = 2. That is, four thirds of three halves, is the same as four halves that is Two, Or (in other words to the same sense) the proportion of 4 to 3, compounded with that of 3 to 2, is the same with that of 4 to 2,m or 2 to 1. And, consequently, the Difference of those Two, which is that of a Tone or Full-Note, is that of 9 to 8. For 4/3)3/2(9/8 that is, three halves divided by four thirds is nine eights; or, if out of the proportion of 3 to 2, we take that of 4 to 3; the Result is that of 9 to 8.
Now, according to this Computation, it is manifest, That an Octave is somewhat less than Six Full-notes. For (as was first demonstrated by Euclide, and since by others) the Proportion of 9 to 8, being six times compounded, is somewhat more than that of 2 to 1. For 9/8×9/8×9/8×9/8×9/8×9/8=531441/262144 is more than 524288/262144=2/1.
This being the Case; they allowed (indisputably) to that of the Dia-zeuctick Tone (La mi) the full proportion of 9 to 8; as a thing not to be altered; being the Difference of Dia-pente and Dia-tessaron or the Fifth and Fourth.
All the Difficulty, was. How the remaining Fourth (Mi fa sol la) should de divided into three parts, so as to answer (pretty near) the Aristoxenians Two Tones and an Half; and might, altogether, make up the proportion of 4 to 3; which is that of a Fourth or Dia-tessaron.
Many attempts were made to this purpose: And, according to those, they gave Names to the different Genera or Kinds of Musick, (the Diatonick, Chromatick, and Enarmonick Kinds,) with the several Species, or lesser Distinctions under those Generals. All which to enumerate, would be too large, and not necessary to our business.
The first was that Euclide (which did most generally obtain for many ages:) Which allows to Fa Sol, and to Sol la, the full proportion of 9 to 8; And therefore to Fa sol la (which we call the greater Third,) that of 81 to 64. (For 9/8 × 9/8 = 81/64.) And, consequently, to that of Mi fa (which is the Remainder to a Fourth) that of 256 to 243. For 81/64 ) 3/4 ( 256/243 that is, if out of the proportion of 4 to 3 we take that of 81 to 64, the Result is that of 256 to 243. To this they gave the name of Limma (λεῖμμα) ​that is, the Remainder (to wit, over and above Two Tones.) But, in common disourse (when we do not pretend to speak nicely, nor intend to be so understood) it is usual to call it an Hemitone or Half-Note (as being very near it ) and, the other, two Whole-Notes. And this is what Ptolemy calls Diatonum Ditonum, (of the Diatonick kind with Two full Tones.)
Against this, it is objected (as not the most convenient Division,) that the Numbers of 81 to 64, are too great for that of a Ditone or Greater Third: Which is not Harsh to the Ear; but is rather Sweeter than that of a single Tone who's proportion is 9 to 8. And in that of 256 to 243 the Numbers are yet much greater. Whereas there are many proportions (as 5/4,6/5,7/6,8/7,) in smaller numbers than that of 9 to 8; of which, in this division, there is no notice taken.
To rectify this, there is another Division thought more convenient; which is Ptolemy's Diatonum Intensum (of the Diatonick Kind. more Intense or Acute than, that other..) Which, instead of Two Full tones for Fa sol la; assignes (what we now call) a Greater and a Lesser Tone; (which, by the more nice Musicians of this and the last Age, seems to be more embraced;) Assigning to Fa sol, that of 9 to 8 (which they call the Greater Tone:) and to Sol la, that of 10 to 9, (which they call the Lesser Tone:) And therefore to Fa la (the Ditone or Greater Third) that of 5 to 4. (For 10/9=9/8 =5/4.) And consequently, to Mi fa (which is remaining of the Fourth) that of 16 to 15. For 5/4) 4/3 (16/15. That is; if out of that of 4 to 3, we take that of 5 to 4, there remains that of 16 to 15.
Many other waies there are(with which I shall not trouble you at present) of dividing the Fourth or Dia-tessaron or the proportion of 4 to 3, into three parts, answering to what (in a looser way of Expression) we call an Half note, and two Whole-notes. But this of 16/15 × 9/8 × 10/9=4/3 is that which is now received as the most proper. To which therefore I shall apply my discourse. Where 16/15 is (what we call) the Hemitone or Half note, in Mi fa: 9/8 that of the Greater-Tone, in Fa sol and 10/9 the Lesser-Tone, in Sol la.
Onely with this addition; That each of those Tones, is (upon occasion) by Flats and Sharps (as we now speak) divided into two Hemitones or Half notes: Which answers to what by the Greeks was called Mutatio quoad Modos (the change of Mood;) and what is now done by removing Mi to another Key. Namely 9/8=18/16=18/17×17/16; and 10/9=20/18=20/19×19/18.
​Thus, by the help of Flats and Sharps (dividing each Whole Note, be it the Greater or the Lesser, into two Half-notes or what we call so,) the whole Octave is divided into Twelve Parts or Intervals (contained between Thirteen Pipes) which are commonly called Hemitones or Half-notes. Not, that each is precisely Half a Note, but somewhat near it, and so called. And I say, by Flats and Sharps; For sometime the one, sometime the other, is used. As, for instance, a Flat in D, or a Sharp in C, do either of them denote a Midling Sound (tho' not precisely in the Midst) between C and D; Sharper than C, and Flatter than D
Accordingly; supposing Mi to stand in B fa b mi (which is accounted its Natural seat) the Sounds of each Pipe are to bear these proportions to each other, viz,
| B | | C | | # | | D | | # | | E | | F | | # | | G | | # | | a | | # | | b |
| mi | | fa | | # | | sol | | # | | la | | fa | | # | | sol | | # | | la | | # | | mi |
| | 16/15 | | 18/17 | | 17/16 | | 20/19 | | 19/18 | | 16/15 | | 18/17 | | 17/16 | | 20/19 | | 19/18 | | 18/17 | | 17/16 | |
| | | | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | | | | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad } }$ | |
| | | | | 9/8 | | | | 10/9 | | | | | | 9/8 | | | | 10/9 | | | | 9/8 | | |
| | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | | | | | |
| | | | | | 4/3 | | | | | | | | | | 4/3 | | | | | | | | | |
| | | | | | | | | | | | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | | |
| | | | | | | | | | | | | | | | | | 3/2 | | | | | | | |
| | | | | | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | $\scriptstyle {\underbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$ | | | | | | | |
| | | | | | | | | | | | 2/1 | | | | | | | | | | | | | |
And so in each Octave successively following. And if the Pipes in each Octave be fitted to sounds in these proportions of Gravity & Acuteness; it will be supposed (according to this Hypothesis) to be perfectly proportioned.
But, instead of these successive proportions for each Hemitone; it is found necessary (if I do not mistake the practise) so to order the 13 Pipes (containing 12 Intervals which they call Hemitones) as that their Sounds (as to Gravity & Acuteness) be in Continual Proportion, (each to its next following, in one and the same Proportion;) which, all together, shall compleat that of an 0ctave or Dia-pason, as 2 to 1. Whereby it comes to pass, that each Pipe doth not express its proper Sound, but very near it, yet somewhat varying from it, Which they call Bearing. Which is somewhat of Imperfection in this Noble Instrument, the Top of all.
​It may be asked, Why may not the Pipes be so ordered, as to have their Sounds in just Proportion, as well as thus Bearing?I answer, It might very well be so, if all Musick were Composed to the same Key or (as the Greeks call it) the same Mode. As, for instance, if in all Compositions, Mi were alwaies placed in B fa b mi. For then the Pipes might be ordered in such proportions as I have now designed.
But Musical Compositions are made in great variety of Modes or with great diversity in the Pitch. Mi is not always placed in B fa b mi; but sometimes in E la mi; sometimes in A la mi re, &c. And (in summe) there is none of these 12 or 13 Pipes but may be made the Seat of Mi. And if they were exactly fitted to any one of these cases, they would be quite out of order for all the rest.
As, for instance; If M be removed from B fa b mi (by a Flat in B) to E la mi: Instead of the Proportions but now designed, they must be thus ordered;
| B | | # | | C | | # | | D | | # | | E | | F | | # | | G | | # | | a | | b |
| fa | | # | | sol | | # | | la | | # | | mi | | fa | | # | | sol | | # | | la | | fa |
| | 18/17 | | 17/16 | | 20/19 | | 19/18 | | 18/17 | | 17/16 | | 16/15 | | 18/17 | | 17/16 | | 20/19 | | 19/18 | | 16/15 |
Where 'tis manifest, that the removal of mi doth quite disorder the whole series of Proportions. And the same would again happen, if mi be removed from E to A (by another Flat in E.) And again if removed from A to D. And so perpetually.
But the Hemitones being made all Equal; they do indifferently answer all the positions of Mi (though not exactly to any:) Yet nearer to some than to others. Whence it is, that the same Tune sounds better at one Key than at another.
It is asked, Whether this may not be remedied; by interposing more Pipes; and thereby dividing a Note, not only (as now) into Half-Notes, but into Quarter-notes or Half quarter-notes &c.
I answer; It may be thus remedied in part; (that is the Imperfection might thus be somewhat Less, and the Sounds somewhat nearer to the just Proportions:) but it can never be exactly true, so long as their Sounds (be they never so ​many) be in continual proportion; that is, each to the next subsequent in the same Proportion.
For it hath been long since Demonstrated, that there is no such thing as a just Hemitone practicable in Musick, (and the like for the division of a Tone into any number of Equal parts; three, four, or more.) For, supposing the Proportion of a Tone or Full-note to be 9/8 (or, as 9 to 8;) that of the Half-note must be as $\surd 9$ to $\surd 8$ (as the Square-root of 9 to the Square-root of 8; that is, as 3 to $\surd 8$, or 3 to $2\surd 2$,) which are Incommensurable quantities. And that of a Quarter-note, as $\surd qq\,9$ to $\surd qq\,8,$ (as the Biquadrate root of 9, to the Biquadrate root of 8,) which is yet more Incormnensurate. And the like for any other number of Equal parts. Which will therefore never fall-in with the Proportions of Number to Number.
So that this can never be perfectly adjusted for all Keys (without somewhat of Bearing) by multiplying of Pipes; unless we would for every Key (or every different Seat of Mi) have a different Set of Pipes, of which this or that is to be used, according as (in the Composition ) Mi is supposed to stand in this or that Seat. Which vast number of Pipes (for every Octave) would vastly increase the Charge. And (when all is done) make the whole impracticable.
These are my present thoughts, of the Question proposed to me, and upon these grounds.
You will please to excuse me for the trouble I give you of so long a Letter.
I thought it necessary, to give a little intimation of the Ancient Greek Musick compared with what is now in practise; which is more the same than most men are aware of: though the Language be very different. But I was not to be large in it, Those who desire to know more of it; may see my thoughts more at large, in that Appendix which I have added at the end of my Edition of Ptolemy's Harmonicks in Greek and Latin.
The two Eminent Sects amongst them, the Aristioxenian and the Pythagorian, differ much at the same rate as doth the Language of our ordinary practical Musicians, and that of those who treat of it in a more Speculative way.
Our Practical Musicians talk of Notes and Half-notes, just as the Aristoxenians did; as if the Whole Notes were all Equal; and the Half notes likewise each the just Half of a Whole Note. ​And thus it is necessary to suppose in the Pipes of an Organ; which have each their determinate Sound and not to be corrected, in their little Inequalities, as the Voice may be by the guidance of the Ear.
But Pythagoras, and those who follow him found (by the Ear) that this Equality of Intervals would not exactly answer the Musical Appearances, in Concords and Discords: just as our Organists and Organ-makers be now aware; that their Pipes at equal Intervals do not give the just desired Harmony, without somewhat of Bearing that is, of some little variation from the just Sound.
The Pythagorians to help this, changed the notion of Equal Intervals into that of due Proportions. And this is followed by Zarline, Keppler, Cartes, and others who treat of Speculative Musick in this and the last Age. And though they speak of Notes and Half notes (in a more gross way) much as others do, yet declare themselves to be understood more nicely.
And though our present Gam-ut take no notice of this little diversity; yet, in Vocal Musick, the Ear directs the Voice to a more just proportion. And, in String Musick, it may in like manner be helped by straining and slackening the Strings, or moving the Frets. But in Wind Musick the Pipes are not capable of such correction; and therefore we must be content with some little irregularity therein; that so they may tolerably answer (though not exactly) the different Compositions according to the different placing of Mi in the Gam-ut.
Now the Design of Mr. Harris seems to be this; either (by multiplying intermediate Pipes) to bring the Organ to a just Perfection: Or else (if that cannot be done) to rest content with the little Imperfection that is; which though, by more Pipes, it may be somewhat abated, yet cannot be perfectly remedied. And in this I think we must acquiesce,
I am SIR Yours to serve you
Oxford June
27. 1698.
John Wallis.
This work was published before January 1, 1929, and is in the public domain worldwide because the author died at least 100 years ago.
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