IVOR DARREG, Composer and Electronic Music Consultant
CONTENTS:
This is as good a time and place as any to give a resume of the contents of the seven previous XENHARMONIC BULLETINS:
Back Issues of Xenharmonic Bulletin:
No. 1. March 1966
Change in attitude at that time, hence reasons for starting the Bulletin. Electrctronic instruments and computer music--new possibilities thereby.
No. 2. May 1974
Further progress summarized. Second-order or tempered temperaments; widespread use of systems which deviate from theoretical equal tempering.
Tables: Intervals of importance in theory; the Harmonic Series.
No. 3. November 1974
General remarks & updating. Review of Harry Partch's Genesis of A Music, 2nd Edition, enlarged. Appraisal of the 24-tone equal temperament, or quartertone system.
What the piano did for and against quartertones; quartertone notations; good and bad instruments for demonstrating or composing in the quartertone system.
Guitar re-fretting table for the 19-tone system in the most popular size, 650 mm active string-length.
No. 4. February 1975
Darreg's wall-charts of comparisons of temperaments and just intonation, described. Harmonics, overtones, and such.
Metric measurements for musical instruments.
"Illegal" sharps and flats: microtonal deviations and tone-bending in actual performance but not written down.
Review: music teaching text--George Whitman's INTRODUCITON OT MICROTONAL MUSIC (along lines of Haba system).
Review: UCLA concert of HarryPartch's THE BEWITCHED.
22-tone organ piece: ON THE ENHARMONIC TETRACHORD. 19-tone guitar piece: PRELUDE IN E MINOR.
No. 5. May 1975
New moods. Article on the emotional effects and uses of different non-12 systems, an important reason for having several non-12 tunings.
Descriptions of Darreg's new instruments. Steel-guitar type, in just intoantion throughout.
No. 6. September 1975
The notation question: some of the many many ways of writing down non-twelvular compositions.
17-tone composition: pentatonic study. 17- or 34-tone composition: LULLABY FOR A BABY COMPUTER.
No. 7. April 1976
Invitation for readers to comment. FRETTING AND FRET-CHARTS: how to fret or re-fret guitars, etc., to new scales; how to make charts for steel guitars and similar fretless instruments; FRETTING TABLES for the popular 650 mm active string-length, approximately 25-5/8", for the following systems: 17, 19, 22, 24, 31, and 34. Of course, 24 includes 12.
Description of the Hobnailed Newel Post, a new justly-intoned instrument with 64 strings, 16 on each side.
because of its length, 16 pages, the music score for four guitar parts in the twenty-two-tone system, COLORLESS GREEN IDEAS SLEEP FURIOUSLY, will be sold separately and done only to order.
Nos. 9 & 10 & 11 now available.
Xenharmonic Bulletins 1 and 2 were bound into Xenharmonikon 1; Xenharmonic Bulletin 3 was bound into Xenharmonikon 2; Bulletin 4 was bound into Xenharmonikon 3; Bulletins 5 & 6 were bound into Xenharmonikon 4, while Bulletin 7 was bound into Xenharmonikon 5.
During the last 30 years, Ivor Darreg has produced a long list of publications, a few of which bear upon xenharmonics and musical theory. The revision and updating of someof this is planned as time and circumstances will allow.
In addition, there are xenharmonic scores, but most of the music will be left in audio form as tape recordings which can be duplicated and copied, again as time and circumstances permit.
Phone number? No, not quite right for that. Government debt? I don't blame you for asking, but not that either. Well, what, then? Factorial 12.
479,001,600 is the number of different orders in which 12 things can be arranged, provided of course that there are no duplications. In 19th-century England, this number was the theoretical number of changes that could be rung on twelve bells, although the rules of change-ringing might cut that down somewhat. It would take almost 38 years to ring that many changes, no time out for coffee-breaks, eating, or sleeping.
479,001,600 is also the number of Schoenbergian "tone-rows" possible, and so the 12-tone serialists bring up this number once in a while to boast how enormous the resources of 12-tone serialism are.
So long as it remains on the "aboutness" level, and so long as attention is riveted on the balck notes on white paper, this seems reasonable.
However, these millions of tone-rows do not sound as different as they look. There are several reasons for this, one being that so many of the resources of ordinary music had to be renounced.
On top of that, anything that will imply traditional melody and harmony, to which the public has been conditioned, has to be avoided. The listener, more than the performer or composer, has to overcome his years of conditioning, and this creates a subconscious strain. As pointed out in Xenharmonic Bulletin No. 5, the mood of the 12-tone temperament is restless. Atonality and serialism increase this restlessness by avoiding the tonic and certain other concepts.
This restricts their powers of expression, and makes the abstract mathematical argument about factorial-12 much less cogent.
This should serve as a warning to us that more abstract calculations or tabulations of the possibilties of this or that tuning-system may be misleading.
Because 24-tone has more sounds per octave does NOT necessarily mean that it has more musical resources than 19 or 22.
How these systems and methods actually SOUND is what counds, and this cannot be calculated in advance, anymore than you could evalulate paintings witout seeing them, by means of calculations based on the cost of paint, canvas, brushes, etc.
When someone writes an article or book either for or against a tuning-system, they consult everybody except those composed who have tried it out!
Over and over again, with the most tiresome regularity, the scholarly articles and books bring up the difficulty of writing this or that non-12 system, or how troublesome the extra tones would be for the KEYBOARD performer, or the contention, which has considerable truth to it, that Bach, Haydn, Mozart, et al. got along very well with what they had at their fingertips, which usually included keyboard instruments with only 12 tones per octave.
That two centuries have passed, and that the obvious resources for composers have been exploited, and that the danger of a new composition merely rehashing the cliches of yesteryear has continually increased, is carefully ignored.
The composer isn't supposed to have any pride anymore, but instead to be content to being reduced to the status of an arranger, or pasticcio expert, or a (subconscious) quotationist. If that is to be the future state of affairs, you might as well replace him with a computer right now, and get the miserable business over with.
If the rules deduced from 18th or 19th century compositional practice are to be slavishly followed forevermore, just for the sake of performers and instrument-manufacturuers, then the computer can obey them far more perfectly than any mere human being could ever hope to: this has been proved already.
The rules of 12-tone serialism are to some extent a complementary reflection of the earlier rules just mentioned. In fact, they depend on them, ina strange sort of way, and they are all the more strict, since they are arbitrarily constructed by people conscious of logical principles and patterning, rather than being passively deduced from a certain period's composers' actual practice.
Happily, there is a "free atonality" which doesn't accept all the new strict rules, and isn't tied down to the set of 479,001,600 tone-rows either. We can at least tell these persons about xenharmonics and the enormous variety of systems suitable for atonality, but so far virgin territory.
Whether it was practical or not in 1705 to build a harpsichord with more than 12 tones per octave and keep it well tuned, has no bearing on what should be composed today.
Whether a piano tuner can or cannot, will or will not, tune a keyboard instrument ot the 13, 19, 18, 21, 23, 22 or 31 system, no longer matters in this Electronic Age, even though it mattered as recently as 1945 or 1963.
Whether today's performers can or cannot, will or will not, perform non-12-tone music must not stop a composer form using non-12 systems: it is no longer necessary to interfere with the Musical Establishment's prejudices in order to hear a new composition. They can be left alone to continue their old conditioned, programmed ways.
That is to say, the XENHARMONIC BULLETIN is not a crusade to abolish the 12-tone system. How could it be, when I have been involved in it for 53 years and have a long list of piano and organ compositions? I do not have to burn scores or wreck instruments as some people actually did, because I am adding to the status quo, not uprooting everybody for the sake of "revolution."
With xenharmonics, there is something with which to CONTRAST 12-tone. So long as you remain inside 12-tone tuning, you cannot be aware of its personality or mood. Discovering this contrast is in itself a novel, unsuspected experience.
Nobody could predict it by silent, visual mathematical calculations.
Serialism is valuable for the peculiar way it sounds, and of the range of emotional and other effects it can produce. It is not limited to the 12-tone temperament, and it should be explored in non-dodecaphonic systems. That is to say, don't take the preceding paragraphs as a putdown of atonal serialism, but as an invitation to join the xenharmonic movement and hear some non-cliches.
If you happen to be an atonal or serial composer, why not try "mapping" something you already have written, onto a scale other than 12-tone equal? It might be a pleasant surprise.
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Some people are going to feel that I am presuming too much here: if so many otherwise talented and capable persons can't master the 12-tone tuning routine for orthodox pianos and organs, how dare I expect anybody to master a routine for microtones?
Frankly I don't. When it takes 11/2 to 2 hours for an ordinary tuner to tune a piano (more if the customer is watching television), and pipe-organ tuning can run into a matter of days, you cannot be blamed for getting frightened at the difficulty of tuning some instrument to the 41- or 53-tone system.
XENHARMONIC BULLETIN NO. 7 has already given one way of escaping this problem: use fretted instruments, adn the mathematical calculations which place the frest ensure correct tuning once and for all at the time the instrument is built or modified.
Well enough as far as it goes; but fretting an instrument to more than 31 tones to the octave runs into all kinds of difficulties, and playing the instrument after you get it made becomes increasingly difficult...and this latter problem escalates rapidly!
The previous article dealt with fretting to the various equal temperaments, not to Pythagorean or just intonation or something in the meantone family. A justly-intoned fretted instrument will require at least some frets spaced a comma (81:80) apart, and possibly other small intervals--the fret-spacing for a comma is equivalent to that of a 55-tone equal temperament, so if you happen to be a guitarist, I am fairly confident this will make you shudder.
Inevitable errors in placing the frets, and the errors entailed by pressing strings down against such frets, also limit the practicable fineness of dividing the octave by this means. So, while the fretting or re-fretting approach is ideal the beginner in xenharmonics, and for practical performance ina number of systems affording various moods, it will not take care of everything.
Systems with smaller numbers of tones which do not have good approximations to the conventional harmonic intervals (13-tone, for instance) will be well served by fretting, whether on fretted instruments themselves, or using such instruments as secondary standard to tune others to.
At first blush, it might appear that the new electronic tuning devices had solved all the problems. I wish they did! I wouldn't have to write this article. For one thing, most of them are quite expensive. Prices may come down eventually, but we're talking about now.
In many cases, such devices will have to be rented or borrowed, or the instrument taken to where they are. What do we do in between times or when a xenharmonic instrument is used somewhere that power lines are not available?
Some of these devices are biassed in favor of the twelve-tone equal temperament; others may be more or less tied to a certain starting-point, usually A at 440 Hz.
Two complementary electronic devices are becoming more generally available and less expensive in recent years: the digital readout frequency counter and the frequency synthesizer.
The frequency counter was developed to measure radio frequencies, at first in the broadcast range of around a million counts per second, and then extended into the billions, as manufacturing techniques got better and better. One part per million accuracy is commonplace.
Frequency counters work well with sustained musical tones, such as those from an organ, but would be worthless to determine the pitch of a high note from a xylophone or banjo.
A frequency synthesizer can be set by dials, pushbuttons, adding-machine-style keys, or similar means to a derived frequency, and then it will output this frequency as a low-level signal, which can then be converted to visible or audible form for comparison with whatever is to be tuned or checked.
Its accuracy is to some degree spoiled by the conditions of the comparison: tuning by ear or with some visual aid. However, this error of comparison may not be too serious in the practical musical situation.
SInce musicians do not think in term of '220 Hz' but rather of 'A below middle C' or 'second ledger line below the treble staff,' frequency tables are necessary in order to use either frequency counters or frequency synthesizers.
This is true also of some of the other tuning devices and electronic aids for tuners.
Some recent electronic organs have a fixed 12-tone runing device built right into their tone-generator circuitry, so they never have to be tuned; everything is 'locked' in. Retuning to non-12 systems is thus out of the question. So be forewarned if you plan on getting organs for xenharmonic purposes!
This is probably the best place for me to repeat my pet protest: one tuning-system is not enough, now that 12 has been worn threadbare. If you did not agree with this, you probably wouldn't be reading the present article.
Very well: two tuning sytems, i.e., 12 and 31, or 12 and 19, are not enough either. In order to encompass the full range of moods, a number of different systems must be available. You don't need every system; there are too many--but my composing so far and my trials with over 200 listeners indicate that at least ten systems will prove sufficiently different in effect to make it worthwhile to have such a variety.
I don't expect you to have time for practicing and performing in many systems, nor the funds for a whole arsenal of 'dedicated' (one-system each) instruments, but I can ask you to collect recordings in as many different systems as possible.
Each system has a characteristic mood, and the choice of moods will have some connection with your personality and that of the composer, along with what has to be expressed in music at the particular point in time.
I don't think I am being unreasonable if I ask that at least four (which four is strictly up to you) systems be available in the well-appointed xenharmonic studio.
Please note that two systems like 17 and 34, whether 34 contains 17, or 12 and 36, where 36 contains 3 circles of 12 fifths each, ought to count as 11/2 systems rather than 2. Aftrer all, few people would argue that the whole-tone scale (the 6-tone system contained within the ordinary 12-tone one) would count as a completeley independent second tuning-system.
The preprogramming of tuning-systems is already reality: the Scalatron, for instance, can store the necessary information required to retune it to any of several systems at the touch of a button. No doubt this principle will be extensively applied in the future.
Again, I have to say that we are talking about Now. How does one tune an isntrument to the new scales in the home, or a small studio? What is feasible or practical when you are just beginning to explore this terra incognita?
We read constantly of feats of fantastic, incredible precision. The man-in-the-street is appalled by abstruse scientific statements about extremely accurate measurements, and often wants to discount them as though they were exaggerated advertising claims.
You really can't blame him, there has been so much misuse of extra decimal places where they do not belong. So, with healthy skepticism, let us ask: how much precision does xenharmonics really call for?
I have to weaselize here: it all depends.
What system do you have in mind? What instrument? Above all, how fast is the music being played? What tone-quality? Is the scale, and/or the music, harmonic or inharmonic?
Where does the main interest lie, in the pitch-range? Middle register? High treble, like the glockenspiel? Low-bass like organ pedalboard--or tenor/alto like the viola?
The way it works out is that a 31-tone organ is very much worth while, but a 31-tone banjo would be a ridiculous waste of time; a 17-tone banjo would be dynamic and brilliant, but a 17-tone harmonium would be unbearably harsh.
We have had several generations of people influenced bythe piano, and an enormous piano literature holding things in place till quite recently. This means inertia (or momentum, if you happen to be in favor of the status quo) and subconscious biases.
On the one hand, the convenience of tuning or having tuned, two pianos a quartertone papart, encourages many people to experiment witht he 24-tone equal temperament. On the other hand, the piano's timbre is such that all the new intervals of the quartertone system sound 'funny' or queer or odd, and thus most of the above-mentioned experimenters abandon their quest.
If the guitar had been as popular during the periods of quartertone-piano experimentation (such as the mid-1920's) as it is today, and if electric guitars as loud as pianos had been available then, the quartertone story would have come out quite differently!
For the purposes of this article, that means that I can be selective, and not spend too much time on unsuitable combinations such as a 22-tone piano and how to tune same, or how to fret your mandolin, ukulele, or banjo to 31-tone.
I don't have to take up the subject of how pipe-organs are tuned, ebcause few people could afford a xenharmonic pipe-organ nowadays.
I definitely do not want ot encourage anybody to tune an incomplete set of a system's tone, such as a piano, to only twelve pitches of the meantone temperament, 12 out of 31, 12 out of 19, nor even 19 out of 31.
There has been far too much of that sort of thing. What it amounts to, is that those who do not compose any new pieces are forbidding composers to explore such tuning-systems. Usually the motives for say, 12 tones only of the meantone system, were legitimate enough at the time and within the context in which they were acted upon, but this does not help the cause of xenharmonic progress. The implication of a 12-out-of-meantone tuning is that the 'wolf' is an inevitable part of the meantone system, when it is nothing of the kind. Instead, it is a compulsory wrong note caused by the potent combination of mechanical difficulties with expense.
The wolf (wrong interval resulting from the circle of 12 meantone fifths failing to close) has even been extolled and praised by antiquarians wanting early keyboard music to be authentically performed. On the surface, this seems innocent enough, but as a composer, I can't imagine anybody wanting a diminished sixth such as G-sharp/E-flat or an augmented third such as A-flat/C-sharp while writing the normal fifths and fourths such as A-flat/E-flat or G-sharp/C-sharp in the score.
When instrument-builders, dealers, performers, historians, and theorists gang up on today's composers and tell them they must not write or play or hear G-sharp and A-flat sounded simultaneously on a meantone, 31-, or 19-tone instrument, that is going too far! Conversely, there isn't much use in a modern composer trying out a retuned piano on which there will are only 12 different pitch-classes.
You are just wasting your time and moreover frustrating yourself in you try to explore a more-than-twelve-tone system on an instrument which only has 12 tones. You can't imagine the difference between G-sharp and A-flat in certain systems unless you can hear it and play it, and then ake advantage of it by actually using it in new compositions. I don't care three broken jellybeans whether certain 17th or 18th-century musicians had their keyboard instruments tuned in this or that version of meantone or other temperament, because this did not necessarily determine how violins or viols were played, and no matter what they did then, it must not influence how 1976 composers write new music on new instruments.
Getting back tothe topic of precision, Bach and his contemporaries didn't have accurate frequency-standards nor accurate frequency-measuring equipment, so even though certain temperaments could be accurately calculated by the theorists, that does not mean that clavichords, harpsichords, etc. would present these tunings in the stable precisionform taht today's electronic organs and comptuer performances can.
I can't help thinking of the Hollywood movie studio that had to scrap an expensive sequence because a couple of the Roman soldiers in the battle-scene had their wristwatches on. This is the kind of anachronism some music theorists indulge in when they extrapolate 1976 precision back to Renaissance or even Ancient Greek isntruments.
Another similar matter concerns the materials and tools availabel before the 19th-century industrial revolution and the 20th-century perfecting of mass production and interchangeable parts and refinement of measuring-standards.
Today's steel music wire, used for all kinds of springs and parts of non-musical mechanisms and very similar to drill-rod in character, is quite a different sort of thing from 17th- or 18th century harpsichord wire, and extremely unlike the handmade gut or other strings used by the Ancient Greeks!
Today's nylon guitar and harp strings are fresh from the modern chemical laboratories.
Who knows what materials the reserachers will perfect next year or next decade?
Ditto for materials going into other kinds of musical instruments. For instance, the pipe-organs of Bach's time were not blown with turbofans driven by electric motors.
The temptation to extrapolate our precision back to earlier instruments is hard to resist.
Because an electronic organ or a computer performance can now have precision beyond any human being's ability to hear, there is even greater danger that one will demand impossible accuracies from other kinds of instruments that will be used to play the new scales.
It's your valuable and irreplaceable time, money, and energy that I'm talking about here. I am assuming that readers of this article are considering and mulling over which scale-systems and which instruments they are going to make or use, and naturally one is afraid to have expensive 'white elephants' around.
Some of you are already familiar with the standard routines for setting the 12-tone equal temperament on pianos and organs. Others of you will have read about it, or watched and listened to a tuner at work.
Nearly all public libraries have books on piano-tuning and sometimes also on organ-tuning. Some of this information is relevant to us who are going in for xenharmonics; other of it need not bother us...such as the peculiarities of wound and double-wound bass strings on pianos, or the special skills involved in tuning the several kinds of organ-pipes and types of reeds.
You may be one of those who say: "I don't need to learn how to tune by ear; I bought an Elektronik Ghizzmo Super DeLux." Mark VI, no less. Or, "I'm a guitarist. Why not let the frets do it for me?" Or, "let the computer do it!"
It is possible to get along without knowing any tuning-routines at all, but there are some real advantages to studying them. You may want to check the tuning of an isntrument: has it drifted? The octaves stretched or shrunk? Was it put in an equal or unequal temperament? If just, is it just throughout?
Even though you have a tuning-device, there may be some unexpected chance to check an instrument--and no!--you don't have it along.
Then there's the financial angle: even in these days of unemployment and all, tuning is in demand, and knowing the 12-tone routine can be well worth the trouble.
No point trying to take the place of the standard books on the subject here; but some discussion of the orthodox twelve-tone routines is quite relevant.
Beginners in tuning often ask: "Why not just get a set of tuning-forks for all the 12 semitones, so that there will be no need to bother with a tuning-routine at all?" Such forks are available; so are sets of reeds and other fixed tuned devices.
If you depend on gadgetry to this extent, the gadget will own you, and you will be its slave. This can be quite embarassing on occasion, for a mere machine, or the lack of same, to humiliate you in public. So it's a matter of pride.
An obvious disadvanatage of tuning by ear with a routine is that it is imprecise; every step is prone to error, and it takes several months to practice and learn a routine.
The compelling advantage is that one memorized, it becomes second nature, and it is valid for a number of different instruments and regardless of the actual pitch, whether above or below standard. The routines are to some degree flexible; slight variations are sometimes made.
Pianos often are too far below pitch to be safely raised in one operation. Orchestras often demand a higher pitch than normal, such as A at 442 or 444 Hz. Many organs are built for A = 435 Hz and cannot be rasied, or there are chimes or 'harp' stops that cannot be altered. Musical historians rightfully demand that certain compositions be performed at original pitch.
The 12-tone routines are based upon beat tables showing the number of beats per second for certain intervals within the range used for setting the temperament. For pianos this is most often F below Middle C to C above Middle C; for organs this is usually Middle C to the C third space treble staff; a useful compromise is the A of the tuning-fork down to the A an octave below, or 440 to 220 Hz at usual standard pitch.
Beat-tables for several systems are found farther on in this Bulletin.
Let us consider the beat-rate of the fifth, D to A or 293.66 Hz and 440.0 Hz.
The same beat-rate belongs to the fourth below it, A - D or 220 and 293.66 Hz.
The table says 0.99 beat per second, which of course means just about once a second outside the laboratory.
This beat-rate is for the 12-tone-tempered intervals shown above in conventional notation, and be it noted that a common beginners' mistake is to get the right beat-rate but the wrong side of just, i.e., the fifth sharp instead of flat. It's only 2 cents, 1/600 of an octave, so this is understandable!
With the aid of watches, pendulums, and now various electronic apparatus, as well as metronomes, the would-be tuner can become familiar with one beat per second.
Theoretically, the beat-rate for other fourths and fifths, such as D-sharp - A-sharp or B - E
should be read off the table and memorized by practicing hearing it in time to some time-keeping device, but obviously this is too much for human memory, and such precision is useless in the real world out there anyhow.
So what actually happens? Why, one accepts a compromise: one uses a beat-rate in the middle of the tuning-octave for all intervals of that same class above or below it.
This obviously does not produce a perfect twelve-tone equal temperament. I call it a second-order temperament or a tempered temperament. But once you compromise by tempering at all, you've already 'sold out,' so why quibble? No one will ever know.
Thus the errors in guitar-fretting spoken of in previous articles have their counterpart in the errors inherent in the customary organ and piano tuning-routines. Furthermore,these have still another counterpart in the methods used to ap proximate tempered tuning in electronic instrument, or in digital-computer synthesis of music.
How was the table calculated? By finding the lowest pair of coincident harmonics the interval would have if it were just, and taking the amount by which the harmonics of the tempered tones fail to coincide--their difference.
In the above example, consider the Fourth A-D, beloved of piano tuners: if it is just, the 4th harmonic of the A coincides at 880 Hz with the 3rd harmonic of the D, and there are no beats. In the twelve-tone case, the D is about 1/3 Hz sharper than just,and so its 3rd harmonic is three times its frequency and therefore about 1 Hz sharper than the 880 Hz 4th harmonic of the A at 220 Hz. THe 880 and 881 are what beat.
How were the tempered pitches, upon which the beat-table was based, calculated? Through the use of logarithms, which are a kind of tempered arithmetic. So you mathematicians out there, don't you dare look down your noses at the mere musician. The tempering principle is everywhere.
The engineers' approximations and the tuners' routines are cousins-not-so-far-removed.
Put a slide-rule near a fretted fingerboard and see what I mean.
While a watchmaker or a quality-control inspector in an IC factory might find the standard tuning-routines too crude for his taste, the people at the average repair shop or contracting firm would think of them as rather fussy.
If you decide to learn tuning-routines, don't try for more precision than they can deliver.
While Alexander J. Ellis did give a routine for the 53-tone temperament in his famous Appendix to Helmholtz' Sensations of Tone, that does not mean that anybody should try to learn and use such a routine in this day and age. back then, it was the only game in town; today, there are so many alternatives that no-one who would afford a 53-tone instrument would put up with the tedium, inaccuracy, and boredom of such a procedure. Its only value now would be to check a few notes suspected of having drifted.
The user of some kind of tuning-device might want to check the accuracy of transfer of each tuning-device note to the instrument, or for the sake of greater objectivity, have an assistant do so. In this case, a beat-table would help, so beat-tables for many-toned systems have their uses.
Beat-tables will be published in this and future issues of Xenharmonic Bulletin, but please don't expect everything at once!
Fortunately, it is not necessary and not even feasible to construct tuing-routines for all the usable tuning-systems--we would never get done if it were practicable, and nobody would master very many routines anyhow.
We might point out that the beat table for meantone temperament will differ slightly from that for the 31-tone equal tempearment, and in theory a meantone beat-table would never be completed, since it contains an infinite number of notes and intervals.
Inevitable errors in tuning and the very nature of tuning-routines would, however, make any of the existing meantone routines do just as well for 31-tone--to close the circle of 31 fifths correctly would require some kind of help besides the routine itself. Meantone fifths and fourths beat fast enough that one beat-rate will not do for the entire octave as it will for the twelve-tone routines.
N.B.: This remark applies also to the 19 and 22 systems!
There are very many ways in which to vary the process of tuning 31-tone or meantone, and still have a tolerable result. Much ingenuity has been squandered on trying to cover up or mollify the 'wolf' produced by the 12-out-of-meantone scheme, and there are a host of unequal temperaments, with wolves in or out of sheep's clothing.
Surely you would save time and headache and annoyance by taking the pitches from a 31-tone electric guitar and then adjusting any mistakes in transferring the tuning by means of a beat-table. The only way to get greater accuracy is the electronic tuning-device.
That is to say, balance the value of your own time learning and practicing the 31-tone tuning routine against the cost of having a guitar refretted to the 31-tone scale with my table, and against the cost of an electronic tuning device.
This may help you to understand why the 31-tone innovators have had such a hard time of it for so long, despite Huygens' stature, and yet the meantone traditionalists are doing well enough since the harpsichord revival.
12 tones of meantone represent a slightly easier tuning routine than 12-tone equal, whereas the remoter reaches of the 31-tone cycle, when attained by ear in the old-fashioned way, have almost certainly picked up and accumulated excessive errors.
It is true that some of these remote tones can be checked by chains of major thirds and/or subminor sevenths, but these also are harder to tune by ear. I must therefore put 31-tone into the class of systems very difficult to tune by ear-routine methods. This will also explain why piano manufacturers will hate you to pieces if you start talking about the 31 system. No tuner would have had the patience even if such pianos had been made. It hurts me to think about such a tuner's problems. The thirty-one-tonists should be most grateful to the electronic organ, which unblocks these difficulties.
All right; where will by-ear routines work?
Most imortant point: routines-by-ear are limited to only certain qualities of tones! Certain harmonics must be present, and that beyond a minimum percentage.
Inharmonic partials must be held to a very small percentage. The tone must be sufficiently sustained. The timbre requirements differ for the different systems. This is too often ignored!
Twelve-tone equal temperament owes its success to the late-19th-century ideal of piano tone, and to such isntruments as nylon-string guitars and soft flute-tone organ stops--and since the 1930s a variety of electronic instruments with deliberately-dull tones. Marimbas, xylophones, vibraphones, etc., have had some part in making the 12-tone thirds and sixths endurable. The electric-clock-motor inspired tone-wheel, driven through special gearing, used in the Hammond organ till quite recently, had a group of special features designed tomake 12-tone temperament bearable: the imitation-harmonics were tempered, and therefore could not include the 7th; the use of tempered imitation-harmonics almost eliminated beats, so these tones could not be evaluated by a piano-tuner or organ-tuner in the regular manner; they are abnormal.
Such instruments (there was a transistor organ on this additive-tempered-synthesis principle) do not have the normal mood of the 12-tone temperament, so this has been a factor in conditioning the public for the last 35 years or such a matter.
'Background' music systems in stores,offices, factories, etc., attain their blandness (cf. Muzak's advertising of 'heard but not listened to') by tone-controls reducing high-frequency components of the sounds and hence reducing beats and dissonances of the originally-normal tones employed.
This kind of dulled tone is not as serious a barrier to tuning-by-ear-routine, but it can make tunign of some elctronic organs difficult. It could make piano-tuning difficult when the instrument has tired listless aged strings.
Xylophones could never be tuned by a routine because the harmonics to beat are almost absent, and even if they were present the tone does not sustain enough. If someone built a keyboard banjo, the tones would have the right harmonics, but routine-tuning would be out of the question, since no tone would last long enough to count the beats.
Metallophones and bell-like affairs such as the celesta or glockenspiel might sustain enough, but do not have hte harmonics for using a tuning-routine--but there is a trick: if each of the two notes involved in one step fo the routine could be separately electronically picked up and routed through separate distorting amplifiers on the 'fuzzbox' order, the necessary harmonics would be present.
Similar reasoning applies to the sinewave oscillators which experimenters sometimes use.
You could adapt a routine to visual tuning of such oscillators with an oscilloscope, but whether it would be worth the trouble is a moot point.
There is an important class of tones, such as clarinet and its imitations, square waves from tone generators, dividers, or synthesizers, and stopped organ-pipes, which lacks even harmonics.
On most synthesizers these odd-harmonics-only tones are readily available, being derived from a square wave in most cases. Some modern electronic organs have them, often as the basic stuff from which all their other tones are made; while in other commercial electronic organs this 'hollow' or 'clarinet-like' or 'stopped flute-like' timbre is lacking. Generally, this timbre is considered to be an essential ingredient in organ ensemble. In the symphony orchestra and the wind band, it is closely approximated by the clarinets.
The pianist never has it, since so far as I know, pianos have never been made with the hammers striking the strings in the middle!
The electrical engineer is familiar witht he odd-harmonics-only waveforms, because alternating-current machinery generates symmetrical waves, which could not be symmetrical if they contained even harmonics. Therefore hum picked up from power lines contains only odd harmonics until or unless it is distorted in some way, as by an amplifier.
At many music stores and toy-shops you can buy what is called a song whistle, which is essentially a stopped organ-pipe with a movable stopped controlled by some kind of handle. This is an inexpensive and pleasurable way to become familiar with odd-harmonics-only tones.
Such stops on a pipe organ cannot be used for setting the temperament; theyhave to be tuned from other stops by unisons. If you are a piano tuner and want to go into organs, pipe and/or electronic, remember this point!
Warning: there are electronic tuning-devices now onthe market which output a square or other odd-harmonics-only wave. They are fine for tuning unions by ear, but useless for tuning something else an octave above them, since there is no octave harmonic to beat with!
If you must use one of these handy doojiggers, you could pass the output through a distorter which could add the missing even harmonics.
The same remarks apply, even more so, to the electronic tuning-aids which output a sine wave, which is fundamental only with no uppper partials at all. This needs a distorter to tune anything other than unisons by ear.
The reason I have to harp on this topic here is that the visually-oriented electronic expert has a love-affair with the sine wave, which can indeed be used to tune octaves or other simple musical intervals via Lissajous' figures on an oscilloscope and in certain other special visual apparatus. So he thinks it ought to work gorgeously jim-dandy for audible tuning too. Sorry about that, Mister, it never ain't can't don't!
Two sine-wave tones are endurable when they form the interval of a major seventh, whereas normal musical tones sounding this interval are dissonant. (Assuming a normal middle pitch-range, of course.)
To recap: sine-wave tones, such as from tuning-forms or harmonic-free oscillators, can only be used to tune unisons by ear, although of course they would beat with the octave harmonic of a normal tone.
Odd-harmonics-only tones can be used to tune unisons or intervals whose ratios do not contain an even number; so they would beat for the major sixth, 3:5, but not for the major third 4:5 nor the minor third 5:6.
The above remarks, of course, apply to cases where both tones lack even harmonics. When one tone has them, the ratios having an even number can be tuned if the normal tone happens to be on the 'right side' but not otherwise.
What makes all the above so relevant is that tuning-routines either set the temperament with fourths, fifths, and octaves--ratios 1:2, 2:3, and 3:4, all of which contain an even number, or they check with those intervals.
The ideal timbre for tuning by routine is not necessarily the ideal or even a desirable tone-quality for a particular musical purpose and that should be borne in mind if you are building an instrument to be tuned by ear.
For instance, the very charactertistics which make it easier to set 12-tone temperament on a harmonium or reed organ than on piano, by the same token make 12-tone-tempered harmonium music less agreeable.
Those who would like to experiment with the special tone-qualities proposed for 19-tone instruments by Joseph Yasser in his Theory of Evolving Tonality--timbres in which the 3rd and 6th harmonics would be totally absent--would not be able to use any practical tuning-routine. No fifths, no fourths, no minor thirds, no major sixths. Minor sixths and major thirds would still be available to beat, but no way of checking them; so errors would pile up. One would either need and electronic tuning-device, or to transfer the tuning form another instrument of normal timbre.
Paradoxically, this article is appearing, and I have been constructing routines, just when they are declining in importance, due to the electronic tuning devices and the new ease of fretting as well as new means of building electronic organs in such a way that they are their own programmable tuning devices.
Don't let this or anybody else's article discourage or prevent you from using those systems which cannot be tuned by routine, or for which routines are difficult. Examples: 13, 14, or those with large numbers of tones.
But the decline of routines will be very gradual, not sudden; and besides that, not only the routines themselves, but the way they are constructed and taught, are of considerable theoretical interest. They produce what I have dubbed tempered temperaments.
Since music is a sort of language of emotions, it is hardly strange to find all the practical, theoretical, and mathematical and engineering issues connected with musical instruments and the tuning thereof, beclouded by years of arguments over trivia, wild polemics, hurt and wounded pride, savage condemnations of this or that system, petitio principii, argumenta ad hominem, trying to have it both ways at once, and such-like.
It's now time to get down to brass tacks. The routine for 12-tone equal temperament has to come first, so that other routines can be compared with it. Fortunately, thank goodness, I don't have to go on at length: the libraries are full of books describing innumerable variations.
I give here the routine I follow in almost all cases. Before you snarl and snap at me, the choice of A 220 Hz to A 44o Hz as the tuning octave is a compromise betwen the F-to-F of piano specialists and the C-to-C of organ specialists, and is not being imposed or foisted on you if you prefer the other starting-points.
Use a tuning-fork or modern electronic substitute to sound a' = 440 Hz (but this will work for other pitches, if such be necessary).
Ideally, the tuner would look up all the beat-rates in one of the published tables such as the 12-entries in the Quartertone Beat Table in this issue, and eventually memorize them. Actually, he memorizes some of them and the result is a tempered temperament which averages out.
The above order goes in a flatward direction around the circle of fifths, but there is no reason why you can't go in a sharpward direction if you prefer. It takes the average person a few months to close the circle on the first try, so don't get discouraged.
If you are tuning a master-oscillator-and-divider electronic organ, that's all there is to it; the octaves do themselves automatically. If you are doing a piano, harpsirchord, or pipe-organ, you've just begun!
Tune the bass octaves first, then the treble end.
As soon as you have don the bass, check the major sixths chromatically from low C for the beat-rates, which should rise gradually and quite evenly:
Don't get too mad if you goofed: the error may well be in the octaves--which is why somebody thought up this test in the first place. Even on the octave-divider organs, this is still the best range to check major sixths.
Be a nasty mean critic: have fun--or rather, Schadenfreude. Go around comparing the actual present tunings of 12-tone instruments with what the table says they should be. You won't have any friends left after this--certainly not among tuners--but think how smugly and knowingly you can smirk at the musical world!
Actually, I am suggesting this course so that you will realize how far real musical instruments in the condition they are kept by average owners, deviate from the abstract Platonic concepts of the ivory-tower theorists. I hope it will be a salutary shock! Even on philosophical grounds, the idea of a Perfect Amount of Imperfection is questionable.
Also, it may keep you from knocking yourself out trying to tune better than the professionals can; and make you realize that instruments go out of tune, so that by a month or two afterwards, untraprecision or the effort to attain it has gone down the drain.
These deviations and instabilities of pitch, so offensive to the abstract intellect, are tolerated to an amazing degree in the actual performance of music. Consequently, only certain styles of composition are going to tolerate great precision--which in turn will profoundly affect one's choice of tuning-systems and instruments.
These considerations save us a lot of time and trouble: all the routines we don't have to construct because they are not worth using. Masochists must look elsewhere.
With that out of the way, let's do something about the 24- or quartertone system. Since the circle of twelve fifths is closed, you can't have a circle of twenty-four equally-tempered fifths. So, the simplest way is to get a fork or other standard for either a quartertone higher or lower than 440 Hz and do the 12-tone routine all over again at this other pitch: you simply pretend that A-semisharp 452.9 or A-semiflat 427.5 Hz is A, and take it from there. Voila!
What if you don't have such a fork, or it would take too many months to get it?
According to some tuners I questioned a while ago, they interpolate the quartertone by ear--which of course means by guess and by gosh. I think I have a better way: if the instrument being tuned is in a place having ordinary power-lines, you can adjust a radio or amplifier or the like to hum at double the power-line frequency. Virtually everywhere in the U.S., the ac power is 60 Hz, so the double-frequency transformer hum is 120 Hz, while the quartertone pitch B-semiflat is 119.956 Hz, flatter than it by less than one beat in 20 seconds, so we can safely ignore that.
So all you would have to do is to come up an octave to 240 Hz,call that B-semiflat, and enter your 12-tone routine at B-flat.
For the 50-Hz power lines in other parts of the world, you would take the 100 Hz hum as G-semisharp, come up one or two octaves as your 12-tone routine demands, and enter the routine at G or A-flat as you please.
For reasons which you should be able to figure out for yourself, the 100 Hz is not accurate a quartertone as the 120 Hz, so you might want to make the G-semisharp one beat per second (almost) sharper.
Tuning the 24-tone system by ear is frustrating in a way in which the 19, 22, and 31 systems are not: few people are familiar with the intervals which contain odd numbers of quartertones and link the two circles of 12 fifths each.
If you have not heard the intervals whose just ratios are 4:7 and 8:11, which I call the subminor seventh and semiaugmented fourth respectively, no amount of written description by me is going to give you what they sound like or what they feel like!
How the blue blazes, then, can I expect you to look up the beat-rates for them in my 24-tone beat-table, and do a good job of tuning the quartertone imitations of those two intervals (the imitation of the 8:11 is excellent; that of the 4:7 is kinda punk).
So if you are not prejudiced against electric guitars with quartertone frets, have one made up (a simple matter of intercalation, with my fretting table) and become familiar with all the quartertone intervals. Then you should be able to tune other instruments to quartertones.
Besides guitars, you will find that harpsichords, polyphonic synthesizers, certain electronic organs, and psalteries are far superior to pianos for the quartertone system. I really would prefer that you not use pianos for 24-tone, although they are quite satisfactory for 17- and 19-tone.
Without further ado, let's go on to the 19-tone system. This is easier to tune than the 12-tone, at least for me; and I have had hundreds more 12-system tunings than 19. You might even learn it first, if you can get a suitable instrument built.
If all you intend to tune is 12 tones out of the 19 on a twelve-tone instrument such as a single ordinary piano, I don't want to help you! You'd be more of a roadblock than a trailblazer. That's sabotage, not progress!
That is to say, when I say that the piano will work in 19-tone, I mean two pianos with the five tones F# C# G# D# A# in common and seven natural on the first piano and seven flats on the second. This disposition was used by M. Joel Mandelbaum in the compositions for his 1961 thesis.
This disposition of 12 12-tone keyboard also works for other instruments, such as harpsichords, electronic organs, harmoniums, or synthesizers. Special keyboards for 19-tone have been invented and the 19-tone system fits well on many 'generalized' keyboards.
On synthesizers and some kinds of organs, the cheapest and quickest way is merely to map the 19 tones onto the conventional 12-tone keyboard, so that 3 octaves of 12 give you not quite 2 of 19, or 5 octaves of 12 give you just over 3 octaves of 19, (this is what I do on my specially-built electronic organ).
This is impossible on master-oscillator-and-divider-type organs, or keyboard stringed instruments, for obvious reasons. But where it is applicable it will save you agonizing waiting and grieving over unfulfilled dreams.
Most master-oscillator-and-divider organs would have the reserve power to run 7 addition divider chains with their oscillators, and while rebuilding the organ, some kind of 19-tone keyboard could be made. This, naturally, is an ambitious understaking, but some of you might be intrepid enough.
I have done enough composing and performing in the 19-tone temperament on different instruments that I am confident the system is worth it, otherwise I wouldn't bother with this informaiton now being dispensed. 19-tone has a peculiar mood and personality and melodic and harmonic properties that no other system has.
A few decades ago, Augusto Novaro of Mexico constructed what he called cajas de afinacion--let us say 'tuning boxes'--which looked like autoharps or zithers, but really were psalteries. This is still a very practical idea, and probably less expensive than Harry Partch's Harmonic Canons.
You will see from my 19-tone Beat Table that the minor thirds and major sixths take as much as 17 seconds to go through one beat. One beat in 5 seconds would be far too slow to count. The fraction-of-a-cent error is way beyond the limits of hearing or hte stability of tuning of most instruments, so forget it, and consider all major sixths and minor thirds as perfect, just, and beatless. 19 minor thirds in a chain fail to close the Circle by only 3 cents!
19 fifths of Francisco Salinas' 1/3-comma temperament miss closing the circle of fifths by 0.94 of a cent! This is indistinguishable from 19-tone equal. So, our routine might as well be: tune fifths and fourths as shown in the beat-table, three in a row to get to the next major sixth or minor third, then check each such minor third or major sixth as you go: starting-tone against 3rd link in the series, 1st fifth-up against the 4th-tuned tone, and so on. 2nd against 5th, 3rd against 6th, &c. Starting from A at 440 Hz, it might be written:
If you are experienced in 12-tone tuning, this or any other 19-tone routine may confuse you. Nothing much I can do about that, except what I said above about getting a 19-tone electric guitar and becoming familiar with the new intervals and some of the chords, just as I advised for the quartertone system.
I can forewarn you of difficulties: the major thirds are flat--repeat that to yourself, please, the major thirds are half as flat as the twelve-tone major thirds are sharp. Memorize that!
The fifths are almost the same flatness as the major thirds, in cents, that is; but that does not mean they will beat at the same rate, as you can see from the table. It does mean that hte minor thirds come out almost perfect.
The beat-rates are easier to count than those for 12-tone. This is a great blessing.
The twelvically-indoctrinated tuner will feel peculiar when he gets to the twelfth note to be tuned from the starting-point and it is not the same as the starting-note at all! Seven more to go. The mere name-change from E-sharp to F-FLAT or B-sharp to C-FLAT will croggle him.
The problems are mostly psychological, matters of attitude, not matters of fact. How the tones are represented by nomenclature or notation is trivial; such questions should not be allowed to discourage and stop people from hearing the effects of the new systems. Remember: the critics of such systems as 19-tone are almost all those who have never heard them!
Since the whole affair is psychological, rather than a matter of moral or physical principles, you won't master the non-2 tuning routines if you are prejudicd against them (i.e.,. have the wrong attitude) or if you don't have good auditory memory-images of the system in question.
Happily, there is a quick and inexpensive way now to get these: have a guitar fretted to the system.
If, before you try to tune the intervals, you already know how they should sound, you will make fewer mistakes in the routine. If, on the other hand, you mentally measure everything against 12-tone, will seem wrong wrong wong, as you will resent every step and do it badly so that you can tell everybody how horrid it is.
I very strongly suspect that the reason there has been so little done about certain systems is that no tuning-routines were ever made up or published for them. I wonder how many people believe that there is no possible routine other than that particular 12-tone one they have learnt or observed.
As with twelve-tone, there are certain chords which can be used to check the 19-tone tuning after some of the octaves are tuned (or right away, in the case of a locked-octaves organ).
Inthe case of other organs and other instruments, the 19-tone system allows stretching of the octaves to a much greater extent than 12 does. Special routines and tables will be provided in the future.
The preceding only scratches the surface, of course. The important point is not to check with any music written by someone who never heard the nineteen-tone scale in all his life. That would be egging on all hecklers to come and do their worst, which they gladly will!
Now we come to the 17-tone system. It will be more difficult to learn and practice the seventeen-tone routine because there are good checking-intervals along the way. Instead of restful, just-about-perfect minor thirds and major sixths, and passable major thirds and minor sixths, their 17-system counterparts are very harsh and dissonant. This turns the rules of harmony upside-down!
Mind what you are about. The fifths are SHARP and the fourths are FLAT.
Note that A-sharp is C-flat and E-sharp is G-flat. Also, A-sharp is sharper than B-flat, or a sharp is two seventeenths of an octave.
Find a brilliant melody for solo violin or trumpet, and play it in 17-tone. Then try pentatonic melodies and chords-in-fourths.
Paradoxically, when the routine is constructed for the 34-tone system, it will make it easier, because major and minor thirds and sixths will "bridge" both circles of 17 fifths and thus tie in at many points. That will provide the checks which 17 cannot have.
When we come to the 22-tone system, notation and nomenclature pose more problems than the routine for tuning it. I have tuned this system in a number of ways which again are difficult to describe but if anything easier than ordinary twelve-tone routines. The thirds and fifths and septimal intervals interlock in a mutually-checking manner.
For instance, one may proceed much as one would to tune a just major scale on the starting-tone. Then add the 7th harmonic as imitated in 22-tone, then a chain of three major thirds to get the "quartertone" between major seventh and octave. This done, there are a number of ways of filling in the remaining tones. That is, it is not necessary to doggedly struggle through the very long circle of 22 fifths. The beat-table works just as well, whatever the chosen routine.
The errors of the 22-tone major third and fifth add instead of cancelling out as they do in 19; therefore the minor third is considerably sharp, but the major seventh adn diatonic semitone are within 3 cents of just values. This alone dictates a different kind of tuning-procedure from that for 12, 17, or 19.
The approximations available in several intervals unfamiliar to most tuners,provide additional mutual-checking-points with which to sew the twenty-two-tone fabric together. It will help considerably if you can sound and become familiar with the subminor third whose just ratio is 6:7, the subminor triad whose just ratio is 6:7:9, the supermajor third whose just ratio is 7:9 and which is approximated only one cents away by 22-tone; and the semi-augmented fourth whose just ratio is 8:11.
We now are at a point where cold reasoning and dry theory and arugment from traditional 12-tone principles are worse than useless. There is no way to put down on paper what the 22-tone equal temperament sounds like; and most emphatically, no way to describe the 22-tone mood other than to play some worthwhile music on an instrument so tuned.
There isn't space in this issue to set out routines in full. Why not get a guitar refretted in 22-tone, so you can become familiar with the effect of the system, and its intervals?
Xenharmonic Bulletin No. 7 contained fretting tables for 17, 19, 22, 24, 31, and 34 tones per octave. This means that there is no longer any pressure on anyone to learn how to use routines or set temperaments by ear. We will nevertheless give routines for 22-tone, 34-tone, meantone, 31-tone, and some stretched-octave systems in future Bulletins for those who might like to try them, or who prefer to check tunings arrived at by other means.
With all of the routines, the commonest mistake is to tune an interval to the right number of beats per second, but on the wrong side of beatlessness: sharp instead of flat, or flat instead of sharp. Remember to tune every interval just, before tempering it.
The peculiarities of piano strings are such as to make the partial tones sharper than the true harmonics, so that stretched-octave tuning has become a custom. The stretch itself is stretched, at both ends of the keyboard. Now, the 19- and 31-tone systems could do with stretched octaves, regardless of what instrument is to be tuned. Applying these systems to pianos will compound the situation, obviously.
On the contrary, such systems as 17, 34, and 22 might do better with shrunken octaves. This will sound 'queer' on pianos, and personal prejudices should be taken into account.
Organs and harpsichords are ordinary not stretched-octave-tuned, or only slightly; but in the cases of 19- and 31-tone (meantone also) this question must be re-examined.
You will notice that I have not said much about a very large family of tuning-routines which consists of unequal temperaments, variations on 12-tone-equal, variations on 12-out-of-meantone, muzzled wolves, schemes with different-sized fifths, with a few just fifths along with other tempered ones, etc. This has been deliberate, intentional and on purpose!
I do not want to encourage the tuning of systems which have only twelve different pitches per octave, no matter how these may be spaced, because none of them offer today's composers anything different enough from 12-tone equal, because anything different enough will need more notes in order to be complete. That is what xenharmonics is all about!
As a composer I will not accept the restrictions of a bygone age, arising from mechanical and financial problems which now have disappeared. Laudable though the attempt to get old keyboard music performed in a better or more authentic intonation may be, this is not the central objective of the Xenharmonic Bulletin.
That is to say, xenharmonics is not the re-exploration of forgotten bypaths, but the blazing of new trails! Nor is it the search for exotica via ethnomusicology, the mere copying of distant cultures in the hope it won't sound like warmed-over Rimskii-Korsakov or Debussy.
