A Kind Word or Two About

ALEXANDER JOHN ELLIS

ON THE OCCASION OF THE CENTENNIAL

1885-1985

OF THE SECOND ENGLISH EDITION OF HIS

TRANSLATION OF AND APPENDIX TO

HELMHOLTZ'S SENSATIONS OF TONE

Ivor Darreg (1985)

Why celebrate the second edition of anything? (Actually, the English translation of the Fourth German Edition of 1877, the First German Edition having been in 1862.) Why, indeed, celebrate a translator and his translation? At first blush, this seems rather strange.

Principally, because the time for certain ideas of both Helmholtz and Ellis seems to have come at long last. The means of achieving a better intonation than the standard 12-tone equal temperament, both for existing music and for new composition works, is at last available in many forms, undreamt-of and quite impossible at the time this 1885 edition was prepared, let alone back in 1862 when Helmholtz penned his first edition.

Also, because the mid-19th century was a time of profound disagreement and dreadful misunderstanding between the practitioners of Science and Art--never before were the two disciplines further apart, and never since, thank goodness. Some of the breach has been healed, though not enough; the divisive Specialism of the turn of the century is giving way (though not enough) to Generalism and Interdisciplinary Studies and Colloquia.

In a way, the Helmholtz book and its translation and appendix by Ellis came out at a bad time, because the Romantic Period in art and music was going very strong. There was no "peaceful coexistence" between the two factions, and polemics flew fast and furiously. The very time of the appearance of these editions was also the very time when the Pianoforte reached its zenith--the point beyond which it could not be improved or changed without losing its personality, its spirit, its unique qualities. Everything since about the piano has been downhill--so slowly at first that nobody noticed it, but gaining momentum decade by decade as the Twentieth Century progressed. Only now, in the 1980s, can you find people willing to admit this, and even they fear the wrath of the Musical Establishment. Only recently have the new electronic musical instruments and the programs for computer music reached a point where they can challenge the Pianofortic Empire.

We can finally do something about what Ellis strove for and spent so much time and trouble tabulating and calculating and describing and pleading for. The means are HERE NOW. They have not been hitherto; that is why so many attempts to follow Ellis' lead have aborted. That is why so many of Ellis' followers have been ridiculed and thwarted. It suffices to cite the late Harry Partch's lifelong struggles to get a hearing for his system of applying Just Intonation.

Why concentrate on Ellis, the translator, rather than the firs author, Prof. Helmholtz? Helmholtz is secure in his place because of many other works and achievements besides what he did for acoustics and the scientific side of music. He doesn't get the venom and spite and slander and malicious misunderstanding that Ellis has suffered for an entire CENTURY! Furthermore, Ellis' works under his own by-line and self-propulsion were in another field: phonetics--we may cite here his Early English Pronunciation With Special Reference to Shakespeare and Chaucer, which sought to recover the ability to declaim those early works in English in the same speech-sounds that those bygone authors originally spoke and heard. He was heavily involved with phonetic transcription schemes and with Pitman of shorthand fame whose system is still used by court reporters and others. Those who know of Ellis' work in phonetics and pronunciation are not apt to make much of this Appendix to Helmholtz' book and his copious footnotes and comments which continue through the many fine-print pages of Helmholtz' text. It is unlikely that European readers of Helmholtz know much about Ellis, either.

That is, we are dealing with an ironic situation bearing a cousinly relation to that of the Ghost-Writer. Unthinking or casual or careless readers of Sensations of Tone frequently quote Ellis on something and say that Helmholtz said it. Rarely, vice versa. In the commonly-available Dover and other reprints of this Second English Edition, we have 430 pages of Helmholtz's work and some 126 pages of what Ellis modestly called Additions By the Translator. Now to this add hundreds of footnotes and comments enclosed in square brackets [ ] and most of which are in extremely small print, and if you could add all that together, you surely would have a big enough book by Ellis in its own right. Another sad story of over-modesty, perhaps. Alas! This was compounded all during the past century by critics who thought no invective or denunciation or literary trick too base to inflict on Ellis. Frankly, I know of no instance of unjust criticism and prejudice and dismissal without any real consideration or investigation, worse than this hundred-year-long attack upon Ellis.

Is there any more misunderstood monograph or scholarly document in the field of music? The garbling and misquotation in citations from Ellis are incredible. One author called the Duodenarium (a table of modulations in just intonation) "a musical nightmare." Many writers accused Ellis, with or without naming him, of proposing impossible instruments with too many keys on them, or asking impossible changes in music, when Ellis included all the proper disclaimers and was in touch with reality. I thought music was supposed to soothe the savage breast, but in this case it brought out the worst in certain professors. We can't afford to waste more space on the cruel criticism, but had to call attention to its long existence.

We should acknowledge how hard it must have been for Ellis to do all the extensive theoretical calculations, for there weren't any electronic calculators--he didn't even have motor-driven mechanical office calculators that graced early 20th-century business offices. Golly, what he could have done with a computer! Or even one of those pocket calculators you can get at the local shopping mall. It must have meant hundreds of lonely solitary hours with pen and paper and tables of logarithms and reference books.

Very well: let's get down to cases. Ellis introduces the measure of musical intervals called cents. A cent is a 1200th of an octave or .01 semitone of the ordinary 12-tone equal temperament. This method of stating the size of musical intervals is now widely accepted in a number of countries, and is the basis for calibrating most of the new electronic tuning-devices now on the market. (France seems to be the holdout with savarts, 301 to the octave.)

With cents, one can gain an idea of how much the just intervals differ from those of 12-tone, and how much the intervals of the non-12-tone temperaments, equal and unequal, differ from our familiar 12-tone intervals. When we say that a just intonation perfect fifth has 702 cents, and the 12-tone one has 700, we can see what the tiny difference is; then go on to the 12-tone major third of 400 cents vs. the just one of 386 cents--now there is a difference worth worrying about. The 31-tone temperament has a major third of 387 cents and its difference from the just or untempered one is inaudible--less than one cent but we rounded it off to one cent for simplicity. And so on--cents make possible the compilation of tables comparing a wide range of tuning systems.

It takes very favorable conditions for the average person to hear the interval of a cent. Ordinarily it would be when one is using sustained tones and listening to the beats between them and counting the beats. It certainly would never occur on ordinary instruments int eh course of a musical performance. But the new electronic devices often are correct to a fraction of a cent and thus for the first time in history instruments can be accurately tuned and kept so.

Ellis in some of the theoretical calculations carried them out to the thousandth of a cent: for instance the 702 cents cited above is 701.955 and the 386 cents cited above is 386.314. This has to be done when computing the accumulated values when a remote interval is calculated. For example how far is the B# 12 fifths up from C from coinciding with C? Then the thousandths would add up. But ordinarily we don't need that much precision. Profound difference, however: today we can attain that laboratory precision and in Ellis' day is was tantamount to science-fiction.

On page 463 of the Appendix we are currently discussing, appears the Duodenarium, a table of modulations which frankly appears formidable. Many musicians refuse to look at it or even admit it has a good purpose. Other people in the field have misunderstood it, and have ignored the disclaimer and observation which follows on page 464: "Of course it is quite out of the question that any attempt should be made to deal with such numbers of tones differing often by only 2 cents from each other. No ear could appreciate the multitude of distinctions. No instrument, even if once correctly tuned, would keep its intonation sufficiently well to preserve such niceties."

In over 50 years since I was introduced to this book. I have been unable to find any references in other books which acknowledge that statement. Maybe there are; maybe someone has been kind, but I can't locate them. Ellis has stood accused of things that no reasonable person should have taken seriously. If I can set one person right it will have been worth the trouble.

There are 117 pitches in the Duodenarium. No point reproducing it here when the real thing is in so many libraries. But there is good reason to analyze it a bit and to quote parts of it and explain how it was constructed and what it is for. This is especially important now when some followers of Partch's system don't go back to Ellis to find out where some of Partch's ideas came from, and how Partch expanded them for certain purposes of his own. Don't throw away the ladder which got you up there! The Duondenarium is important when considering how to use just intonation for rendering existing music and also Ellis' schemes for getting rid of enough of those 117 pitches to bring things down to earth and make a better intonation practical to use.

To those already knowing Partch: the Duodenarium is a 5-limit chart. On another part of page 464, Ellis details what would happen if one goes to what Partch called a 7-limit--more than doubling those 117 pitches. The two-dimensional chart goes to three dimensions. Theoretically, if one considered Partch's 11-limit, it would have to be plotted in 4 dimensions! Partch did not choose to modulate as far from his starting-note as conventional music often does (if, that is, it were to be played in just intonation) so got along with 43 pitches per octave in most cases (there were certain exceptions). But that meant he could not play the existing musical literature in just (non-tempered) intonation. I would like to preserve continuity with existing music, so demand that just intonation include all reasonable modulations.

"5-limit" means that no prime factor larger than 5 is present in the ratio numbers--a 5-limit array of tones moves by only fifths and thirds. A "7-limit" system moves also by subminor or harmonic sevenths. An "11-limit" system moves also by the semiaugmented fourth involving the eleventh harmonic. And further if one is venturesome enough. 13-limit has been tried. There is a 3-limit system, called Pythagorean intonation usually--all notes are obtained by taking an endless series of perfect fifths, ratio 3:2. 12-tone equal temperament is a fairly good imitation of 3-limit, but very poor at 5-limit and the 7 and 11 ratios are beyond its powers. Ellis wrote about this in his own language and limited by the technology of his time--but it is amazing how much he was able to conceive without ever hearing a synthesizer or any computer music accessible to us today. For this great vision he has been excoriated and denounced lo to this day. A few kind words are in order!

Conventional harmony books ignore a very important distinction which occurs in just intonation. This is the Ptolemaic or Didymus or syntonic comma (the name varies with the textbook author or music theorist). So conventionally-trained musicians are often denied even the information that the comma exists. Now in order to understand Ellis, you have to become aware of this comma and how it arises, and the fact that most temperaments are means to get rid of commas.

There is another comma called the Pythagorean comma and this is the comma explained to piano-tuners. It is larger. Hence much confusion, and as usual Ellis gets raked over the coals.

The piano or organ tuner is taught a routine for tuning by ear. Basically, the ideal of 12-tone equal temperament is approached by tuning a series of fifths and fourths arranged to stay within one octave and distorting them by approximately 1/600 octave (2 cents) so that instead of the B# 12 fifths removed from C being 23.5 cents will coincide with C and you will pretend that you have merely names for the same pitch--actually, if you went he other way from C you would get D-double-flat and distort every fifth on the way to it so that that, too, would coincide and be just another name for your starting-note. (3/2)¹² = 531441/4096. 12 fifths exceed 7 octaves. At this point, the average musician will have an attack of the Callithumpian Flibbertygibbets and get real mad and refuse to face the music-theory and inflict terrible curses upon all mathematicians and scientists and stomp out of the room in a Blue Huff.

It doesn't take much imagination for you or me to picture the scene when Ellis or any other theorist or acoustician tries to explain this twelfth power of a fraction of some piano-tuner or music-teacher or Conservatory Professor. I know all about this first-hand. How to Lose Friends and ALienate People. One is treated either like a sinner committing a sacrilege in some new fanatical cult, or like a boor committing a Breach of Etiquette in the 1920s when Emily Post ruled supreme. Ellis merely wanted to help. He gave a routine for tuning ordinary 12 and a little table showing how close the use of a by-rote routine came to the actual exactly-equally-tempered pitches.

It took a century to get us to the point where silicon chips can put tuners out of business--where the 12-tone temperament can be built into an organ or other keyboard instrument at the factory. Actually it is an approximation, but so was the tuner's routine, so let's don't have the Pot Calling the Kettle Black! No fair.

It is a mere coincidence that the difference between the Pythagorean comma and the other comma whose ratio is the much simpler 81:80, should be about 2 cents and within a gnat's eyelash of the error of the 12-tone tempered fifth. 1.955 cents vs. 1.9537. Real hairsplitting. Now we get into another needless hassle: Ellis was a linguist and phonetician and knew about the Classics. So he called this tiny difference between the two kinds of commas, a skhisma from the Greek word meaning division and spelt it this way to get the correct pronunciation by the average reader. Bu the ordinary book-publisher or magazine office won't let Ellis' common sense prevail: they want to spell it schisma as though Ellis didn't know what he was doing. That of course calls up the religious term schism pronounced "sizzem" and injects wrong images and irrelevant notions into the discussion. Why won't anybody leave well enough alone? Ellis was only trying to help.

The syntonic comma 81:80 is of more musical importance than the Pythagorean comma. WHo is going to modulate to the key 12 fifths away? Who, listening to the music thus generated, could follow the modulation and remember the original pitch? But to get to the key a syntonic comma away--that's another matter. From C major we can go to D₁ minor through F major. From C major we can go to G major or minor and then to D minor. These two D minors in just intonation and in certain temperaments are different. Closer to home: the chords built on the second degree of the major scale in just intonation are several, not just one. Taking the usual tuning written about in textbooks that mention just intonation at all, the triad on the second degree of the scale is badly out of tune!

Or to put it another way, D to A in the just intonation scale of C major is a false very flat fifth that sounds like a big mistake. Now there has to be some way of writing about this unambiguously--we can't have D meaning two different pitches and confusing us before we start our little family quarrel. Horitz Hauptmann who was a musician in Leipzig in the early part of the 19th century used various means to distinguish the comma-apart notes--lines above and below the letters, capital vs. small D d were used by Ellis in an earlier edition of the Helmholtz book. For the second English edition whose centennial we should be celebrating. Ellis settles on super- and subscripts, this way: D without a number meant the D obtained by tuning fifths from C, or 9/8 to C in Partch's system. The D₁ obtained by going to F bears a subscript thus: D₁. (Unfortunately, some books on music and acoustics use subscript and superscript numbers to denote what octave a note is in, and they do not agree on it at all, so that the confusion is compounded and someone learning from the textbooks in question can't understand Ellis at all.) Partch got impatient with that and scrapped all letter-names, which really louses things up. So now fruitless arguments rage among persons interested in just intonation and the conventional musicians poohpooh the whole thing and heap scorn on it. Just because I use Ellis' system here doesn't mean you have to if it hurts you too much, but I might ask you to bear with me for the duration of this article.

Another way to get commas (81:80) is to tune fifths and tune major thirds. So Ellis, after Oettingen and Hauptmann and some others, constructed a Web of Fifths and Thirds, or two-dimensional lattice if you will. By studying a table of modulations proposed by Riemann, he got the idea for the Duodenarium. Now, the three systems for which the traditional name-system works, or the solfa syllables work--the Pythagorean tuning, the 12-tone temperament, the meantone temperament--all ignore the syntonic comma 81:80, getting rid of it. This makes it next to impossible to explain the comma to people indoctrinated in any of the three systems abovementioned, or those using the 19- or 31-tone temperaments, for that matter. Ellis and many others are so highly technical about it, that no wonder so many misunderstood.

If you are never allowed to hear just intonation or even good imitations (53-tone or certain unequal tunings or some new possibilities) then any kind of comma seems unimportant. But hear it and all is quite different. Ellis persuaded a harmonium maker to make available the Harmonical, a special reed-organ tuned to order. How many of these ever were sold is questionable. Today, it is a matter of programming a computer or building or modifying a polyphonic synthesizer to do far more than the Harmonical could.

Now for the other side of the argument: for lively music, allegro or faster, the comma distinction has no time to be heard. So in the rendering of many pieces of music the comma would be a mere nuisance, very much in the way. As a cellist of 54 years' standing I should know. In certain qualities of tone the comma would not be clearly defined. It is mechanically impossible to fit pianos with enough keys to render commas and tuners would not tune them and pianos don't stay in tune well enough to bother with them. To say nothing of the fantastic cost. So run-of-the-mill musicians don't want to read about it even.

Therefore Ellis gives tables and data about meantone temperament and mentions it in passing. He also details several ways to get rid of the tiny skhisma while retaining the comma.

Let us set forth the way Ellis wrote the C major scale:

C D E₁ F G A₁ B₁ C

That is to say, the E, A, and B are a comma lower than the E, A and B in the Pythagorean version of the C major scale as might be played on an unaccompanied violin. To tune a Pythagorean C major you would simply tune perfect beatless fifths in the chain F C G D A E B.

But to tune a just C major you would only have F C G D in the primary chain of fifths, and would take E₁ as a just major third from C and A₁ and B₁ by fifths from that. That is the way you might deduce from some books which mention just intonation at all, but it is not the end of the story, nor it is enough. To get a consonant chord on the second degree of the scale you need another D, which Ellis writes D1.

To get a harmonic seventh chord you need septimal intervals and to get the chord of the ninth on G which resolves into the C major triad you need an unmarked A. The higher kind of A is in the Duodenarium but the septimal lowered B-flat and lowered F are not. Let's make a map:

A (imagine these septimal notes to

be above the plane of the paper,

D overlaying C and G)

G B₁ ⁷F

C E₁ ⁷Bb

F A₁

Bb D₁

That's just for C major and trivial modulation to F major or getting into a transient modal chord. No wonder Ellis didn't publish the three-dimensional and four-dimensional versions of the chart! C minor adds notes a comma up instead of down, viz., A¹B E¹b B¹b, while A₁ minor needs notes two commas down such as F₂# C₂# G₂# and a ⁷D₁ to boot. No wonder Ellis got brickbats.

Fortunately, when these arrays of tones are carried out a little further, near-identities appear. For instance, the Fb in a series of 8 fifths down from C, is only 2 cents flatter than the E₁ or just major third, and there just is no possible realistic musical performance where that would matter. Carry the array far enough and the 7th-harmonic and 11th-harmonic-derived tones get near neighbors also. Ellis tabulated the 53-tone equal temperament and some unequal temperaments based on such ideas, including Helmholtz's own 24-notes-per-octave scheme. The purists won't like it, but what the heck.

Do we want to hear new kinds of musical performance, or do we merely want to play around with silent numbers of paper? The choice yours. Today, there is another way out, and it has been used: build into an instrument, or program into a computer, certain counting schemes and you get within a cent or even a fraction of a cent of any pitch you could possibly use. Theoretically and strictly, this is temperament of a kind Ellis didn't mention. Practically, let's pretend. No-one will ever know. This is way beyond any errors in tuning, and the old-fashioned tuner-by-ear has no right to complain. Indeed, such counting methods put organs and synthesizers in 12-equal at the factory.

So most of these complexities resulting from the use of just intonation can now be alleviated or even evaded with our new technology. Waage patented an automatic method for getting just major thirds automatically from the keyboard of a conventional organ, and supposedly there are computer programs for similar purposes of automating most of the un-tempering process for conventional music at least. This will not help composers of really new music, but it is certainly a welcome step regardless. It is similar to the choices now available to photographers: you can study and use an adjustable camera requiring skill to set it each time; or you can trust the automatic focusser and automatic f-stop mechanism and automatic shutter and/or flash device.

Ellis thus was not a specialist in just intonation to the exclusion of everything else. He set out the principles of how to temper and the mathematics of several temperaments, and also listed the non-harmonic scales of certain foreign cultures. Much of the world even today does not use harmony, so the question of basing melodies on just ratios (small integers) does not even arise. He wasn't crusading to eliminate all the "unjust" scales, nor to convert everybody on the remote islands to Western European systems. But eh gets accused of this,a nd too many newcomers to the field of non-12 acquire an unseemly missionary zeal.

His APpendix contains a valuable Table of Intervals Not Exceeding One Octave. In his footnotes to Helmholtz's text through the book, he takes up such matters as plotting out the upper partials of the tones comprising just intervals, showing which harmonics of the two tones coincide and which beat because they are dissonantly close together.

In other place the favorable and unfavorable positions of common major and minor chords are discussed. That is to say, on an ordinary 12-tone-tempered organ there isn't that much difference between different inversions and close vs. open layouts of a given chord, but in just intonation it makes a big difference and adds a subtle resource to harmony. This fact is usually ignored.

Ellis gives tables of historical pitches, and in another book he wrote extensively on this subject. Pitches have tended to rise to give more brilliance, till they become too high for singers to handle, when the pitch sometimes is taken down, only to rise again some years later. At present, the standard A = 440 Hz has begun to yield to A = 442 or even 445. Performers of earlier music often try to take earlier pitches such as A = 420. His tables show that A has been everything from about F# up to D!

He may have been ahead of his time, but now we can do his memory justice and give this pioneer his due.