In the seven years since Xenharmonic Bulletin No. 10 came out, there have been remarkable advances in electronic musical instrument technology.
I knew this would happen someday, but 'someday' kept receding as the 60 years since 1929 crept past. Quite regardless of the pessimists in the musical world, inventors kept on inventing, and engineers kept on reducing ideas to practice.
What has happened in the last few years is that a cross-over has occurred: as the cost of electronic keyboards and equipment has come down to affordable figures for many persons, the cost of pianos and other conventional 19th-century instruments has escalated and then gone through the roof!
Read the ads in the papers and see the signs where you shop or look around; you don't have to take my word for it. Unfortunately, many musicians and people who write about music and musical events refuse to see what is staring them in the face. They insist it is still the 19th Century in Central Europe and their REligion takes pianos as Idols and the Romantic and Classical composers as Saints.
They hate progress in music with a fierce intensity. So I don't think they will be reading Xenharmonic Bulletin. My reason for bringing them up here is simply that economic conditions have changed in our favor, not theirs.
The objections brought up hitherto simply do not obtain anymore. The problems adduced as regarding Xenharmonics are now pseudo-problems--in particular, the matter of providing more than twelve pitches per octave on instruments, available AT ALL TIMES.
Not 12 pitches out of just intonation nor 12 pitches out of 19 or 22 or 31; ALL the pitches of the system being used beyond our familiar twelve, up to 24 or even 31 in many cases. Of course it may be impractical to have all pitches of a system like 53 or 41 tones per octave on all instruments, but even there, it is only a pseudo-problem, a red herring.
We have computers. Not huge mainframes of the 1950s but microcomputers on office desks and in many homes. Not just prosaic business applications or dry-as-dust scientific data, but games and graphics and art and software and MUSIC.
Not the impersonal remote computerized systems in some giant corporate building, but contemporary home appliances and many of the new synthesizers and other musical apparatus contain computer insides. You can walk over to your corner drugstore and get a pocket electronic calculator for a few dollars. It has pieces of computer circuitry inside it!
Some of you are wearing digital watches right now, that have counting circuits within them. A conventional windup watch of the 1930s might tick 5 times a second, but many digital watches and quartz analog watches some of you may have on your wrist, "tick" 32,768 times a second for the older quarts crystals and an incredible 4 million + times a second for newer models!
You may ask, What has this million-vibrations-per-second thing or Megahertz as that is now called, to do with music, or with Xenharmonics?
Maybe it didn't when you were a child, but it sure does now! Frequency is the reciprocal of time. Controlling the number of ticks per minute or hour on a cluck so it will keep correct time is much the same affair as putting any musical instrument in tune or keeping it in tune or restoring its tuning. Music is a multi-ordinal pattern in time. The mechanical clocks and watches of our childhood or the previous generation worked at infrasonic frequencies such that a watch-tick was 5 or 6 octaves below a note in the middle octave. The quartz crystal digital watch mentioned above might be 2 octaves beyond audible range the 4,194,304 vibrations-per-second crystal in many modern watches will be 8 to 9 octaves beyond the keenest ear of any human.
We said "octaves"--music anticipate the binary system of counting now used in digital equipment! The ratio 2:1 is built right into almost every part of a computer and into digital watches. Those numbers up there, 32,768 and 4,194,304 are powers of 2--2^15 and 2^22, respectively.
So, fellow-musicians, use this ammunition when computer buffs try to intimidate you! Octaves are powers of two,a nd the infrasonic frequencies used in music as rhythmic elements are also powers of 2 or of 1/2--whole, half, quarter, eighth, sixteenth, thirty-second, and sixty-fourth notes, so there.
While we are about it, a very common vibrato frequency is 6 to 7 pulsations per second, and this works out to 2 to 2/12 octaves below that Note #1, the bottom or subcontra A on the piano keyboard.
That is, music actually counts elements to make patterns in time ranging from the slow infrasonic affairs like sections and phrases and measures and beats through ever more rapid notes to vibrato and tremolo frequencies to contra-bass notes through middle and treble and altissimo frequencies up tot he limits of human hearing.
And musica counting is mostly binary like the hardware inside computers. Music may not use the 1's and 0's mathematicians use to express the binary counting-system, but the open and closed note-heads and flags and beams on more rapid notes amount to the same thing. Yes, there are triplets and other non-binary notes, but they are the minority and binary are the majority.
There is an electronic element called a flipflop which counts every other vibration of what is fed into it--or, otherwise stated, a flipflop outputs 1/2 the input frequency. It is not tuned in the way a string or organ pipe is tuned, but it does provide one octave below whatever pitch goes into it--however, it will also count every other vibration even if the vibrations put in are irregular or get faster or slower. Some electronic organs had similar divider circuits which were fed a note in the top octave of the organ and stepped it down octave by octave all the way to the contrabass. What that meant was that tuning the 12 tempered pitches of the top octave automatically tuned every note on that organ without the laborious tuning of each and every note--each and every reed or pipe of a conventional organ.
In fact, the two Farfisa transistor organs provided me in this studio by Buzz Kimball of New Hampshire are of this master-oscillator-divider type and thus 24 notes per octave are available in a total of 6 octaves, by tuning only 24 oscillator-coils by turning core-slugs with a screwdriver. No tedious business of having to tune all 146 notes.
So going from one tuning to another of the more than 100 possible tunings of these twin organs is a matter of 40 minutes or so, not of all day and into the night.
no wearisome back- and arm-straining elbow grease like tuning two PIANOS would be! Well, if you tried to tune two pianos to most of the tunings these organs have in for my compositions, you would break strings and never be able to afford either the time or the money to replace those strings! There, in a nutshell, is why xenharmonics has had to wait for modern electronic technology even for the tuning of new acoustic instrument to our new scales.
This is why I am writing this article and why you have to know certain things about digital counting circuitry.
Let's put in something else here about piano-tuning. With 49 years' experience I should be worth heeding on that. It takes 6 months for the average person to learn how to tune a piano the common old-fashioned way, by ear. Some people never learn; they lack the talent.
This means learning tuning well enough so that regular musicians can perform standard piano music properly. It does not mean real perfection. This means learning ONLY a system which is very close to, but not exactly, the 12-tone equal temperament. Just one system, not meantone or other members of the meantone family; not he unequal temperaments proposed by various theorists; just something close to the equal ideal.
Back to the electronic alternatives to conventional instruments. Keyboards are now PORTABLE. This makes a tremendous difference. Electronic instruments are now longer tied permanently tot he light-socket, since many of the new ones run of dry batteries, rechargeable batteries, or plug into the automobile battery.
Now why all that fussing & grumbling about piano-tuning? Because as I already said but cannot repeat too much, you cannot demand that a tuner learn several new tuning-systems that conflict int heir very essence. You'll get a bad job and the tuner will take his frustration out on you, and have to charge more.
The piano will be put out of whack by retuning it too far from the 12-equal for which it was designed. It will try to go back and it will do that soon and ruin your experimental recordings. It's bad for the tuning-pin block and for the frame.
OK! Now with the electronic yattis-frammis and gizmos and flibbertigibbets, retuning doesn't strain its innards. True, many recent keyboards are permanently tied to 12-tone-equal at the factory. Others are either retunable or can be modified.
If conventional manufacturers and synthesizer companies won't make special retunable instruments, others will; and there are enough broad-minded people in various kinds of electronic businesses that somewhere somehow enough will be built and there will be do-it-yourself kits and there already is some computer software because I have HEARD it, and I have some sample tapes.
now back to those flipflops that I mentioned earlier. Automatical octaves means as I said, only the need to tune a top octave and all other octaves fall in line, and even respond to retuning. When I put the Farfisa organs both of which were designed for ordinary 12-equal, into unequal tunings, into just arrays, into Pythagorean, into 24 notes of Meantone, into 17, 19 or 22, all the dividers in them AUTOMATICALLY obeyed the master oscillators and only 24 things had to be tuned, not 146.
Certainly not the four hundred twenty strings that two pianos would have! This is the crux of the matter, the reason why there has been so little progress toward new scales for 200 years--i.e., since the piano took over from the harpsichord and drove other keyboards out of business till electronic instruments could mature.
Now for another practical consideration: it is not enough to have just one non-twelve tuning. You need several. Where would anybody put a dozen pianos? Why, many piano stores these days don't keep a dozen of them on the floor at one time.
No sweat with electronics: there are instruments, and I have one right here, that will do over forty different tunings by just turning one knob! It takes and average or 15 seconds to go from one tuning to another. IT defaults to 12 if you turn the knob all the way. But it stays int eh tuning you last set if you leave the knob alone.
Now really: if you can have an instrument that will retune in 15 seconds, why wait two hours for piano merely to be corrected for its drift away from string 12? Two pianos for 17, 19, or 24-tone? Who's got the time these days? Even if they had the money?
No wonder all this has been delayed and delayed. Now we are free to forge ahead. Lots of lost time to make up for.
Let's go back to those new clocks and watches for a moment--they have the built-in counters which take the inaudibly high vibration of 32,768 or 4,194,304 or whatever it may be, and bring it down an octave at a time through the audible range to the infrasonic range which is inaudible low--on vibration per second, then it counts the second to show minutes and hour sand then days--for nearly all of them have a calendar with day and month. Sometimes the battery which is about the size of a dried pea, runs down before it can count through the years.
So new musical instruments often have a similar counting system, not all that different from the way a computer counts and the little pocket calculators count.
This is how we can get rid of the necessity to rune instruments. MOREOVER, to tune acoustic instruments and those which do not contain digital counters built in, we now have tuning-devices which contain some kind of counting mechanism. The user only has to tune unisons or in many cases, set the dials on the device and read and indicator that tells when the external instrument is in unison with the device.
The usual thing is to fake a 12-tone tempered scale on many organs and synthesizers and new keyboards.
Temperament is itself tempered! Here's how: an inaudibly high frequency, just like those watches and clock ,is generated by a stable oscillator. Someone does the arithmetic to find divisors for this large number of cycles per second, whose quotients will come near the frequencies for the 12-tone equal scale for the top octave on the instrument. The larger the high frequency chosen, the closer together the divided frequencies can be, so it is generally somewhere between 2 and 5 megaherz. This determines the accuracy of the audible frequencies. An exact 12-tone equality is impossible, since one cannot do the twelfth root of the two and its powers digitally, but it can be approximated within a few cents or even a fraction of one cents, i.e., see below the ability of any human ears to distringuishf or perfect temperament.
It is possible to have something very close to just intonation, subject to a tine error of a cent or so; it is possible to have any temperament one might long for by similar means; just pick the right divisors and set up counting circuits, or program a computer to produce these countings. This is not the only way to get rid of all necessity of anyone having to tune an instrument, but it is one of the commonly-devised ways.
So, if you are at all interested or concerned with the theory of tuning, you should know the above.
Nothing at all alike this was dreamt of in the 19th century or early 20th. For us in xenharmonics, it is one of the important spinoffs from computers.
There are synthesizers of the analog variety which do not get their pitches that way, and we will go into this some later time.
Millions of electronic keyboard being sold! How could they be any good if humans had to keep tuning all of them?
However much the above may displease a theorist, it is a practical matter that has to be solved somehow in the real worlds. Pianos generally do not get tuned in the 1980s very often, partly because there are not enough tuners and partly because many owners cannot afford it. So the decline of the Pianofortic Empire is inevitable,sob, whimper, crocodile tears, anyone?
This was one of my motives for includes the article Metal Tubes and Bars in this issue: once you tune a set of those, it will stay in tune for a century. Once a set of them has been made in a given just-intonation array or equal or unequal temperament, anybody else who wants such bars or a marimba or other stable acoustic instrument can copy its pitches and enjoy the same permanent tuning.
Tuning pipe organs or pianos or harpsichords or harps or psalteries is the labor of Sisyphus--all your effort and skill and dedication goes down the drain in a few months or even sooner. What a bummer!
So it's not just the difficulty of learning a whole variety of new tunings-systems...it's the fac that many conventional instruments do not say in tune, and so your new-scale instrument of that kind would be a useless white elephant pretty soon.
With fretted instruments the spacing of the frets, which can be worked out on a comptuer or pocket calculator of the "scientific" variety, sets the temperament or the just array and only 6 strings have to be brought up to pitch the aid of those frets.
Here comes the Commercial: I have made up about 110 tables for fretting instruments of different string-length to different systems.
In various other issues of this BULLETIN, I have published frequencies and beat tables as well as some fretting tables.
It should be obvious enough at this point that I was trying to connect the idea of pitch or musical use of audible frequencies to the the idea of keeping time in both the musician's sense and the clockmaker's sense.
We can wind up this discussion by considering attainable accuracies today. A quartz crystal oscillator may often be accurate to one part per million--that is, a nominal 1 MHz crystal might stay between 999,999 and 1,000,001 Hz if properly mounted and temperature-controlled. In some cases, better than that. What does this mean in terms of timekeeping? One second per day variation would be one part in 86,400. One second per week would be 1 part in 604,800. One second in 11 days would be about one part per million. Of course, a clock or watch in the home or office wouldn't enjoy such a constant temperature and freedom from outside disturbances, but it generally does a pretty good job compared with regular watches and clocks that have pendulums and gears or springs.
That is to say, for the first time in history, the average person can get a clock that surely will do one part in a hundred thousand.
now let's try to evaluate the tiny interval of one cent (.01 semitone, 1/1200 octave) in terms of clocks. About the gain or loss of one second in 28.5 minutes would be a tuning-error of nearly one cent. If your clock or watch gained or lost that much, you no doubt would complain. But a musical mistuning of that magnitude would not be audible except under very good conditions.
Double the error to 2 cents and many people would be able to hear it--that is about the normal threshold in the middle of the scale, with sustained tones. Still much too small to matter in any musical performance, but when you are tuning something and paying attention, it may be heard. Now we are talking about losing or gaining a second in 14 minutes or so.
This is about the error of the fourth or fifth in standard 12-tone tuning. But let's turn to the major third or minor sixth. Error nearly 14 cents. Now we are talking about losing or gaining a second in two minutes or so. Well, that wouldn't be much of a clock. No way to run a railroad. This error is clearly audible.
Indeed, the way that the 12-tone equal temperament ruins both sixths and third (major AND minor) is the chief reason why newcomers get interested in xenharmonics. Yet again the introduction of sustained tones through organs and synthesizers, as opposed to the percussive noise of piano hammers and the dying-away tones, altered the situation with regard to a heightened awareness of tempered tuning errors.
The way is wide open for me and anyone else to explore new scales, so I am already doing that and helping others explore without getting in one another's way...since there is plenty of room in the multitonal world, unlike the tight oppressive competition of the orthodox musical Establishment.
As I pointed out some time ago, the new scales all have new MOODS which could not be predicted mathematically, so that is why the experts and teachers and theorists missed them.
I can't write down the MOODS, but I can record compositions in them and copy the cassettes for you. So I did and they are now available.
It's your turn.
As this is being written, the Sonic Arts Galley in downtown San Diego is showing an exhibit called TUBULONGS, that being Ervin Wilson's name for new musical instruments constructed out of metal tubes.
Glen Prior, Jonathan Glasier, Ervin Wilson, Ivor Darreg, and others have been building instruments with aluminum, steel, brass and copper tubing for several years now. At the Gallery, all those materials are represented, along with other instruments made of steel bars not too different in sound from the conventional glockenspiel.
Since the physical principles governing metal bars and tubes (mounted "free-free, i.e., not clamped at the end but supported at the nodes of the principal mode of vibration without too much constraint) are similar, we can consider bars and tubes together.
Why here and now, in Xenharmonic Bulletin Number Eleven? One of the reasons for the long time between No. 10 and No. 11 of this journal has been the opportunity to do more practice of xenharmonics--composition, adaptation of existing music, performance, recording,and now and then the availability of materials to build instruments and/or to repair instruments, amplifiers, etc.
So much theory has been written about possible new tunings and what might be done in the Sweet Bye and Bye, that he time has come to correct a severe imbalance between theory and practice! This article is about what HAS BEEN DONE, not about "vaporware" as the computer people call those airy prognostications.
Some people look around of instruments to adapt or convert to new scales, but this often leads to disappointing results or even failure.
In particular, people want to make a piano over. That might have been the Only Game in Town in 1920, but this is 1988--well into the Electronic and Computer and Space Age, so let's leave Pianoforte Modification to incurable nostalgics and museum repair shops.
One very practical beginning is to use guitars refretted to non-12, as detailed in Xenharmonic Bulletin No. 7. That issue and other publications are still available and will continue to be.
Also cassette and other recordings in various new scales are now obtainable through various channels.
Now let's get into the reasons for describing metal tubes and bars. The main reason is that they don't cost too much to make, and the material is available at many stores and it is even possible to recycle used metal for instruments.
The cutting and mounting and other operations are within the ability of many persons and even if you personally don't have many such skills, it is easy enough to find some home-workshop person or even group willing to make metal tube and bar instruments.
The main problem for average people would be, do you have enough floor and storage space to keep such instruments after you have made them? Not for dwellers in trailer and furnished rooms and hotels, obviously!
However, there are enough meeting-halls and classroooms used only part-time and garages and backyards and other places to set up and play these instruments, and they are not too hard to transport or store, since they can be easily set up and taken down.
Best of all, they stay in tune much longer than other instruments. They are that different from tuning-forks, accepted the world over as pitch-standards. Quite likely, a well-made set of bars or tubes will be in tune a century form now. Here is your chance to make something that will live after you.
True, the pitch of a metal bar or pipe will go up and down with temperature, but if returned to that temperature at which it was tuned, it should be right on the button. A group of bars or tubes will go up and down together, and remain in tune with themselves.
Why use bars or tubes for new tuning-systems and non-12 scales?
In recent years, percussion instruments have become much more popular then they had been int he past, and so there are new magazines for percussion players and more teachers of the various instruments. That is, it wouldn't make too much sense to build as your first project something so alien that you wouldn't be able to get beginner's help from books or regular musicians.
The principles of playing mallet instruments are the same even for these new tuning-systems, and in almost all cases, the new scale will have at least one pitch in common with commercial instruments such as vibes, glockenspiel, marimba, etc. We recommend that A = 440 Hz or its octave 880 be that note in common.
If you are so inclined, you could take lessons on a conventional mallet instrument--be cautious, though: do not tell them too much about what you are building--they may try to stop you using non-12-tone scales. Why borrow trouble? It's your future, not theirs, that is at stake, and no use trying to convert others against their will!
You may wonder why this article doesn't feature wooden-bar instruments such as marimbas. Simply that it will be easier to get the pitches for a marimba or xylophone from a metal instrument than the other way around. The metal instrument will sustain longer and can be tuned more accurately to the tuning device or other pitch-standard. Especially if you are a beginner. Also, lower notes on marimbas require resonators in most cases, and that's another building problem that can wait.
No doubt another question is already in your mind: with the new Sampling Keyboards, couldn't you avoid building metal tube and bar instruments and let the keyboard imitate them? Well, almost but not quite, and some of these keyboards are locke to the 12-tone scale at the factory. Besides, isn't it prudent to have one instrument at least in your collection that will still function when the power is off or you want to take and instrument out in the country for a day?
WARNING: Some samplers as of 1988 would take a sample of a bar or tube, then shift its pitch to other notes of the 12-tone equal temperament, not the new scale to which you will have tuned the bars or tubes! This would defeat your purpose. In years to come, this problem will be alleviated.
Comparison of tubes with bars: Much of the tubing commercially available has welded seams. Some tubing is seamless. This is important because a tube having much of a seam (usually only noticeable by looking down its inside surface) will be somewhat irregular in its modes of vibration, such that it will beat with itself.
For the ordinary 12-tone scale and some other scales such as 13 through 24 tones per octave, this may often be acceptable. Tubing having a seam may vibrate at a different frequency depending on where it is struck around the circumference. The most favorable or steadiest sound can be marked by placing a dot or other mark at that point after completing the set.
For critical situations especially for more than 24 tones per octave and where just intonation is the ideal with tine comma intervals and steady beatlessness, it might be better to consider using flat bars.
On the other hand, for scales like 13, 14, 18 or certain ethnic tunings like those used with the gamelan, beats may even be sought.
The main point is: don't spend a lot of money on enough tubing to build an instrument without first testing a sample tube for beats.
Tubes are generally loud enough for about three octaves from A=440 to the usual top C around 4200 Hz that they don't need resonators or amplification. That is the main advantage over metal bars, which (size for size) do not have as much sound-radiating surface exposed to the atmosphere as tubes do. At the top end of the scale, tubes may get too short, and if they have thin walls, they will not sustain enough. Length of sustain is dependent upon mass and stiffness, i.e., amount of energy that can be store dint he bar or tube from the energy imparted by the blow.
Even with tubes, it gets harder and harder to radiate enough sound as one goes to lower notes, i.e., longer wavelengths and more power needed to give the human ear an adequate sensation of loudness, hence our recommendation of note trying to take a set of tubes below A=400 or even C=528 Hz.
It would be nice if one could give tables of scales here with exact length to cut the tubing, but this impossible. There are too many different metals and alloys suitable for tubing, and and too many different sizes and thicknesses of the tube-walls on the market. A factory may quite making a particular kind of tubing, or change their alloy. The longest tube in one set here, sounding A 440 Hz and made of galvanized steel electrical conduit, is just over 23 mm outside diameter and is 570 mm long. This might serve as a typical dimension of the bottom note. A flat bar sounding the same note and made of similar steel might be 199 mm wide, 5 mm thick, and 230 mm long, but it must be added that this bar as been ground int he middle to make it thinner there.
This variation does not make your task hopeless. Once you have your first bar tuned to a definite pitch and stock of similar material on hand, then you can estimate how much shorter or longer the net note above or below should be.
Begin that with a Rule-of-Thumb which isn't perfect but close enough: twice the length is TWO octaves lower, not ONE octave as is the rule for strings and for air-columns in flutes or organ pipes. And likewise, half the length is TWO octaves higher, not ONE octave higher, as it would be for strings or air columns. This does not apply to resonator tubes if you need to make resonators for bars--those resonator tubes are normal air-columns just like organ pipes and obey the organ pipe rules.
It would be very rare for the air-column in one of the metal tubes we are talking about here to be in tune with the note we are obtaining by striking that tube with a mallet when it is supported at the two nodes of the vibrational mode generally used with free-free bars or tubes. If it ever happened to coincide and resonate it would make that note louder than its neighbors since it would be a fortuitous coincidence of the one- and two-octave rules just mentioned.
Now let's discussed nodes. If a tube or bar be struck at the very end or int he center, the points of no vibration for the normal mode of vibration will be approximately two-ninth of the length of the bar from either end. That is where the tube or bar should be supported, preferably by some soft resilient material such as foam rubber or certain foam plastics; felt has also been used for the heavy glockenspiel bars. The 2/9 rule is approximate: 0.224 of the distance is closer, but that would be hairsplitting in the case of these instruments.
We are concerned with the modes of vibration used in tubular chimes or the kind that imitate bells. Those follow still different rules.
Now comes the topic which differs greatly form any ordinary article on tuned percussion: for which scales are these tubes and bars suited? Is there some special reason for combining the use of these instruments with the exploration of new scales?
Yes, there is! It has to do with the Harmonic Series of normal overtones. Just intonation, and the equal temperaments with 19, 22, 31 tones per octave, and number of unequal temperaments of certain kinds, take advantage of the Harmonic Series and it works well with them. That is, such common intervals as the fourth and fifth and major and minor thirds and sixths depend for the consonance and their definition upon the presence and relative amounts of harmonic overtones in the sound of the instruments playing them.
But the sounds from these bars and tubes under discussion here do not have harmonic overtones. They usually have inharmonic overtones which are not members of the harmonic series and part of the playing technique for these instruments is learning how to keep the loudness of inharmonic partials down in relation to the fundamental.
It is possible to reshape some bars so as to make certain overtones harmonic or nearly so. A handbook published several years by Chris Banta of Pasadena CA treats of this subject.
For certain new scales that are worth trying, this absence of harmonic partials and the exotic color given by the presence of inharmonic overtones is a definite and positive advantage. Take, for instance, the equal temperament with 13 tones per octave. All its intervals save the octave are weird and wacky. Ordinary musical notation is almost useless for 13-tone. The only good way to deal with it is to number the tones.
Conventional rules of harmony will not work in 13, but it is melodically interesting and effective for atonal music. If we try to play 13-tone in ordinary timbres like organ, the chords can get very hairy, so many experimenters have given up on it. But suppose we tune a set of mellow metal or wooden bars to 13--much better! We have now tamed this wild exotic creature and are ready to explore a new corner of the tonal world.
Take the 17-tone system. Its fourth, fifths, and major seconds are more brilliant than those of ordinary twelve-tone; the leading tones are more decisive and snappy than those of 12, so that even conventional pieces such as unaccompanied violin solos or regular songs take on an added color. This new brilliance is purchased at the price of harshness when the usual kind of harmony is attempted in 17-tone, because of the dissonant nature of the thirds and sixths in this system.
However, the near-absence of harmonic overtones int he sound of tubes and bars being discussed here, tames 17-tone so much that we don't have to worry about harshness when playing most familiar music on a seventeen-tone tube or bar instrument; just go ahead and enjoy the melodic brilliance.
Similar reasoning applies to some other tuning-systems that can be used with these tubes and bars, such as 14, 15 or 21 tones/octave.
Followers of the Harry Partch tradition have constructed many bar instruments with various just intonation arrays. In recent years, there has been a growing interest in the Indonesian gamelan idea and other exotic tuning-systems found around the globe. This has contributed to the expansion in tuned percussions.
very well: tuning. How do you tune bars and tubes? By sawing, filing and grinding,mostly. Yes, but! You probably didn't mean that.
You tune them to what? That's more like th answer you wanted. Back in the 1930s an article like this would have been unrealistic and pointless. Tuning two or three octaves of bars to an unheard-of scale like 17 or 19 tones per octave, or to a selection of that many pitches from Just Intonation, well...no way for the average person! Tuning them to ordinary 12 would have been difficult enough, unless one had access to a commercial already-tuned metal-bar instrument.
Without a series of harmonic overtones, even a trained piano-tuner should not set the 12-tone temperament on the bars by ear, since the beats would not be there.
It would do you no good to take a piano- or organ-tuning class somewhere, since the 12-tone routine for tuning by ear will not work for the tubes or bars. However, in the 1940s and 1950s, electronic tuning-devices such as the Hale, Conn, Peterson and others were out on the market and many but no tall of these units can be used to tune bars to non-12 sclaes event though they were not designed for this job.
Most of these devices are expensive, so you may have to rent or borrow from somewhere. There are signs that the cost is coming down and in a few years it would become reasonable.
To use these tuners for non-12, you need a Cents Table for the tunings-system in question, and some such tables have been published byIvor Darreg and othwrs and more will appear soon. The above-mentioned tuners will tune 12-tone by turning dials to the wanted semitone pitch-class, but outside 12-tone equal temperament you must have a +- cents table. I.e., the pitches of the non-twelve tuning have to be presented as some number of cents below or above some standard 12-tone equal-tempered-semitone pitch, usually designated by letter name: F+23, C C#-30, G+21, A-3, F exactly, etc.
You could use a straight cents table like 386 cents for E (a just major third above just C), but then you would have to do oodles of subtracting, which can be mighty confusing when you are swaing and grinding bars! Cents, of course, are one-hundredths of a regular 12-tone-tempered semitone and the designation is due to A. J. Ellis a century ago.
To use a frequency-measuring device that reads out in hertz as a Frequency Counter does, you would need not only a Frequency Table for the tuning-system, but also an oscillator to provide a steady tone to your ear and to the frequency-counter, which cannot work with percussion tones that die away. Hence the adjustable sustained-tone oscillator with an amplifier and speaker is necessary for the frequency-counter method.
Signal generators where you dial your wanted frequency may become affordable later on. At present they are high-priced laboratory treasures.
Again, if you had an electronic keyboard instrument that wasnot locked in 12-tone equal temperament, you could set it to 13 or 17 or whatever and tune your tubes and bars to that. But this is only recently becoming affordable and available, so you might have to wait a couple of years or so for that.
When tuning, have a thermometer handy! Metal expands with heat and bars and tubes go flat; metal contracts with cold and bars and tubes go sharp. Considerable heat is produced by hacksawing, filing, and grinding, so it is easy to overcut at the final stage of tuning. Keep a record of room temperatures when you work on the instrument so that is will be in tune with itself when finished.
Other insturments, such as bowed stringed instruments or electronic instruments or the woodwinds, will go up or down differently with temperature than the percussions. Metallophones and xylophones are often tuned slightly sharp, sat to A = 442, for this reason.
IMPORTANT: A very common auditory illusion is that tones above 1100 Hz, when tuned to perfect octaves wtht heir counterparts an octavelower, will seem to be flat to almost every listener. It is almost universal to tune all notes on a piano higher than High C second ledger line above the treble staff, sharper than theory calls for, and as one goes to the top the amount of sharpness increases.
For metal bars and tubes the degree of sharpening will be somewhat less than on a piano. This illusion obtains, regardless of the new tuning system you may be using.
Last but not least: do not begin a project like this unless you have the patience and energy to see it through. Nothing is more pitiful than an unfinished instrument!
Most readers of the Xenharmonic Bulletin are familiar with the life and work of the late Harry Partch, who pioneered the actual use of non-12-tone-tempered intervals int he face of dreadful and fanatical opposition over a very long term.
His ghost is not allowed to rest either: the usual courtesy of De Mortuis nil nisi bonum is not often extended to him. Small wonder, then, that he reacted against his assorted opponents in the musical establishment.
One form this reaction took, which has not been commented on in the literature to my knowledge, was his rejection of many musical standards--well we might call them norms or trade-customs or textbook precepts--or in the case treated below, engineering standards.
Future publications may treat of these other rebellions against the standards, but for now the question is: what pitch did Partch take as a starting-point for tuning his instruments, and why?
In Xenharmonic Bulletin No. 11 we are discussing the trio of equal temperaments 41,43,46, and of course Partch is famous for a 43-tone system which the Great Uninformed Majority obstinately keeps calling 43-tone Equal, when it IS NOT!
Honestly, college-trained and wordly-wise people ought to know better! Don't they care? We can pardon innocent ignorance, but these writers bull themselves as seasoned experts. They know whom their favorite composer slept with and how often. They know which side of Tachikovsky's bread it was buttered on. They know how many buttons were on Schubert's coat.
But when it comes to American composers--and this includes Canadians and Mexicans--they don't even want to know anything, and are quite willing to let miniformation go through uncjecked.
Very well: does it matter? Am I nitpicking or fussing over trifles? In particular, why should it matter now?
First, because Partch's instruments are getting old,a nd for quite understandable reasons very few people have had access to them; and since Partch was careful to give many details of their construction in his book, the only way the average newcomer is going to duplicate the sound of Partch's music is to build and tune instruments to Partch's directions or to get the same set of pitches on a computer or electronic instrument, and if one attempts to do so witht he 43-tone Equual Temperament, it would be a terrible waste of time and possibly money, since 43-equal does not sound like Partch's 43 UN-equally-spaced Extended Just Intonation System.
The 43-tone equal temperament was invented long before Partch, and for a purpose having little or nothing to do wtih Partch's theory or practice. 43-equal is a variant of meantone temperament, and a very close approximation to what is called 1/5-comma meantone, and so far I have been unable to heary anybody's composition using all 43 of its tones. If you must use an equal temperament to approximate Partch's tuning, try forty-ONE equal, not forty-three. This was investigated some time ago by Ervin Wilson.
Actually, 43 unequally-spaced tones is merely Partch's norm or nominal set of pitches--early in his work he used 29, and later on had to use more than 43 occasionally.
There is nothing sacred about the number 43, and frankly, I could not perform most of my justly-intoned compositions, using his selection of 43 pitches. Nor could most other composers who use just intonation do theirs either. He had his own private reasons for his idiosyncratic choice and we need to copy him.
Since most of Partch's reasons for choosing 43 just pitches and not some other array of 43 just pitches (hundreds are possible) are given or implied in Genesis of A Music, I needn't recap them all here. Just what is relevant to the present-time situation is the huge amount of mininformationa nd harbled reports getting into the public press these days, and they get further and further from the turth year after year, and as we get further removed from the memory of live Partch performances.
"Live" in the sense that Partch coached the performers and was present to see that it went off right.
Do you want a parallel? Take bach--his harpsichord or clavichord might be at a different pitch from the organs his church pieces had to be played on, and many 19tha nd 20th century performances have been at very different pitches from those. The temperaments used also differed; and there is now a huge literature of speculation on what is and what is not correct tuning for Bach works. Incredible amounts of energyand time have been expended over arguing about that instead of playing the music.
Too manypeople would rather argue with each other or write about minutae, than get good performances or arrangements HEARD.
There are many followers of Partch and some of them forget how unique he was and that certain equipment and facilities and possibiltiies simply did not exist during Partch's lifetime, but are available to them today, so imitating all the dreadful privations and constraints he went through, or refusing to take advantage of today's technology, will NOT help Partch's music to survive and will not help their own composing either. His instruments and notation and nomenclature and selection of a pitch-array and a pitch-standard were not for you or me, just for HIM. It should be clear enough, reading his book, that he wasn't trying to impose it upon us...outside of performances of his own compositions, of course.
I am alarmed by the way some people treat him as a religious leader and use his system where it does not apply, instead of using it as he inteded.
The late Peter Yates, who spent a lot of time and trouble promoting Partch's works, used the term "private jobkes" to describe the way in which Partch was trying to get even with the Musical Establishment. This concept of Yates' is relevant here,f or good or ill, but please don't blame me for it! What Yates meant was that Partch continually needled the standardized, Classical and Romantic, European musical tradition and its instruments and performers and teacher sand academics.
Some of this protest and complain is obvious and clear and readily understood,but the "private" portion of it is almost encoded in an esoteric jargon, or has tobe read between the lines. His deliberate incompatibility, such that almost no existing music and practically nothing by other contemporary composers including me can be performed on his instruments, might not be clear enough to the music critic listening to Partch recordings or attending a Partch performance, then or now. This incompatibility extends to the tuning of his instruments, what they can and cannot do, the total elimination of traditional pitch-names A B C D E F G and the sharps and flats, or even new accidentals for small itnervals that some composers have invented, and his replacing of them by a system of FRACTIONS expressing ratios. All these fractions are "improper" or in the older musical-theory jargon, "superparticular"--what is meant is that these fractions must have values between integers 1 and 2 nor more and no less.
The above-mentioned integers are written in the forms 1/1 and 2/1, so that all his ratio-expressions will have the best uniform appearance. The larger number is always the numerator; i.e., the practic's value is always unity or greater, but it may not exceed 2/1.
Intervals greater than on octave, i.e., larger than 2:1, are reduced to the same octave--OK for him, but don't ask me to do this in my own writings! I want the freedom to writer 1:2 here and a 2:1 there, and to use simply 2 here are to write 0.5 somewhere else when that is the best way for my purposes.
He abolishes the familiar names for intervals, such as Fifth or Second or Major Third, and uses only the fractions as stated above. Good for his book because he had to avoid the confusion in English between the fraction 1/5 and the interval called a fifth because it occupies five successive degrees counting inclusively on the staff or five successive letter-names in ordinary nomenclature.
I almost wish he had written in German where there cannot be any confusion bewteen Funftel and Quinte. So far, so good. He gets rid of what could have ruined his chances and nobody would have followed him. But he introduced a few fly int he ointment that affords us no relief at all! This is his treating the set of ratio-fractions between 1/1 and 2/1 as both names of intervals regardless of where they are in the range of hearing, down in the sub-basement or up in the stratosphere; and at the same time these same fractions have to do double duty for absolute pitches, such that 1/1 = 392 Hz.
This reflects the solfeggio in English- and German-speaking countries where DO RE MI FA SOL are movable to any pitch you please, keeping their relations or rations; but in the Latin countries, DO is a FIXED pitch the same as our C and RE is always D (or D-sharp or D-flat by adding the proper name for accidentals in the language).
I just said "absolute pitches" but I must immediately qualify: he does not express octaves, so if 1/1 denotes 392 Hz it can also denote 784 or 196 or 98 or 1568.
Other composers who use Partch's fractions to name absolute pitches sometimes take their 1/1 to be C or A or something else, sucha s the old Physical Pitch C=256 Hz (or 512 or other octave), or some such standard as A-435 or A=440 or C=261, etc.
If you want your instruments to be tuned to unison with Partch's instruments, take G=392 Hz. That still leaves the question, Why? Where did he get that from?
This is where that Private Joke business comes in! Here he rebels against traditional music, pianos, the names of notes, the fact hat orchestras tune to A and bands to B-flat in many cases, the A=440 tuning-forks piano-tuners carry, the 12-tone equal temperament in general use, and a lot of other standards.
Some people tune to A, other to C, some band instruments to B-flat, and guitarists to E in most cases. A=440 is almsot an international standard in the 1980's, but htere is some use or lower and higher pitches such as A=435 and A=442 and A=444 or higher.
G as 1/1, the starting-point from which to tune everything else, has some justification--it is the note symbolized by the G or treble clef, for one thing. All the violin family have G strings (but they do not have C strings on violin or doublebass). As a compromise between C and A why not G? So that is not the Private Joke.
The joke is that here is Partch, the champion of Just Intonation and indefatigable enemy of all tempered scales and especially implacable denouncer of the twelve-tone equal temperament, choosing 392 Hz for this G which is almost exactly the 12-tone G for the standard A=440 Hz. He ranted and raved against 12-equal, then let is in the back door!
Oh, I know...the table for 12-tone equal temperament gives 391.99545 Hz for G or, more usually, 391.995 to 3 places. Now before quarrelling over split gnat's eyelashes, realize that this deviation or discrepancy is about 1/60,000 octave and most people would be happy if their clocks and watches kept time to better than one second per day. Tuning forks for G would be stamped 392.0 to nearest .1 Hz.
Now, tuning-forks and even electronic apparatus will shift in pitch with temperature and this will be considerably more than one-tenth of a vibration per second.
So the above difference between the value of the sixth root of two, times A=440, and rounding-off by Partch and the Fork Manufacturers to an integer, is of no importance is the real world. This should be proof enough of Partch's common sense.
392 Hz does look a little peculiar still. Why the 12-tone G, not the just G, from A=440? My best guess is this: there is more than one Just G for A, viz. the one 8/9 the frequency of A, and the other one 9/10 the frequency of A. The lower G has a frequency of 391 1/9 Hz, while the higher G has a frequency of 396 hz. The interval between them has a ratio of 80:81 or as Partch would have written it 81/80, but no matter--same difference. It's called a Didymus or Syntonic Comma if you must call it something. Frankly, it's a nuisance to musicians and pianists never understand it and piano tuners know only the other comma, whose ratio is 531441:524288, called the Pythagorean comma, and not relevant here.
But Partch knew and demanded Just Intonation, which has an infinite number of notes only a comma apart. Now suppose he was thinking about the cello or viola tuned in fifths, C G D and A, and the fact that the C which is a perfect major sixth below A is higher in pitch than the C you would get on a viola or cello by tuning it the conventional way.
The C a perfect major sixth below A=440 is 264, while the C belonging to a series of perfect fifths is 260.74+. Can you imagine the problems explaining this over and over to musicians of various kinds who could care less? They don't want to hear about such mathematical minutia and were already hostile to Partch and he's got enough trouble as it is, so if he were to get any more assistance or sponsorship or help in his career, better not to make enemies of everybody!
Helmholtz in Sensations of Tone, and his translator Ellis, chose A to be 440 Hz, anticipating much of Europe and America by decades, and doing better than they knew--435 and many other pitches for A were held to, throughout the musical world, and real standardization had to wait a very long time--and even now the pressure is mounting to raise the pitch to 444 or higher. At least, we do have a rather official paper 440 standard in effect at present.
Now Helmholtz and Ellis chose C to be 264 and 528 Hz, a perfect major sixth below A and a perfect minor third above it, so that the A=440 was really what Ellis wrote as A1--i.e., A-one-comma-down-from-the-A obtained by taking perfect fourths or fifths from C. A normal, plain, unnumbered A would then have to be 445.5 Hz.
So, with C and A being standard notes to start counting from for most musicians and instrument-makers and tuners, if Partch had been a good little boy and had chosen C as his fundamental basic resting-point for his system of tuning, this subscript and comma business would have gotten him into terrific trouble and would have crushed him. Maybe I'm botching this job of explaining. Maybe I too will get into a peck of trouble by this thankless task of attempting to ferret out Partch's motives for choosing G instead of C or A and then building into his pitch for G the 6th root of 2, rather than some nice neat pure just interval such as 9:8 or 10:9.
It may not be immediately obvious why abandoning C and going up a fifth to G would help him in his dilemma. In ordinary 12-tone equal temperament, and in Pythagorean intonation often used for melodic purposes, especially by string players, there is only ONE major second. But in just intonation and in some equal temperaments that have a small interval functioning as a comma, such as 22, 34, 41, 53, 72, &c., there will be at least two intervals entitled to the name of major second or whole tone and major third can be divided into unequal parts. Unequal systems also exist that have his property.
Now if Partch chose C=264 Hz as Helmholtz and Ellis in fact did, and made his starting-pitch C and kept all the letter-names for the conventional major and minor system, the nagging problem of the comma (81/80 in Partch's notation) immediately appears!
Which pitch, 293 1/9 or 297 Hz, shall be called D? Why, both! But how can D be two pitches so dreadfully out-of-tune with each other? Go to any library or store and find a whole shelf of harmony books and books on how to arrange or compose or play some instrument, and there will be no mention of commas and nothing to suggest any problem about how to tune the second degree of the C scale, major or minor.
Nothing to suggest that if you tune the standard just C-major scale, the D will make a horrid beating with the A. Nothing to suggest that the chord D F A is NOT a D-minor triad, but a discordant jangle. Early investigators of just intonation such as Moritz Hauptmann, whom Helmholtz used as one source for his work, solved this problem by using small d for the lower pitch and capital D for the higher.
Ellis went along with that for a while in the English translation of Helmholtz's book, but it created typographical problems, since as soon as you go to flat key and minor scales you need more comma-distinctions, so Ellis borrowed the sub- and super-script notation used in mathematics and then the two D-notes are distinguished as D and D1.
Fine and dandy, if you are a music theorist writing a scholarly article, or an instrument-builder trying to figure out some kind of special instrument that would be capable of playing conventional harmonic music in just intonation and modulating to a number of different keys. (Say some kind of an organ having 24 notes per octave.) This or another comma-notation will take care of what Partch called the 5-Limit.
Yes, it will be on paper. But try and get anywhere with it if you are confronted by any ordinary musician or music-teacher or piano-tuner! You will get thrown downstairs on your hind end, or told to Get Lost, or they might send for the young fellows in white coats with handcuffs and straitjackets.
Ellis has been slandered and misunderstood for a whole century and there has been no respect for the dead in his case but rather vicious unfounded attacks in book after book and the most unkind and unreasonable accusations and falsehoods. All because he tried to explain and straighten out just intonation and other tuning-systems which would be useful for progress in musical compositions and performance and instrument-construction.
I know of no greater ingratitude.
Partch surely was aware of happened to Ellis even beyond the grave, so he sought some artful dodges or way or escaping Ellis' fate. Now, suppose he chose G instead of C as the starting point. Then project the standard Just intonation scale upon G as tonic. By standard I meant hat form of a just major scale which can be found in physics and acoustics books and which was officially set forth as a standard in some engineering publications. No! Look what happens. The ratios are: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1.
Based on the just-intonation C = 264 Hz for A = 440, the above scale projected onto C as tonic would have D out-of-tune with the standard A = 440. Instead of a perfect fifth, ratio 3/2 in Partch's notation, we would get a false "fifth" of ratio 40/27 in his notation.
Now if we want to keep the letter-named and still want to show what is going to be out-of-tune, we could use Ellis' notation and show the false "fifth" as D A1.
But that would go against every musician in every orchestra and all the music-instrument manufacturers in the world, and y our name would be MUD if you tried that.
Worse yet, some people use subscripts and superscripts to mean octaves, such that A1 would be 110 Hz or 108.75 Hz for A=435, or A1 might even be 55 Hz an octave lower than the other fellow's A1 and neither having the slightest thing to do with commas. To make the whole mess even more nasty, the A which is now 440 Hz has been called A3 by some people and A4 by others. Don't try to mediate this eight- or nine-cornered dispute--they are all angry with one another and will never give in.
Funny how the persons concerned with Harmony and especially the very smooth harmony of Just Intonation, write the angriest polemics! They show no mercy and don't even bother to check the facts.
Back to the standard just major scale on the preceding page. If G were chosen as tonic, then A would no longer be a false fifth but now a true perfect 3:2 fifth from D. The set of perfect fifths C G D A on cello or viola open strings would now be a full member of the G-major just scale. SO partch chose G. All right, A is no longer the starting point, and A=440 Hz calls for G 391 1/9. But Partch wanted Unity, 1/1, as he writes, to start his calculations, and wouldn't it look odd if his starting pitch were a fraction, or worse still the decimal expression of that number, 391.11+?
Besides: somebody wants a tuning-fork for Partch's system. To get forks for everybody for 391.1 or 391.111 or whoever they might be marked, would be a special order that I am sure no American factory would consider. Eureka! How about G=392, which was available from a few manufacturers? Even if only as part of a 12-tone set. 392 is an integer and so that makes it much more acceptable as a published standard. Besides, it might be available as one of the metal bars in a vibraphone or other metallophone instruments, such as a school or popular band or orchestra might have.
I can't guarantee that this is how Partch's reasoning went back in the 1920's or 1930's but nobody has come up with anything to contradict it. I suppose that now some angry purist that wants to take all the fun out of life will try to start a fight with me about it. Frankly, I haven't time.
At least part of the above educated guessing has to be true. Should one or two points be proved mistaken, it doesn't really matter.
This isn't quite the end of the affair, however: Partch simply threw out the customary letter-names and their accidentals (and presumably that would mean that he threw out the Spanish, French and Italian fixed pitch names also, such as RE = D and MI = E and FA = F, because they do not take care neatly of his 43-tone 11-limit just system.
Up to this point I have not brought in the real problem Partch faced in addition to the comma above, which makes say an E1 derived as tuning a perfect 5/4 major third from C different from an E tuned by taking four just fifths upward from C as C G D A E. His system is not a 5-limit one: i.e., his intervals cannot all be derived by tuning just fifths and thirds 3/2 and 5/4--5-limit because the only prime numbers if the calculations are 2 (for what we call octaves and he would not), 3 and 5. No, his is an Eleven-Limit System because calculating frequencies or string-lengths for it involves prime factors 2,3,5,7, and 11.
Alexander J. Ellis, referred to above, could denote commas above and below the conventional pitches easily enough but his methods for dealing with sub-minor sevenths (Partch's 7/4) and other intervals based upon them, such as 7/5 or 7/6, are awkward, and when one gets to the eleven-based intervals and/or to what Partch called Utonalities (extensions of the minor chord idea to more complex intervals and sort of mirror-images of the upward-reckoned intervals), Ellis almost gave up on it. It would be a printer's nightmare and very puzzling on the music-staff. So Partch didn't attempt to supplement conventional names with three sets of additional signs.
Nobody would have understood him if he had, and so it would have hindered his composing and building instruments and teaching others how to play them. It would not have helped preserve his music for posterity.
We needn't go into the details of the special set of 43 just pitches here, since they are diagrammed in his book, Genesis of a Music, along with some hints on how to tune them.
Partch also gave many dimensions for duplicating his instruments, and instructions on how they were played, and he set for the various performance-notations for them. Actually many of these notations are tablatures.
His instructions on how to tune are of course based on his expert ear and long practice; it might be difficult or impossible for the average person to do it well enough.
Fortunately , he has a table of cents values for his intervals and related intervals. From his table another table can be constructed and used with any of the new electronic tuning-devices which are based on the ordinary 12-tone system, but which divide each ordinary semitone into 100 cents, and because they have the 12 semi-tones to begin with, will have Partch's 1/1 starting pitch as G = 392 Hz. So in this respect, his choice of 392 Hz wasn't such a bad deal! It was a lucky accident.
Otherwise there would be no hope of people living thousands of miles away, being able to use his tuning-system, and being sure they had it right.
We don't have to do EVERYTHING the HARD WAY just because Partch didn't have our measuring equipment and portable electronic keyboards and computers with software that can be used to make music in almost any scale. Some people take Partch's system as some kind of masochistic religion when he didn't want a cult. Others become highly intolerant of me and lots of other composers because we won't compose all of our music with exactly the same pitches, intervals, and instruments that he used. He didn't want any clones!
Few people will have the money, the strength and skill, the time, or the floor and storage space to copy his instruments, practice on them and maintain them and get other performers to use them also.
But why not use modern electronic means to keep his music alive and not let it be forgotten? Sure, I will be denounced for heresy; but his legacy to us is in danger of perishing unless somebody does this by the new methods.
Now that we have computers and recording machines and all kinds of sound-altering instruments, each of us can do our OWN thing.
It would be stupidity rather than loyalty to stay within his scale when our compositional talents may and should run in quite different directions. There are many thousands of extended just intonation systems possible. We don't have to stay inside his Eleven-Limit, which was imposed by practical considerations of the time in which he lived.
He couldn't modulate as much as we now can, simply because there would have been nowhere to add the needed tone; but for us those restrictions have vanished.
His instruments are incompatible with most existing music, and with much that is being done today, whether just or tempered. That is, the tones are his and not necessarily ours.
With versatile instruments and computers also, we can have his system to play his music without giving up our systems--just change the tuning-memory or something like that.
I didn't write this article merely to make you long for something you couldn't do or can't have because you are too old or have no money or haven't enough time, or have nowhere to put a bass marimba or no time to practice or learn his system or whatever.
I wrote it in the hope that some people who have access to suitable equipment can get Partch's music recorded in the new media such as cassette, DAT, CD, computer software, whatever, and also to prevent such expensive mistakes as someone doing it in the 43-tone EQUAL temperament or for that matter, thinking that Partch's tuning doesn't matter and daring to ruin it by performing it in 12-tone equal tempreament.
This last danger is very real. I have seen articles and essays flatly deny any value in Partch's tuning. There are opera-oriented people who want to present or perform or record his singing and drama, but would just as soon play all the music on a piano or other 12-tone instrument. I don't know how they can be stopped.
Why this article in this issue of the Bulletin? Because in No. 11 we are focussing upon the set of temperaments 41-43-46, and many publications have carried misinformation about Partch's music to the effect that it is in 43-equal, when in fact it is 43 unequally-spaced pitches that are not tempered at all. Also, Partch's music is definitely xenharmonic--i.e., it does not sound like 12-tone equal temperament and thus is relevant to this Bulletin.
A while ago, someone tried to imply that just intonation was not a part of xenharmonics, but it is xenharmonic in most cases. Exceptions are rare--a few cases where old music is being improved by playing it in just intonation but the harmonies are so tame or the melody so simple that the differences betweent eh just and 12-equal intervals are not clear.
The particular kind of xenharmonic effect in Partch's music is produced by man intervals involving 7 and 11 in their ratios: something completely alien tot he ordinary 12-tone equal temperament. The presence of these intervals is hard to ignore because Partch has so ordered and patterned his set of tones and the way he uses them that they occur frequency throughout a performance. Most composers in just systems use a very different array of pitches and most or even all their pitches lie within the 5-limit. Or they have a 7-limit with fewer 7s than Partch has, and no 11s. Partch's array forces him to sprinkle the music with lots of 11s and 7s, while someone else's array might not contain the very peculiar constraints which Partch built in. A few people are not trying 13 and 17. Partch discusses 13, but decided that a 13-limit would complicate his instruments too much and of course cost more.
Theoretically, the number of just pitches is INFINITE. But in the Real World he had to make do with 43 and occasionally just a few more. With modern electronics we could go into the hundreds, far more than we could hear.
Partch did not let his theory spoil his practice. There were some pitch-bends and sliding tones, for instance, not part of the theory, so in any performance, even by computer, this must be retained, in order that the affair be live instead of "dead." Purist insistence on uptight theory could spoil it.
The unusual peculiar unique timbres of the Partch ensemble are just as important as the pitches! And they are in considerable danger of being forgotten and lost as the memory of performances during Partch's lifetime recedes into the past.
Ironically enough, it may be easier to preserve the visual aspects of the instruments and performances than it will be to preserve the SOUND.
An important lesson to be learned form his system is: do not be seduced by the Standardization and Mass-Production Ideal of our society! Do your own thing and don't worry about wether or not it can be turned out in millions of copies.
In Xenharmonic Bulletin No. 9 I discussed the trio of systems 31, 34 and 36. These lie on an important boundary: where the single degree or unit-interval, viz. 1/31, 1/34, or 1/36 octave, is becoming too small to be a telling melodic interval. Systems like 17, 19, and 22 have decidely distinctive unit-intervals that work fine melodically; but when you get beyond 31, the harmonic differences for one-unit note-changes in a chord remain striking but the melodic impact of the single unit or degree is less than the harmonic effectiveness of a one-unit alteration in a chord.
Now with the 41, 43, and 46 tone systems being considered here the melodic value of one unit is further attenuated. The unit is still larger than a comma and still can make a noticeable harmonic difference.
41, 43, and 46 tones/octave are still suitable for special keyboards--and a number of designers, notably Ervin Wilson, have drawn up various plans for such keyboards. The most recent 41-tone design brought to me is by Helen Fowler of North Carolina, who labels it a Pythagorean design.
When it comes to fretted instruments, such as guitars of the standard size, problem appear. We are dealing with about one-seventh of the ordinary whole step. (42 tones/octave would be exactly 1/7 whole-step and this is one of the systems that Julian Carrillo had made up on one of his pianos.)
For people with average-sized hands and fingers, going beyond a 31-tone guitar presents problems. Remember that where tow frets are placed on a regular 12-tone guitar, one of these guitars will have about seven!
Getting down the fingerboard to the octave, the spacing is going to have to be real tight. I have had the experience of playing the 41-tone guitar that Ervin Wilson had fretted some two decades ago--he had to have the neck extended by a graft.
This in turn made it imperative to tune the guitar a major second lower; and some sets of strings would not reach far enough to put them on. However, the tone-quality was excellent. And it proved once and for all that the 41-tone equal temperament was satisfactory.
Int he coming 1990s we will have many instruments capable of these three systems (41, 43, 46 and beyond); so nothing to worry about. The MOOD of 41 is not the same as that of 31: it is more assertive and of course capable of subtlety. As we already have stated in another article, Partch's 43-tone just un-equal system is more similar to 41 than to 43 equal. The way in which intervals of the Harmonic series are approximated is more important than the number of tones per octave.
The MOOD of 43-equal is more like that of 31 than is the mood of 41.
41 is a "positive" system; that is, it has a slightly-sharp fifth. WHat this means is that a circle of 41 just fifths fails to close, and so a tiny amount--not quite half of cent--must be added to each fifth in order to make the circle close with the tempered fifths. As we are familiar without our conditioning to the 12-tone system with FLAT fifths, some people may have trouble getting used to the idea of a circle of fifths which requires ADDING something to close the gap rather than SUBTRACTING something as is necessary in 12, 19, 31, and the 43 system to be discussed later.
The 12-tone fifths error of not quite 2 cents works out to 1/614 of an octave, an amount that never could matter in any actual musical performance. The error of the 41-tone fifths is only 1/2479 of an octave, which nobody could hear and it would require some kind of digital instrument to detect it--beats of such a fifth will be very slow and frankly useless for anyone trying to tune 41-tone by ear.
If I had to face the problems faced by someone in 1870 like Ellis or Helmholtz or Janko who was interested in the 41-tone scale, there would be no point my writing about these really microtonal system and I would better have been doing something else.
But we are in the electronic and computer age, the age of precision instruments available now as never before in history. So it behooves us to re-examine these questions which music-theorists took up a century ago and didn't have the means to evaluate as we now can.
Conventional acoustic instruments like violins and horns are hardly suitable for this kind of investigation of tuning-systems that split the octave into such minuscule units. Ordinary musical notation misleads people into believing that the actual sounds produced by performing such notation are extremely precise, right on the nose. Sorry! They are not.
Conventional instruments make lots of noise. They do not stay in tune. Wind instruments are subject to continually-varying wind-pressure. Strings on other instruments keep going flat. In an animated expressive musical performance this doesn't matter very much. Wide tolerances are understood. 12-tone is such a coarse digitalization or quantization of the sound-spectrum that deviations are encouraged by all the room for bending or vibrato or sliding or smearing up or down that exists between the nominal pitch-standards of 12.
But the visual cosmetic appearance of many scores, especially those which get printed and published, and those which are quoted from textbooks, implies a totally unrealistic precision--people (particular those outside the musical professions) get too impressed with what seems to be meticulous care about microscopic details.
Some stuffy pretentious musicians and professors cultivate an image which makes them look as though they really worried over the fifth or sixth decimal place all the time.
In this age of science and High Technology it is all too easy to put that over on the public. Some of the 12-tone serialists a few years back may have been laughing up their sleeves the way people foolishly believed that a performance of their neat-looking scores must give rise each and every time to incredibly accurate and precise hard-edged sounds of a neatness never before achieved in all the centuries of music history. Of course this does not happen; but so few performances of this kind of music get heard by ordinary listeners that the illusion persists.
One consequence of this false image-making is that notation as written and printed becomes far more important than any SOUNDS whatsoever from any instruments. If the Symphony or the Concert Pianist or the Concert Managers and Music Critics refuse to perform their scores, why then the Serialists and other practitioners of Music In Cold Blood will resign themselves to exhibiting their scores as VISUAL art--and indeed the Germans have the right word for it! Augenmusik--music For Your Eyes Only, and save yourself the earache!
Visual music notation is a drastic abstraction from what is heard in a normal performance! It leaves out many things which a performance, whether live or recorded or overdubbed or done with a computer, MUST PUT IN under penalty of the sounds being unacceptable to normal listeners.
I recommend that you experience something of this sort for yourself: try playing one of the new Samplers somewhere. You do not control the sound as you would if you were preforming on the instrument that had been sampled. You are playing back somebody else's nuances. I.e., the sample at one and the same time does too little and does too MUCH!
This experience should show you how much notation leaves out.
Now in the case of the tunings dealt with in previous issues of this Bulletin, the details to be supplied when performing something are not too different from those which must be supplied when performing in orthodox standard ordinary 12.
If you are going to compose or perform in the teen scales, your room in which to bend or slide or have frequency vibrato or make leading-tone sharper as a violist does, is about the same as you have been doing in 12-tone, nearly all the way up to quarter-tone or the 24-tone temperament.
This is why I have been recommending for 20 years so that newcomers should start with 19-tone temperament, even if they are going to use something else and even if they are going to do most of their later work in Just Intonation. It just would be too "tough" for you to have to deal with tighter tolerances and special new considerations while starting to explore beyond 12.
In 19, or in 17 or 22 for that matter, you can employ most of the nuances and deviations and habits like vibrato that you always have been using in 12 and deal with just one new thing at a time--in this case the moods of your first outside-twelve scale.
Jumping from ordinary 12-tone to something like 41, 43, or 46 is far too much to expect of a composer or performer! Even the jump from 12 to 31 is too much. I recommend that those who have fallen madly in love with it, first get a little experience in 22 or 19 before practicing and composing in 31. All the more if you want to pioneer in 41, 43, or 46.
Another good point is that the mood-differences beyond 31 are less and less. The mood-contrast among 17 18 19 20 is tremendous and greatly expands the composer's vocabulary. That between 31, 34, and 36 is more subtle but still of value.
On actual trial with sustained chords and various figurations in 41, 43 and 46, the differences are much smaller and more subtle than any guesses or theories could predict. This comparison test was run with relatively brief passages in each system and not very much wait between systems.
Definitely, the difference in the moods of 31, 34, and 36 is far greater. 41-43-46 still presents some mood-change, but other factors would now be the chief reasons for choosing one or another for a particular composition, new or existing. In Xenharmonic Bulletin No. 12 we will describe the trio of systems 50-53-55 and from some experience with preliminary trials, any mood-difference is doing to be slight.
The reason for discussing mood just now is that the gradual decline in mood-contrast as one goes up the Number of Tones Per Octave should not lead anyone to discount mood down in the regions below 31. Nor should a preoccupation with Just Intonation discourage anyone from exploiting moods. In that case, the Limits and the extent of a Just array will influence mood. Sure, the number of just pitches is infinite, but the number of pitches even on a computer will not be infinite and that on some just instruments is gong to be limited by hardware factors. Also by the time we get toe 41, and certainly by the time we get to 53, very hard practical considerations limit accuracy and stability of tuning--whether live-performance instruments or computers; and in the case of recording and record-copying, the difficulties of keeping speed constant will affect systems beyond 31. Most of us will have to be content with recorded music where these many-small-interval systems are concerned and so the theoretical arguments about a Just Utopia of Infinitely Many Pitches are not realistic for real people listening to real music under normal conditions. The limitations of the human ear come first--we cannot distinguish pitches as well at the ends of the spectrum as we can in the middle.
That means we can get away with mistunings at both ends even if we must be careful in the midrange.
How many pitches per octave could we do with a computer or a special instrument such as a sophisticated synthesizer?
How about 2000 per octave in the midrange? That is attainable on a number of present-day tuning-devices. Those that could do 1000 or 1024 per octave are actually on the market now, with some restrictions as to how to get them heard.
In my Electronic Instrument Article in this issue, I was leading up to this point.
A tuning-scheme as programmed might call for strict just intonation for many pitches,but what comes out of the computer or synthesizer or other sound-producer may be slightly tempered--not enough to be audible under ordinary conditions, but yet enough to offend the mathematician and the stickler for perfection who demands exact just intonation and never mind the cost and never mind whether he/she can hear it or not.
No advertiser or a piece of music equipment is ever going to be that honest. It would be Far Beyond the Call of Duty. Don't expect it and don't demand it.
Some existing instruments and some computers will give you Just Intonation close enough for any conceivable music you want.
You've got toe be reasonable. Progress in music has been halted by zealots as much as it has been stymied by opponents. "With that kind of 'friend' who needs enemies?"
Listening to slow beats in a laboratory or reading dials on a tuning device is not the same thing as sitting in a living room or concert hall and listening to actual music.
So when we come to things like the 41-tone system, and its half-of-one-cent error in the fifth, and its audible error of nearly 6 cents in the major third, as compared to .7 of a cent for the major third in 31 but 5 cents for the fifth, and try to evaluate this situation by mathematics and abstract argument, this doesn't have as much to do with composition and use of new scales for previous music and instrument building and performance as it seems to.
There are other considerations that cannot be fully evaluated in advance. One that I discovered a long time ago when trying the 41-tone guitar was the melodic value of two units of 41.
This is something that no amount of pencil-and-paper or calculating could reveal--you have to hear 31 and 41 without too much time between them. That would lead you to choose 41 when certain melodies were to be played and when other things were to be done, then choose 31 instead.
Similar considerations with 43 & 46 if those were available. There has been too much of the Numbers Game by those who never heard new systems. I had to heard enough of many different systems before I dared write about them.
An important point is that with some computer music programs and many of the instruments that can now do non-12 or be modified so that they could, their version of just be a teeny bit off such that the errors of certain intervals in a system like 31 or 41 for that interval would not count, since they might improve the temperament or reduce the difference between that device's version of that temperament and that device's version of Just Intonation for that interval.
Now these differences will be very tiny indeed. They will hardly ever matter for a musical performance. But they will distort a tuning-device reading, or change the beat-rate of a tempered interval or cause slow beats in what should be a beatless Just interval, so that someone with good test equipment will complain about it. Or if you happen to own a frequency-counter, the readings will be "wrong" enough, above or below, to wonder if the computer program or the musical instrument is defective.
One of the synthesizers here was connected to a frequency-counter to check its pitches for the standard 12. As expected from the methods now used to secure 12-tone at the factory in such instruments, it was slightly off, above here and blow there, and some of the octaves were out, where theory called for perfect octaves in any temperament (this is not a piano, so presumably the stretched-octave practice tuners use on pianos will not apply to synthesizers). Now why where the octaves out? Why did most octaves beat when theory said they shouldn't?
Then I went through the circle of fourths and fifths that most piano and organ tuners use to set a 12-tone equal temperament. Would they beat at the rate they do on a well-tuned piano?
Nope, they didn't--most were too fast or too slow. Again, why should that be? This latter has to do with the counting-down process described on pp. 5 & 6 of our Electronics article in this issue. If the synthesizer only has to do 12, the circuitry is somewhat simpler than if it has to do a number of tuning-systems; the space between successive quotients can be larger and so fewer divisors are needed. That may or may not be the case here. What we have here is a tempered temperament!
In my Xenharmonic Bulletin No. 2 way back in May 1974, I discussed tempered temperaments. Perhaps I should do a new article on its since there weren't so many countdown circuits and special computer methods of approximating temperaments back then.
A tempered temperament is one where the big errors have little errors attached to them. It's kind of like compound interest or chain discounts or a tan upon something already subject to another tax.
Why do it? Because it saves money and it saves time,and permits the mass production of instruments which never have to be tuned--no tuner required. Actually, most of these countdown units tune better than can be done by ear.
The other question I left hanging: why are the octaves out on that synthesizer? Because when there is a slight difference from perfect octave lock-in, the general effect is not so "dead"--this is important with electronic instruments. ALso, when a given note is held, then its octave above or below is then sounded, if the octave is "dead on," it may sound like a change in timbre of the first note, rather than the addition of a new note.
Below 31, this sort of thing may be actually beneficial. It is only when we get up to large number of closely-spaced pitches that the discrepancies arising from countdown might alter the system enough to be a real concern. One advantage of the countdown-dividing method is the ease of going from one system to another, and the fact that just one stable master frequency has to be carefully controlled, instead of many.
The attraction to 41-tone, in theory, lies in its very close approximation to Pythagorean such that one-one could hear a difference, coupled with its good presentation of the Harmonic Series all the way up to the 16th Harmonic.
The possibility of having 13-limit chords on any of 41 starting-points is hardly to be sneezed at. Advocates of Just Intonation will of course complain loudly about the 6-cent errors of major and minor thirds and sixths, just as they will pooh-pooh the 31-tone system for its flat fifths. They often condemn both systems and a whole raft of others, literally without a hearing.
Polemic arguments and denunciations continue to be published in many journals in spite of the fac that with new electronic instruments and computers, having a wide range of temperaments does not rule out also having easy access to Just Intonation with a large number of pitches.
I don't need to debate the matter because the instruments in this studio prove my point right on the spot! Furthermore, cassettes sent me from distant places prove that others have the same ability with today's facilities.
If some Nostalgiac wants to do everything the hard way by using only acoustic instruments, of course I can't intervene to stop them. That's THEIR problem! Not mine at all. I have never been a member of any masochistic cult religion and I don't see why music has to be such a cult either.
The tables have turned: it is now cheaper to use the latest equipment for new scales than to do it the 1880 or 1920 way. If the cost of exploring beyond 12-tone were still at the 1935 or the 1945 level, I would not be getting out Bulletins
like these, since it would be cruelto increase your sadness, hopelessness, and frustration. Besides that, my assorted spoilsports and enemies and nasty mean opponents would be sneering at me for even discussing systems like 41, 43, 46, or Just Intonation! They would win all the argument and leave me in Black Despair.
Now on to 43: so far as I know at present, there are no dedicated 43-tone instruments in existence, but for quite some time there have been experimental instruments with 12 tones of 43 or what is practically the same thing, the 1/5-comma Meantone System. Experimental instruments do exist that have 16 or more tones of 1/5-comma meantone. Now if only that many of the pitches are made available, no audible discrepancy between 43-equal and 1/5-comma meantone would be heard in any musical performance, so it's not worth using space to discuss here. "1/5-comma" refers to how flat the fifth is--4.3 cents as against some 5.2 cents for 31 and quarter-comma meantone (the standard kind).
Would is have been worth the trouble to have built 43-tone instruments with all the tones? I think not. If we didn't have this new power on computers and synthesizers of changing from one system to another quickly, I would have omitted 43 from this article. While it has usable representations of higher harmonics, such as 11 and 13, and practically-perfect major seventh (who is going to quibble over a tenth of a cent, one-twelve-thousandth of an octave?), financial considerations alone would have dictated dedicating instruments to cheaper systems with fewer tones, that still could afford new chords and melodies.
--And a 43-tone guitar? Why the UNAVOIDABLE errors of fretting and of the strings you put on it would wipe out much of the difference between that and other systems.
As Jonathan Glasier was saying recently, "Do you actually enjoy putting your fingers into the pencil-sharpener in order to get between the frets?"
The main reason why I had to include 43-equal here is not the historical business of 43-equal or 1/5-comma authentic performances of certain old keyboard and organ music, but rather the serious confusion between the 43-tone equal temperament and the 43-tone 11-limit just system devised by the late Harry Partch, and quite different in its effect.
Too many magazine and newspaper articles have said "Harry Partch's 43-tone equal-temperament" and it is extremely important to point out that Partch's system was untempered and UNEQUALLY-SPACED through the octave. This is my reason for putting this article in the same issue as the other article on PARTCH'S PITCHES. I can append frequency-tables of both and thus anyone can see the difference.
Many people who now have computers can have BOTH systems on disk and compare their sounds with certain new software that would permit this. Hopefully this will get rid of the shameful confusion that has been repeated in print far too much: 41, not 43, is the closest equal temperament to Partch's system unless you are willing to go to very large numbers of tones. But if you go that far, why temper?
Now for 46: I never heard 46 till this week. February 1989. All I knew about it was from Ervin Wilson's experiments related to me sometime ago when he had a guitar temporarily fretted to 23 and tuned two strings 1/46 of an octave apart to get 46-tone out of it.
The fifth and major third of 23 are so bad that 2 x 23 = 46 simply had to be better. Much as the impossible thirds and sixths of 17 suggest using 34, which has excellent thirds and sixths.
Trying out harmonic-series chords in 46, they are satisfactory enough to make the system interesting, now that it won't cost extra to use it if one has the new adjustable or reprogrammable equipment for other beyond-12 scales.
Also, its harmonic-series performance would be better than that of 48-tone, which has been in the music-theory literature as one of Julian Carrillo's scales for which a special piano and fretted instruments were built, and which also is available for some new electronic instruments.
(In future issues we can deal with Carrillo and other exponents of the multiples-of-twelve and multiples-of-six ideas.)
There is no point in inventing or using special notations for scales with so many tones as these. I've said this before but it needs repeating: progress is RETARDED or even HALTED by all the mental gymnastics required to compose in such scales while juggling three or even ten different kinds of notations with multiple meanings for the same accidentals and with too many rival notations where the same accidental in the same system has multiple meanings. The quartertone situation is a public disgrace!
To compare a group of systems like 42, 43, 46 [and 48], it has to be possible to try over IN SOUNDS this or that chord or melody, and laying a notation problem on top of that would simply stop composers dead-cold! If you must, number the tones as Carrillo did.
The new device called the sequencer will play automatically and solve the problem of finger-technique and strange new keyboards--now you won't have to learn them! We didn't mention scales like 42, 44, 47, 49 -- well, they are not important enough to invest in, and if they intrigue you, a computer will take the hassle off your back and save loads of money and time. One reason why I started XENHARMONIC BULLETIN was to find out and tell people which scales are more useful.
In connection with 43-tone equal temperament, we should mention the French theorist Sauveur, who wrote about it. As a finer division of the octave for paper calculations, he also proposed 43 x 7 = 301. 301 happens to be also the first three digits of the common (base 10) logarithm of 2, so in the old days when people had to use log tables, it was easy to get intervals expressed in terms of 301st of on octave. Cf. the French units named for Savart.
The Hungarian theorist Paul Janko, who also designed a whole-tone-scale keyboard for pianos and other instruments that carries his name, wrote about the 41-tone equal temperament and more about him can be found in some reference books.
Since all back issues of Xenharmonic Bulletin are available on order and will continue to be available, it was not necessary in this Issue Number 11 to keep referring to the other tuning systems such as 5, 7, 10, 13...19, 22...31 etc.: and this may mislead some readers into thinking that we don't consider those other systems so important. This is not the case at all! There are tuning-tables for those systems in back issues of the Bulletin and elsewhere. We have described the new moods of those systems and their relative values, and which ones will be most useful and which others will not be quite so important unless some composer comes up with a new unheard-of idea.
The long gap of about seven years between Issue No. 10 and Issue no. 11 may also raise a few eyebrows. Actually there are dozens of Darreg publications and articles in other journals as well as leaflets and booklets and more recently, and indeed most important, recordings of music in new scales now available. This is more a question of naming publications than any real separation between the eleven Xenharmonic Bulletins and the other writings.
Plans are to issue a book of Tables, where all the different kinds of tables would be gathered together, both published and unpublished. Other persons publishing tables include John H. Chalmers, Jr., Ervin M. Wilson, and Buzz Kimball. Information about their tables on request, and how to order from them.
The practical use of scales beyond 31 tones per octave will usually involve sequencers, synthesizers, and computers so we don't have to cover the subject of how to tune Partch's system, 41-, 43-, or 46-equal, or other systems beyond 46 by ear. This just is not necessary anymore, and who in these hectic days would have the time? Only a masochist would want to undertake such tunings by ear at the present time! You would be so tired out and mentally exhausted that you wouldn't want to compose any music for such many-toned scales nor to perform existing music in them. The Xenharmonic Movement is, we repeat once more, not a religion of self-torture, but a broadening and expansion of composers' and listeners' experiences. Let the new time-saving tuning-devices and facilities do the mind-numbing drudgery for you! That is what has kept non-twelve music from being practical until now. Well, now it's easy enough. Enjoy!
