editing disabled

The field of microtonality is rife with colorful personalities and diverse perspectives, and there are many contradictory philosophies and approaches. However, the literature on microtonality in general seems to over-represent certain perspectives, and this page is intended specifically to represent some of the views that diverge from the more "mainstream" or traditional ideas about microtonality.

Chuckles McGee's "Six Misconceptions of Novice Microtonalists"


This section expresses the views of Chuckles McGee, a famous clown with a bright red shining nose. He is an outspoken advocate for many relatively-discordant tunings (and for the importance of discord in general).

Misconception 1: "More is More"

When you begin in microtonality, usually the first thing that happens is you discover there's a huge variety of potential new consonant intervals, which are not represented in 12-TET. It's only natural to want to try out all these new intervals, but it's a mistake to begin with a humongous tuning system that approximates as many of these new intervals as possible. If you're accustomed to (and proficient in) 12-TET, you are going to have a mental melt-down if you go straight for something like 31-EDO or 41-EDO or triple-BP or something. The sheer variety of new possibilities will short-circuit your brain. The truth is that nothing you have learned in 12-TET has prepared you to juggle such a plethora of new harmonic possibilities, and you will probably try to do a million things at once--and this will probably sound terrible, and you will probably be disappointed, and you will probably lay your shiny new microtonal instrument aside and go back to what's familiar. It is better to begin small--absolutely no more than 24 notes if an equal temperament, and definitely no more than 12 if JI (since JI is so drastically different than equal temperament). You can always expand later, and you will find the novel sounds of smaller systems more accessible and more immediately rewarding. I speak from experience, as someone who did begin immediately in 31-EDO, and have been retreating to smaller and smaller EDOs ever since, lamenting the loss of thousands of dollars and countless hours to tuning systems that looked good on paper but were difficult and confusing (and therefore disappointing) to work with. If I could do it all over again, I would start at 13-EDO or 14-EDO and not read a word of theory until thoroughly accustomed to a single new system. As far as I have seen, nobody in this century who has begun with 31-EDO has been satisfied enough with it to stop their exploration right there and spend the rest of their life in that one tuning; they either move on to JI, or down to the smaller EDOs, or up to the more accurate temperaments (41, 53, 72, etc.)...or they give up on microtonality all together, thinking that if the "best" system didn't work out for them, what hope is there?

It's also true that a JI system can produce a drastically larger palette of intervals than an equally-sized equal temperament. If you are hell-bent on exploring all the intervals within the 15-limit tonality diamond, do not pass go, do not collect $200, do not touch 31-EDO, but go straight to the harmonic series, specifically the scale of harmonics 8-16. In one 8-note octave-repeating scale, you will find all the intervals in the 15-limit tonality diamond (which is A LOT of intervals), although most only occur at one place in the scale. You should absolutely become fluent with the sound of these intervals in this scale before you consider trying out a temperament based on them. Then bump it up to harmonics 16-32 to see what some of the even more exotic identities feel like. Then, and only then, are you ready to start looking at high-limit temperaments. The sky's the limit once you get your sea-legs, but you must get those sea-legs first!

Misconception 2: "Consonance is Rare"

Consonance is not rare at all. In fact it is omnipresent. Especially in the higher ETs, maybe 24-EDO and above, it is almost impossible to find a tuning that is not at least as capable of consonance as 12-TET. Even among the smaller EDOs, it is almost universally true that each one approximates some consonant subgroup of Just Intonation with the same or greater level of accuracy that 12-TET has in the 5-limit. With a little care, all of these EDOs can be made to sound nice enough for the tastes general public. Yes, even 10, 11, 13, and 14-EDO. In fact, even 8-EDO does a fairly passable approximation of harmonics 10:11:12:13:14 as 0-150-300-450-600 cents; it's not great, but it's awesome for such a tiny EDO--no interval is off by more than 18 cents, which is more or less as good as 12-TET.

No, consonance is ubiquitous, practically inescapable unless you insist on using ridiculous scales like 0-10-20-30-40-50-1300 cents repeating every 1300 cents (or something). The strength and quality of consonance may vary from tuning to tuning, but there is nearly always enough to serve effectively as contrast to the equally-ubiquitous dissonance, if only you take the time to understand what the contrast is and how to deal with it appropriately. Sometimes the most consonant harmonies look nothing like major and minor chords in 12-TET, so they can take some searching. But they are almost always there to be found if you know how to look.

It is true that accurate approximations of the 5-limit (let alone the 7, 11, or 13-limit) are rare among small tunings. This should not be surprising, considering that the octave-equivalent 13-odd-limit tonality diamond contains 42 intervals. But consonance does not require the full 13-limit, and subgroups of the 13-limit are plentiful.

Misconception 3: "Tunings Related to the Familiar are Easier to Learn"

Tunings related to the familiar, like 17, 19, 22, 26, 27, 29, and 31-EDO, are easy to learn--if what you want to learn is how to make familiar-sounding music! These tuning all support many of the same patterns and relationships that work in 12-TET, so at first blush it is dead-simple to apply those same patterns and make nice-sounding music. The problem is that this music will not sound a whole lot different than what you're used to. If you want to make music that doesn't just sound like a mild retuning of the same old diatonic cliches, these systems are all a greater challenge than less-familiar ones, because the strong pull of the familiar is difficult to escape from. The truth is, the familiar diatonic scale is about the sweetest-sounding thing in music, as are the familiar 5-limit consonances. When an instrument gives you the choice between familiar and sweet or unfamiliar and sour, it is hard to make yourself choose the latter. Odds are you will keep coming back to those familiar patterns, because they sound nicer and are easier to play--it's as if the instrument is rewarding you for being conservative and punishing you for trying new things.

On the other hand, if you begin with a tuning that lacks the familiar diatonic structure and the familiar 5-limit consonances, you will have no choice but to find something new. At first it may seem that the learning curve is steeper, because your old habits are being thwarted at every turn, but what is actually happening is that the tuning is teaching itself to you. When you don't find sweet sounds in the old familiar places, you have to look in new places, and they are more rewarding when you find them; being "rewarded" helps you stay motivated to continue learning. Eventually the old habits will die, new ones will take their place, and you will be effortlessly making music that sounds new and good.

Misconception 4: "Tunings Related to the Familiar are More Appealing to the 'Average' Listener"

I know how it feels at first. You've gotten your first taste of microtonality and you think it's the greatest thing since sliced bread, but your friends, family, and fellow musicians are totally NOT sharing your enthusiasm. They are utterly failing to grasp why in the hell you would want to split off from the rest of the civilized 12-equal world and play music that "only aliens would like". So you start to have your doubts, and you start thinking that maaaaaybe instead of starting out with something wildly unfamiliar like Miracle temperament or 16-EDO, maybe you should be "easing people into it" with something like extended 7-limit Meantone or 19-EDO, even though what really got your motor running for microtonality in the first place was the really crazy-sounding new stuff. You think that if you can show people, "look, I can still play 'Smoke on the Water' or 'Moonlight Sonata', microtonality doesn't have to sound like alien music!" that this will turn their aversion into fascination and then they will eagerly join your cause.

Guess again, grasshopper. There's a big and obvious problem with this way of thinking. The truth is that people who want to listen to and/or make familiar-sounding music are not going to find anything very compelling about microtonality, because if you are going to make a new and more complicated tuning sound totally normal...what's the point of switching? It's like trying to sell ice-makers to the Inuits--why would they want to spend money for a machine that makes ice cubes when there's snow all around them? It's all cost and no benefit, because all they want is stuff that sounds familiar...and there's nothing more familiar than what they already have.

The truth is that lots and lots of people don't give two turds about microtonality, no matter what form it's in. The only way to win them over is to make music they like that is impossible to approximate with something familiar, and even then the odds of them actually becoming "converts" is practically non-existent. On the other hand, the people who are interested in microtonality want to hear music that sounds, well, microtonal. Sure, there are lots of ways to sound microtonal and not all microtonalists like all forms of microtonal music, but none of them want to hear music that sounds like 12-TET if you promise them something microtonal. There's nothing at all wrong with playing in 12-TET, and most microtonalists still listen to tons and tons of music in 12-TET (it's hard not to!), but you've really got to ask yourself if it's worth the trouble of refretting a guitar, or buying a retunable synth, or learning new fingerings on the bassoon (etc.), if what you really want to do is make music that doesn't actually sound much different.

Misconception #5: "Beatless Harmonies are More Relaxing" and "12-TET Music is Fast Because it's Out of Tune"

This is one that really gets me. Yes, it is true that if you try to play a piece of symphonic music written for meantone tuning in something like 13 or 23-EDO, the results will be harsh, unsettling, and generally nasty, and if you play the same piece in adaptive Just Intonation, it will be much more "restful". Many conclude from this that beatless harmonies are thus inherently more "restful" than those that beat...but this is a regrettable example of wrongful inductive generalization.

The more correct conclusion suggested by this observation is that what determines the amount of "restlessness" a musical stimulus will induce in a normal listener is the sheer volume of psychoacoustic and musical information present. A little bit of information is boring but not unpleasant--think the single drone of a tambura or the hum of a refrigerator--and an overload causes the cognitive faculty to shut down and let the stimuli blur into pure noise--which is also, coincidentally, soothing, at least if it's near pink or brown noise. So at either extreme of the spectrum--monophonic drone vs. noise--we have a sort of soothing "dullness". As we edge away from the drone, the informational content increases, and we develop interest; this can take many forms, be it monophonic melody or subtly shifting overtones or harmonic textures and what not. At some point--a point which is very much listener-dependent--interest (and thus pleasure) peaks, and further increasing the informational content becomes confusing and decreases pleasure. At some point (also very listener-dependent), pleasure becomes negative; this is usually the point where the information is as high as it can get before it becomes totally unintelligible, i.e. before it comes to be heard as pure noise.

Now, as I said, there are many ways to increase the informational content of a piece of music. One of them is to decrease the concordance of the intervals, as this introduces beating and increases harmonic entropy. Another one of them is to increase the level of compositional complexity, i.e. to increase the number of pitches being heard within a given time-frame. The implications of this should be obvious: to maintain a constant level of interest, compositional complexity ought to vary inversely with harmonic concordance of intervals being heard. In other words, music that is "out of tune" will be more pleasant if it is slower, not faster.

If one looks to the meditative traditions of the world that use sound to help enhance meditation, the most common sounds are gongs, bells, and group chants (usually monophonic or even monotonic). The clear trend between all of these is "harmonic impurity", i.e. beating. Most bells produce inharmonic spectra where the overtones actually beat with one another, as do most gongs, and a room full of people chanting the same mantra or hymn will never be in perfect tune--there will always be some amount of beating. In modern times, the phenomenon of "binaural beating" is well-known and quite popular as a method of inducing relaxed states. One thing that is conspicuously absent from all meditative or trance-inducing sounds is beatless harmony played by pure harmonic timbres. Try this experiment for yourself: listen to 10 minutes of a Justly-tuned pipe organ sustaining a 4:5:6:7:9:11 hexad, and see just how relaxed it makes you feel! The truth is, given a static sustained harmony, one that beats is more relaxing than one that doesn't.

Of course, if you want to write music with lots of melodic and harmonic complexity, then by all means, go with the more near-Just harmonies. The spectrum of musical interest is broad and deep, and the most important thing is to develop a sensitivity to the sorts of music appropriate to the tuning you're working with.

Misconception #6: "You Should Listen to the Advice of Experienced Microtonalists"

The world of alternative intonational practices (also know as the world of microtonality or xenharmony) is vast, and when you first enter it, it's easy to be overwhelmed and confused and maybe even terrified. Especially when there are all sorts of people and organizations making all sorts of grandiose claims about this tuning or that tuning. There is almost no consensus, and there is a TON of rhetoric, much of it based on questionable scientific studies or historical sources. In your initial confusion, it will be almost impossible to resist the siren song of someone's particular brand of microtonality, because you need to start somewhere and it's hard to figure out this stuff on your own. The rhetoric is occasionally made all the more enticing by the existence of compelling music based on the advertised tuning system; after all, if composer X could make such cool sounds with this tuning, then surely it must be a great tuning for anybody!

The truth is that no one is qualified to tell you what tunings to you use, because different tunings are good for different things and it's almost impossible to know what you want from a tuning when you've spent your whole life immersed in a single tuning. You will have no idea at first whether harmonic properties or melodic properties are more important, or whether you work better in equal tunings or unequal tunings, or whether you think better in terms of frequency ratios or cents or note-names, or whether you think beating and discordance are really as undesirable as some people say they are. In all probability, no matter what alternative tuning you begin with, you will find something unsatisfactory about it and seek out alternatives. There are very, very few microtonalists in the world who have picked a single tuning system and stuck with it, and just because someone has spent a lifetime trying out every tuning under the sun and finally settled on one, that does not mean you will get the same results. Tuning is a very personal choice, and can be a very deep and enlightening personal journey. Do NOT trust anyone who tells you it has to be a certain way. In fact, don't even listen to me. Every word ever written about microtonality needs to be taken with a heap of salt, including mine.

Now go forth, make mistakes, and learn from them!

Myths and Facts about Xenharmonics by mclaren

(written for this wiki)

Myth #1: "Everyone prefers the natural intervals of the pure perfect harmonic series." This myth remains pervasive. It has been stated by Hermann Helmholtz, in the form "instrumentalists naturally tend to play in the intervals of just intonation." This myth was also repeatedly stated by Harry Partch, who claimed "The ear demands small integer ratios, and accepts substitutes against its will."

These myths have been debunked for well over 80 years. In the 1930s, the music psychologist Carl Seashore first investigated the actual intonation of violinists and other Western performers. He found that they played intervals which were neither just (i.e., small integer ratios) or equal divisions of the octave, but something entirely different. Typical intervals performed by trained Western symphony-caliber musicians are neither just nor equal-tempered. The intervals performed often differ wildly from the putative size of the musical intervals which should be played, yet audiences typically hear these distorted intervals as sounding "perfectly in tune." See Seashore, Carl, Psychology of Music, 1936.

In 1961-2, physicist Charles Shackford published three articles in the then-new Journal of Music Theory examining the intervals actually performed by symphony musicians in live concerts. He found that the performed intervals typically differed by at least 10 cents from the target intervals, and often differed by up to 50 cents -- yet listeners were unable to hear any problem with these distorted intervals. To audiences, these extremely out-of-tune intervals sounded "perfectly in tune" and "entirely musical." See “Some Aspects of Perception, I: Sizes of Harmonic Intervals in Performance,” Shackford, Charles, Journal of Music Theory, Vol. 5, No. 1, 1961, 162–202; also “Some Aspects of Perception, II: Interval Sizes and Tonal Dynamics in Performance,” Shackford, Charles, Journal of Music Theory, Volume 6, No. 1, 1962, pp. 66–90, and "Some Aspects of Perception III: Remarks," Shackford, Charles, Journal of Music Theory, Volume 6, No. 2, 1962. Shackford's work builds on earlier studies which reached the same conclusions: see Paul C. Green, “Violin Intonation,” Journal of the Acoustical Society of America, IX (1937), 43–44; James F. Nickerson, “Comparison of Performances of the Same Melody in Solo and Ensemble with Reference to Equal Tempered, Just, and Pythagorean Intonations,” Journal of the Acoustical Society of America, XXI (1949), 462; idem, “Intonation of Solo and Ensemble Performance of the Same Melody,” Journal of the Acoustical Society of America, XXI (1949), 593.

This research has been confirmed by many subsequent listening experiments. Psychoacousticians have shown that listeners typically cannot hear a difference between pitches less than 15 cents larger or smaller than their nominal values in a real performance (see "Intonation precision of choir singers, " Sten Ternström and Johan Sundberg, Journal of the Acoustical Society of America, Volume 84, Issue 1, 1988, pp. 59-69), and that all musical performers across all cultures (non-Western performers in India, for example, as well as Western symphonic musicians in Europe/North America) tend to perform large musical intervals of the size of a minor third or larger as bigger than they should be (often between 5 to 10 cents larger), while performing small musical intervals the size of a major second as smaller than they should be (typically compressing a whole tone which should be 200 cents to a value as small as 170 cents or smaller) and compressing semitones even more, typically by at least 30 cents (so that semitones, particularly those resolving downward from a supertonic to a tonic or moving upward from a leading tone to a tonic, are often measured with values as small as 70 cents or 60 cents or in some cases even 50 cents or less). See "Intonation Variants of Musical Intervals in Isolation and in Musical Contexts," Andrzej Rakowski, Chopin Academy of Music, Okolnik 2, 00-368 Warszawa, Poland; "Musical Intervals and Simple Number Ratios," Cazden, N., J. Res. Music Educ., 7, pp. 197-220; "Influence of the Time Interval on Experimentally Induced Shifts of Pitch," Christman, R. J. and Williams, W. E., J. Acoust. Soc. Am., 35, pp. 1030-1033, 1963; "Further Investigation of the Effects of Intensity Upon the Pitch of Pure Tones," Cohen, A., J. Acoust. Soc. Am., 33, pp. 1363-1376, 1961; "Pitch relations and the formation of scalar structure," Cross, I., R. West and P. Howell, Music perception, Vol. 2, No. 3, 1985, pp. 329-344; "The influence of pitch on time perception in short melodies," Crowder, Robert G. and Ian Neath, Music perception, Vol. 12, No. 4, 1994, pp. 379-386; "Grouping in pitch perception: evidence for sequential constraints," Darwin, C. J., R. W. Hukin and Batul Y. Al-Khatib, Journal of the Acoustical Society of America, Vol. 98, No. 2, part 1, 1995, pp. 880-885; "Some Observations on Pitch and Frequency," Davis, H., S. R. Silverman and D. R. McAuliffe, J. Acoust. Soc. Am., Vol. 23, pp. 40-42, 1951; "Mapping of interactions in the pitch memory store," Deutsch, Diana, Science, Vol. 175, 1972, pp. 1020-1022; "The Processing of Pitch Combinations," in D. Deutsch, (Ed.), The Psychology of Music, New York: Academic Press, 1982;
"Interference in memory between tones adjacent in the musical scale," Deutsch, Diana, Journal of Experimental Psychology, Vol. 100, No. 2, 1973, pp. 228-231; "Music perception," Deutsch, Diana, The Musical Quarterly, Vol. 66, No. 2, 1980, pp. 165-179; "Dichotic listening to musical sequences: Relationship to hemispheric specialization of function," Deutsch, Diana, Journ. Acoust. Soc. Am., Vol. 74, 1983, pp. 579-80; Dowling, W. J. "The 1215-Cent Octave: Convergence of Western and Non-Western Data on Pitch Scaling," Abstract QQ5, 84th meeting of the Acoustical Society of America, Friday, December 1, 1972, p. 101 of program, and so on.

Moreover, in 1986, Linda Roberts along with John R. Pierce and Max Mathews published a study in which they investigated the actual listening preferences of musical audiences. They found that presented with a choice, 8 out of 9 listeners preferred musical intervals which beat, while only 1 out of 9 listeners preferred musical intervals which were beatless. As in, for example, perfect fifths or major thirds, etc. Roberts, Pierce and Mathews referred to the first group who preferred musical intervals which beat as "rich listeners" because these listeners preferred tunings which made the music sound "rich" and "lively" with a plethora of active beats. The second group Roberts, Pierce and Mathews referred to as "pure listeners" because they preferred beatless major and minor thirds, beatless perfect fifths and perfect fourths, and so on. The interesting fact about this study is the lopsidedly bimodal nature of the distribution. Rich listeners far outnumber pure listeners. See "Harmony and New Scales," M. V. Mathews, J. R. Pierce and L. A. Roberts, in Harmony and Tonality, ed. J. Sundberg, 1987, pp. 59-84.

Notice that these studies present no aesthetic preference. They do not tell us that rich listeners are "better" or "more discerning" than pure listeners. These studies merely inform us that rich listeners outnumber pure listeners in Western musical audiences by a ratio of roughly 8 to 1. There is no indication that musical tunings which produce more beats are any better or any worse than musical tunings which produce fewer beats (just intonation with small integer ratios). As Warren Burt put it, "I don't hear small integers ratios as sounding any better than intervals which beat. I hear a difference -- I simply don't acknowledge that the difference produces any aesthetic superiority." Or, as William Schottstaedt, arguably the greatest living American composer, put it: "I like beats. Beats sound good."

Myth #2: "The small integer ratios like 3/2 and 5/4 are the original intervals from which all other musical intervals are derived." Kyle Gann teaches this provably false claim in his course on microtonality. (Gann's discussion of microtonality is generally scrupulously accurate: this offers a rare exception. See Gann, Kyle, "My Idiosyncratic Reasons for Using Just Intonation" for one of the best explanations of why composers may find just intonation useful.) Or, as Lou Harrison put it, "Just intonation tunings are the only real musical intervals. All other musical intervals are fake musical intervals."

The actual evidence of peer-reviewed published listening tests in the psychoacoustic literature show that there exists a wide range within which listeners recognize musical interval categories like "fifth" and "third" as sounding functional and musically effective. Once again, this has been known for more than 80 years, and documented in a wide variety of peer-reviewed scientific papers going back to 1926.

In "Variability of judgments of musical intervals," Moran and Pratt, Journal of Experimental Psychology, Vol. 9, 1926, pp. 492-500, 1926, researchers found that the range of recognizable musically effective and musically functional intervals ran from a low of 680 cents to a high of 720 cents for the perfect fifth. This conclusion has been confirmed and more supporting evidence piled up by many subsequent listening tests.

Moreover, this conclusion is also supported by ethnomusicological studies which show that worldwide non-Western cultures tend to use a plethora of unequally spaced (or sometimes quasi-equal-spaced) 5- and 7-tone musical scales, with fifths ranging from roughly 680 cents on the low side to 720 cents on the high side.

This conclusion is also supported by the historical record of tempered tunings, which have used perfect fifths as low as 685 cents and as high as 705-710 cents.

More recently, in 1978 Easley Blackwood proclaimed the excellence and musical value of the 15 tone equal tuning, with its 720 cent perfect fifth. Ivor Darreg also concurred in his Xenharmonic Bulletins in the 1970s and 1980s, and Wendy Carlos chimed in to give her enthusiastic support to the 15 equal tuning. Blackwood, Darreg, Carlos and many others have composed notable pieces in the 15 equal tuning, and listeners have founds its 720-cent perfect fifths lively and vividly musical. Likewise, Ivor Darreg and many others have enthusiastically spoken in favor of tunings like 7 and 14 equal, with 685.4-cent perfect fifths. Ivor in particular boosted 14 equal because of its freshness combined with its memorable and impressive musical "mood."

All this evidence converges on the conclusion that within a wide range of about 20 cents lower than, to 20 cents higher than, the just 3/2, perfect fifths sound recognizable and musically effective in actual music. The claim that small integer ratios like 3/2 represent the only real musical intervals that listeners prefer is so far the opposite of the documented facts that the opposite is actually true. As Erv Wilson succinctly put it, "Musical cultures around the world tend to systematically avoid the intervals of the harmonic series." (Wilson, E., personal communication).

Myth #3: "All music derives from harmony, and thus the pure prefect intervals of the 4:5:6 triad are the basis on which we must build musical tunings." Western musical analysis reinforces this misconception by doing an analysis of music which almost entirely boils down Western music to series of harmonic progressions. The pseudo-scientific claims of Schenker reiterate this claim, stripping music down a series of urlinie which amount to little more than harmonic progressions.

In reality, melody proves far more important in music worldwide than harmony. Most of the world's musical cultures do not use triads and have no interest in musical harmony. Most of the world's music has nothing to do with triads, and well over 80% of the world's musicians do not think of music in terms of harmonic progressions. Indeed, the vast majority of the world's musicians and composers have no interest in harmonic progressions at all. Ancient cultures like the Greeks were well aware of the possibilities of producing triads: they simply had no interest in doing so.

The use of triadic harmony and triadic harmonic progressions, far from being a universal basis for music, in reality qualifies as a bizarre fringe case -- a rare exception. We find it only in North American/European music, and then only within a very limited time period (roughly 1490 to 1910). Before that time period, triads and triadic harmonic progressions are simply not used, even in Western music. And later than 1910, triads get used in Western music intermittantly -- tone clusters (Xenakis, Ligeti, Pendercki, Ives, Cowell, et al.) and sound-masses are used at least as much as triads after 1910, and heterophony and dense dissonant counterpoint are used at least as often as triadic chord progressions after 1910 even in Western music.

Myth #4: "We must match the tuning to the timbres, so that harmonic series timbres play music written in the pure perfect natural intervals of the harmonic series." The effects of acoustic roughness and smoothness do depend on the degree to which timbres match tunings. For example, as John R. Pierce and Max Mathews first showed in their article "Attaining Consonance in Arbitrarily Musical Scales," in the book Music By Computers, ed., C. Beauchamp, 1969, and as was further developed by composers like James Dashow and William Sethares (see Sethares' book Timbre, Tuning, Spectrum, Scale," Elsevier, 1992), the familiar effects of acoustic points of rest (relatively beatless intervals) and acoustic points of tension (intervals within roughly 1/4 of the critical band which beat at the circa 30 hz rate first identified by Helmholtz as maximally disturbing) only exist when timbre approximately matches tuning. Bach played on a carillon, for example, sounds confusing, because the normal points of acoustic rest and acoustic tension fail to fall in the places we expect.

However, the experience of composers and audiences since 1969 has shown that musical audiences seem to prefer a wide range of timbres in musical compositions. Digitally modifying timbres so that they perfectly match the musical tuning tends to produce bland-sounding excessively vocoder-like timbres which leave audiences restless.

In fact, the history of modern music post-1970 shows that percussion ensembles have become increasingly prominent in contemporary music. These percussion ensembles typically use inharmonic timbres which utterly fail to match the 12-equal tuning, yet audience love the music produced by these percussion ensembles. The answer to this seeming conundrum is that audience crave variety. We like to hear compositions in which some of the timbres match the tuning, and in which some other timbres clash with the tuning. As with food, eating the same thing all the time day after day makes you sick. You get tired of it. In the same way, musical repasts which feature nothing but harmonic series timbre after harmonic series timbre perfectly matched to the musical tuning quickly grows dull. Audiences get restless. They want some variety, not the same bland vocoded-sounding hum all the time.

Myth #5: "Mathematics forms the basis of music, and therefore mathematical music theory must guide us when we create new tunings." As Paul Hindemith noted in 1937, "Theorists, basing their reasoning on acoustical phenomena, have repeatedly come to conclusions wholly at variance with those of practical musicians." (Hindemith, P., The Craft of Musical Composition, Vol. 1, 1937.) The human ear/brain system stands between the acoustic wavefronts of musical instruments and the music as we perceive it. Our human sensory apparatus and our cognitive processes are highly non-linear and subject to a wide range of cognitive biases. See "Judgement Under Uncertainty: Heuristics and Biases," Tversky, A. and Kahneman, D., Science, new series, Vol. 185, No. 4157, September 27 1974, pp. 1124-1131. Also see Gestalt Laws of Perceptual Organization and "The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information," by George A. Miller, The Psychological Review, 1956, vol. 63, pp. 81-97

We do not hear frequency; rather, we perceive pitch. We do not hear amplitude; rather, we perceive loudness. We do not hear the wavefronts of series of atmospheric compressions or rarefaction; rather, we perceive music. And our perceptions find themselves subject to a vast multiplicity of distortions and cognitive limitations. See Music, Cognition and Computerized Sound: An Introduction to Psychoacoustics, MIT Press, ed. Perry Cook

It has been shown for more than 80 years that our perception of pitch depends on loudness, and contrariwise that our perception of loudness is greatly dependent on pitch. In the first case, it was shown by Harvey Fletcher back the early 1930s that our perception of the pitch of a sound tends to drop with its amplitude, and a louder sound in the mid-high range can be as much as a minor third lower, depending on its loudness. See Fletcher, H., "Loudness, pitch and the timbre of musical tones and their relation to the intensity, the frequency and the overtone structure," Journal of the Acoustical Society of America, Vol 6, 1934, pp. 59-69. In the second case, see the well -known Fletcher-Munson curve, which all audio mixing engineers must take into account. (The Fletcher-Munson curve tells us that low frequency sounds must be greatly boosted in amplitude to sound as loud and mid-range to high frequency sounds. Recording engineers deal with this problem by using compression or hard limiting in musical recordings. Live concerts deal with this well-known problem by placing the bass-heavy instruments closer to the audience and the instruments with higher tessitura farther away from the audience, as a kind of physical equalization curve which decreases the apparently amplitude of high-pitched instruments while enhancing the bass of the low-pitched instruments.)

Mathematics has consistently failed to predict the musical effect or musical utility of new musical tunings. As Ivor Darreg pointed out, "It is absolutely impossible to imagine the sound or mood or a new tuning. You have to hear it. Only then can you imagine it."

The systematic musical failure of mathematically-based methods of musical organization, like total serialism, which refused take the characteristics of the human cognitive system and the human auditory system into account, also leads us toward the conclusion that mathematics fails as a basis for creating new tunings.

"There has been so much theory, so much mathematical speculation about new tunings, and what they failed to take into account is that there is no such thing as a bad tuning, there is no such thing as a useless tuning. Every tuning has its musical uses." -- Ivor Darreg, personal communication.

An overwhelming mountain of evidence from many different fields converges on the conclusion that the only valid way to explore microtonality is by means of experience-based knowledge. As music history shows, composers do weird bizarre things for years, then the theorists belatedly catch up. When theorists try to lead and predict what will prove musically effective, they typically fail.

Xenharmonics offers such a completely novel field of musical exploration that the only reasonable way to press forward involves hands-on experimentation. This is, in fact, the scientific method: the universe typically proves too complex for us to reason our way to correct conclusions sans hands-on experience. We must try things out, make observations, and then compare our observations of the real world with our mental models in order to gain useful knowledge.

Myth #5: "Microtonality produces great theory and bad music." -- Brian Eno.

The internet abounds with information about microtonality and xenharmonic, essentially all of it provably false. In contemporary music as in foreign affairs and economics and most other realms of daily life, those who talk don't know, while those who know don't talk.

Pournell's Iron Law of Bureaucracy states that any institution will tend to harbor two kinds of the people. The first are the people who actually do the work that pushes things forward. The second group are those those who excel in the kind of bureaucratic infighting which advances their own careers and gains them publicity and renown. And Pournell's Law states that the second group will always tend to take power in an institution, write the rules, and end up marginalizing the first group.

This applies to the internet and to academic bodies and prestigious contemporary music institutions (like Wikipedia or tuning discussion groups or Ivy League tenured university professorships or institutions like Lincoln Center) as well as to other other types of bureaucracies. The people who wind up dominating Wikipedia articles about xenharmonics (as administrators with the power to delete edits they don't like) or Ivy League tenured professorships or the concert programmes or high-profile concert venues like Lincoln Center tend to be the people who excel at politicking and bureaucratic infighting...not the people who actually know or have accomplished things. A person becomes an administrator with the power to delete undesired entries in Wikipedia articles about micrtonality by spending 16 hours per day editing Wikipedia. This leaves no time to compose or perform or listen to microtonal music. As a result, the people who spend all their time composing and performing microtonal music get marginalized and written out of Wikipedia articles about microtonality. The same applies to people who attain positions of great power in institutions like Lincoln Center. Such people must spend essentially all their time running and politicking in Lincoln center and navigating the treacherous waters of funding committees and budget infighting with the City of New York, leaving no time to compose or perform music.

Consequently, there is essentially no valid information about xenharmonic on the internet. Moreover, as a general rule, the better-connected any xenharmonic commentator is to the internet, the more elaborate hi/r webpage, the more highly visible and polished hi/r YouTube videos, the less that person knows about xenharmonics. People with elaborate and impressive web pages and superb YouTube videos have attained that level of expertise by spending all their waking hours learning web design and video production. This leaves no time for composing and performing music. Contrariwise, the expert musicians who spending all their waking hours composing or performing music don't have years to take off to learn web design or high-definition video editing and production. Invariably, the expert musician who asks someone "Please design a high-quality professional looking web page for me" or "I need three hundred hours of video of performances edited and titles added and the viewpoints of three different cameras intercut, with SMPTE synchronization" gets the response: "I make my living doing web design/video editing and I charge $50 per hour -- why should I do it for you for free?" With the inevitable result that the web page never gets designed or the video never gets edited and put up on YouTube.

There exists a vast amount of superb microtonal music. Brian Eno has never heard it because it's produced by practicing musicians and composers who spend their time making vividly memorable music, not impressive websites or Lincoln Center concerts or thick gilt-edged books published by prestigious academic publishers. There is a great deal of insightful and accurate writing about microtonality, but it was produced by people like Ivor Darreg who cannot get published by conventional academic publishers. (Peer review generally offers a reliable method of academic quality control except in new fields like xenharmonics. With microtonality, peer-review encounters a vicious cycle of Catch-22: the academic to whom the book on microtonality gets sent for peer review responds "Never heard of this. Deep six it." And because of this kind of response in peer review, academic books on microtonality typically don't get published. But because academic books on microtonality don't get published, academics remain unfamiliar with the subject -- leading to a self-reinforcing closed cycle of lack of information about microtonality in academia.) Meanwhile, the books on microtonality which do get published (viz., Harry Partch's Genesis of a Music) contain enormous amounts of misinformation about microtonality and ignore most of the range of xenharmonic tunings and most of the styles of xenharmonic music produced over the last 80 years.

This appears to be the case in the early part of the development of any new art. For the first few years, the people who are most prominent are those who know the least and have produced the worst music or art. Only slowly, after a period of many decades, do the obscure figures eventually become revealed as the great practititioners, and the previously unpublished writings finally get into (and stay in) print. Henry Cowell's New Musical Resources, for example, was written in 1919 but not published until 1930. it then fell out of print in the 1950s, and stayed out of print for well over 40 years.

Contemporary music finds itself subject to even more violent fads and fashions than bubble-gum pop music designed for teenagers. And just as pop music witnesses transient fashions like The Spice Girls (who at one time sold more records faster than any other group in music history and have now completely vanished from pop culture, never to be heard of again), in contemporary music transient fashions like total serialism gain immense fame, only to submerge into oblivion and disappear from the general consciousness, never to be heard of again. In contemporary music, as in bubblegum pop music, the transient fads and fashions are what grab peoples' attention. The work that stands the test of time only emerges gradually, over the course of many years. (Sometimes the work that stands the test of time was famous when originally produced. But sometimes not.)

Myth #6: "Acoustics forms the basis of all music, and the acoustic laws of physics show that all vibrating objects resonate with natural modes of vibrations which form small integer ratios."

This claim is so diametrically the opposite of mathematical and acoustical reality that it's hard to find words with which to state the sheer wrongness of this claim.

In actual fact, Weyl's Law of Acoustics states that only one-dimensional vibrational systems produce harmonic series vibrational modes. I.e., only one-dimensional vibrating strings, or tubes which exhibit only one degree of vibrational free (the cylindrical tube can be viewed as a rotational symmetry around a one-dimensional line, mathematically speaking, since the air in the cylindrical tube has only one degree of freedom--it can only move forward or back in one dimension).

This means that essentially all vibrating objects produce natural resonant modes of vibration which are non-just non-equal-tempered. If you pick up any object in your immediate vicinity and tap it, you will hear an inharmonic series of partials produced by non-just non-equal-tempered modes of natural vibration.

This tells us that "the chord of nature," if there is any such thing, is a non-just non-equal-tempered set of inharmonic vibrational modes. Essentially all objects in the circumambient universe have three dimensions, and Weyl's Law tells us that any vibrating objects which are not 1-dimensional exhibit inharmonic modes of vibration which are non-just non-equal-tempered. (Even vibrating systems which approximate 1-dimension systems, like a taut string or a cylindrical tube, exhibit slightly inharmonic partials whose inharmonicity results from the three dimensional nature of the system. A cylindrical tube has modes of vibration which depart from harmonicity due to edge effects and viscous air flow friction at the edges of the tube, while a string under tensions has partials which depart from harmonicity because of the mass and diameter of the strings and their tendency to act as vibrating metal rods to some degree, rather than as 1-dimensional string with length but no width or height.)

Ethnomusicology confirms this, showing that well over 80% of the world's musicians produce music using inharmonic instruments like gourds, metallophones, xylophones, drums, beaters, shakers, and so on, in non-just non-equal tempered tunings.

"As to whether the interval 3:2 is common to all of the world's musical systems, as has occasionally been claimed, Fritz Kuttner asserts that the "fifth" in Chinese music is 20 to 30 cents flat. It is apparently nearly as flat in Siamese music..." M. Joel Mandelbaum, Multiple Division of the Octave and the Tonal Resources of 19-Tone Temperament, PhD thesis, 1960, p. 16.

"Two theoretical systems evolved in China, one derived from the Cyclic Pentatonic and the other from the division of string lengths. They are found combined in the highest form of Ch'in music. (..) Methods of arriving at these fifths included the use of twelve tubes... The fifths produced by these tubes were small compared to Western fifths. Various musicologiests place them between 670 and 680 cents as compared to the Just fifth of 702 cents." [Lentz, Donald A., The Gamelan Music of Java and Bali, 1965, pg. 27]

"There are...a number of musical cultures that apparently employ approximately equally tempered 5- and 7-interval scales (i.e., 240 and 171 cent step-sizes, respectively) in which the fourths and fifths are significantly mistuned form their natural values. Seven-interval scales are usually associated with Southeast Asian cultures (Malm, 1967). For example, Morton (1974) reports measurements (with a Stroboconn) of the tuning of a Thai xylophone that `varied only + or - 5 cents' from an equally tempered 7-interval tuning. (In ethnomusicological studies measurement variability, if reported at all, is generally reported without definition.) Haddon reported (1952) another example of a xylophone tuned in 171-cent steps from the Chopi tribe in Uganda. The 240-cent step-size, 5-interval scales are typically associated with the `gamelan' (tuned gongs and xylophone-type instruments) orchestras of Java and Bali (e., Kunst, 1949). However, measurements of gamelan tuning by Hood (1966) and McPhee (1966) show extremely large variations, so much so that McPhee states: `Deviations in what is considered the same scale are so large that one might with reason state that there are as many scales as there are gamelans.' Another example of a 5-interval, 24--cent step tuning (measured by a stroboconn, 'variations' of 15 cents) was reported by Wachsmann (1950) for a Ugandan harp. Other examples of equally tempered scales are often reported for pre-instrumental cultures... For example, Boiles (1969) reports measurements (with a Stroboconn, `+ or - 5 cents accuracy') of a South American Indian scale with equal intervals of 175 cents, which results in a progressive octave stretch. Ellis (1963), in extensive measurements in Australian aboriginal pre-instrumental cultures, reports pitch distributions that apparently follow arithmetic scales (i.e., equal separation in Hz).

"Thus there seems to be a propensity for scales that do not utilize perfect consonances and that are in many cases highly variable, in cultures that either are pre-instrumental or whose main instruments are of the xylophone type. Instruments of this type produce tones who partials are largely inharmonic (see Rossing, 1976) and whose pitches are often ambiguous (see de Boer, 1976)." [Burns, E. M. and Ward, W. D., "Intervals, Scales and Tuning," in The Psychology of Music, 1982, ed. Diana Deutsch, pg. 258]

One-dimensional vibrational systems do not appear in nature. (All objects in our universe have three dimensions.) 1-dimensional vibrational systems are not natural, and objects like stretched strings or perfectly cylindrical hollow tubes which approximate to some degree a 1-dimensional vibrational system must be produced artificially. This means that just intonation is the most artificial and least natural possible tuning, while the most natural tuning would be some form of non-just non-equal-tempered tuning with highly inharmonic partials, like the natural vibrational modes of a struck wooden block or a metal bar or a drumhead.

Musical instruments which produce harmonic series timbres are so rare and so unusual that, to a first approximation, essentially all the world's musical instruments avoid this kind of construction. There is nothing new about this conclusion: A. J. Ellis first stated in 1885 that his survey of world music showed that "The music of most of the world's cultures is not based on mathematics nor or integer ratios, but is very contingent, and arbitrary, and entirely unique to its own society." (Ellis, A. J., "On the Musical Scales Of Various Nations," Journal of the Royal Society of the Arts, Vol. 3, 1885, pg. 536). The mathematical acoustics of most vibrating bodies have been known to be nonlinear and to produce inharmonic partials for most vibrating objects for well over 100 years: see Lord Rayleigh's two-volume Acoustics, 1895, for details.