BETWEEN U S: A HyperHistory of American Microtonalists

BETWEEN U S: A HyperHistory of American Microtonalists

Twelve-tone equal temperament, as this common tuning is called, is a 20th-century phenomenon, a blandly homogenous tuning increasingly imposed on all the world’s musics in the name of scientific progress. In short, twelve-tone equal temperament is to tuning what the McDonald’s hamburger is to food.

Written By

Kyle Gann

There is nothing that musicians take more for granted than the fact that there are twelve pitches to an octave, and that these pitches divide the octave into twelve equal steps. Apparently few musicians question this arrangement, and only a tiny minority can explain whence it arose, why, and from what principles its authority derives. This 12-pitch assumption, however, is far from innocent. Twelve-tone equal temperament, as this common tuning is called, is a 20th-century phenomenon, a blandly homogenous tuning increasingly imposed on all the world’s musics in the name of scientific progress. In short, twelve-tone equal temperament is to tuning what the McDonald’s hamburger is to food.

How can this be so? What is so unnatural about twelve-tone equal temperament?

The basis of any natural system of tuning is that two pitches sound consonant (that is to say, sweet, or intelligible to the ear) when their sound waves vibrate at ratios of relatively small whole numbers. In an octave, for example, two pitches vibrate at a ratio of 2 to 1, one pitch vibrating twice as fast as the other. In a perfect fifth, such as C up to G, the ratio is 3 to 2. In a major third C to E, the ratio is 5 to 4.

The great problem that nature bequeaths to us in the mathematics of tuning – not an obstacle, but a wonderful challenge when viewed the right way – is that these simple intervals aren’t divisible by each other. To illustrate, we need a perceptual measure of interval size. The one invented by the great acoustician Alexander Ellis in the late 19th century is called a cent, and is equal, by definition to one 1200th of an octave, or 1/100th of a half-step.

An octave: (ratio 2:1) = 1200 cents
A perfect fifth: (ratio 3:2) = 701.955 cents
A major third: (ratio 5:4) = 386.3 cents

In the equal temperament we’re used to, three major thirds – C to E, E to G#, G# to C – equal one octave. But as you can see, three pure major thirds of 386.3 cents do not equal one octave, because 3 x 386.3 does not equal 1200. So equal temperament, our McDonald’s hamburger tuning, stretches every major third out to an arbitrarily out-of-tune 400 cents, somewhat the way McDonalds standardizes every patty to a flat quarter-pound of dubious relation to beef. These means that every major third on the piano is out of tune by 13.7 cents, creating busy little beat patterns between the overtones of every major third we hear. Unless you’ve had some exposure to Indian or Indonesian or some other non-Western musical tradition (or authentic barbershop quartet music, the last pure-tuned tradition in America), it’s quite likely that you’ve never heard a true major third in your life, nor a true major or minor triad.

Music schools teach that this Big Mac tuning has been around for centuries and represents an immutable endpoint of progress. It’s a lie. History, even in Europe, has provided many alternatives, Arabic and Asian cultures have provided rich tuning resources unknown to us, and many recent American composers have explored alternative tuning possibilities.

There are many reasons to write in other tunings, seemingly as many as there are composers who do it. La Monte Young seeks absolute purity of pitch so he can explore complex combinations of distant overtones never heard before. Harry Partch wanted to imitate in melody the subtle contours of the human voice, without compromise. Lou Harrison wants to recapture the sensuous presence that true intervals had before the 20th century. Ben Johnston wants his music perfectly in tune so it will have a healthful psychological effect on the listener. Myself, I enjoy the expanded composing resources of 30 or so pitches to the octave, and the option of creating amazing chromatic effects through minimal voice-leading. Some composers are seeking a magical harmonic alchemy written about in ancient treatises. Others just enjoy exotic out-of-tuneness. One of the exciting things about the microtonal field is that, despite its grounding in natural laws of acoustics, its diverse practitioners hardly agree on anything.

For those intrigued but unfamiliar with the wide range of microtonal strategies, this quick survey in four sections will explore several options for escaping equal temperament. We’ll look at forms of historical tunings, take a regrettably brief glimpse at other tunings of the world, and examine tunings devised by several American composers, both in the areas of just intonation and of equal temperaments based on divisions other than 12.
And for those who want more information, there are a lot of Web sites. Book publishers and academic musicians are absolutely convinced that alternate tuning is a strange, esoteric subject that no one except a few weirdos is interested in. If they’d look on the Web, they’d find thousands of tuning aficionados. You can learn everything you wanted to know about meantone at, and Terry Blackburn, Zeke Hoscan, and Stephen Malinowski have excellent pages on the mathematics of different European tunings. There’s a Pythagorean Web page. And one of the most forward-looking theoretical thinkers, with a lot of new tuning conceptions for new composition, is Joe Monzo. The tuning of our music evolves historically more rapidly than people realize, and it’s on the move again.

Just intonation isn’t for everyone. Its plethora of fractions can be daunting, and a lot of composers dissatisfied with 12-tone equal temperament simply can’t think creatively in a free field of fractions. Many of these composers have divided the octave, equally, into more than twelve pitches per octave. The great attraction of any kind of equal temperament is that it allows free transposition to any step of the scale. During the 19th century, transposition became so central to compositional thinking that many classically-oriented composers can’t imagine doing without it. Just intonation doesn’t make transposition impossible by any means, but a just-intonation scale limited in its number of pitches will make certain transpositions available, and others difficult.

It must be said, though, from my experience, that working with an unequal just-intonation scale is like carving in wood – the material has a grain to it that gives the artist something to work against. Working in equal temperament is like carving in plastic: every scale step is the same, and the scale doesn’t suggest very much about how to compose in it. Transposition in an unequal scale can lead to very interesting results, with old musical content expressed in new interval patterns. Many classically trained musicians, however, aren’t willing or ready to think in such terms.

That said, various equal temperaments were the usual deviation from 12-pitch tuning from the 1920s until at least the 1970s. Division into 24 pitches per octave used to be considered, in the first half of the 20th century, the most convenient alternative; this is known as quartertone tuning. Other composers have divided the whole-tone into 5, 6, 8, or 12 equal parts, for 30-tone, 36-tone, 48-tone, or 72-tone scales, each of which offers certain advantages. Other, seemingly more eccentric equal divisionsare actually quite natural, such as 19, 31, and 53 tones per octave.

From BETWEEN U S: A HyperHistory of American Microtonalists

Back in the ultramodern 1920s, quartertones seemed like the next logical step for deeper exploration of pitch language. The theory of acoustical tuning wasn’t taught in those days; thinkers as brilliant as Schoenberg and Cowell went around insisting that the 11th harmonic of C was F# (it’s 551 cents, halfway between F and F#, not 600 cents), and no one really believed that you could hear such tiny pitch differences. In addition, the nature of musical instruments, especially that behemoth the piano, was not going to change any time soon, but if you put two pianos together, you could tune one down 50 cents, and between them you’d have a quarter-tone scale, 24 equal steps to the octave. It was an interval not of acoustic necessity, but of convenience. Thinkers like Ferruccio Busoni theorized about splitting the half-step into three and four equal parts as well. In those days of talk about splitting the atom, it must have been in the air.

And so a number of composers wrote music in quartertones. Chief among these were (for quality) the American Charles Ives and (for quantity) the Czech Alois Hába. Ives’s main contribution was Three Pieces for Quarter-Tone Pianos, completed in 1926, one of his last works. His article “Some Quarter-Tone Impressions” (published by Norton with his Essays Before a Sonata) theorizes about what kind of harmonies quartertones would support. He postulates a triad in-between major and minor, say, C and G with a pitch between Eb and E; the chord sounds more stable, he claims, if you add a seventh halfway between Bb and B. The Quarter-Tone Pieces carry out these theories beautifully. The only recording I’ve ever found, however (and there are several), that really has the pianos exactly a quarter-tone apart is the old vinyl recording on Odyssey. All the others miss slightly.

Alois Hába (1893-1973) wrote a considerable amount of quartertone music. He also wrote string quartets that divided the whole step in to five equal parts (fifth-tone, or 30 equal steps to the octave) and six equal parts (or 36 steps to the octave). Haba’s opera The Mother is in quartertones, and is recorded on Supraphon. Another composer of divided half-steps is the Russian Ivan Wyschnegradsky (1893-1979), who had a vision in the street one day that he was supposed to write microtonal music. Musicians at McGill University, including the Mather-LePage Duo, put out two recordings of Wyschnegradsky’s multiple piano works, one for two pianos tuned a quarter-tone apart, another for three pianos tuned a sixth-tone apart. It’s visionary music, like Scriabin but creepily in-between-the-keys. The Arditti Quartet has supposedly made a recording of Wyschnegradsky’s string quartets that was released in Europe, but I’ve never found it.

Mexico’s Julian Carrillo also made a career out of what he portentously called “the Thirteenth Tone,” although what he actually did was to divide the half-step into four parts for 48 equal steps per octave. His Preludio a Cristobal Colon, published in Henry Cowell’s New Music Edition, is written for an ensemble in his special notation.

Ezra Sims of Boston has gone even further, writing in a special notation for 72 pitches per octave. Sweet, haunting, sometimes folk-music-based, Sims’s music sounds natural but is very careful about its intonation, not compromising on commas and raised and lower leading tones.

Each of these divisions has certain acoustical features in its favor; the more divisions, the more acoustical accuracy and the less convenience of notation and performance. Quartertone tuning captures several 11-based intervals, intervals based on the 11th harmonic:

11/8 = 551 cents
11/9 = 347 cents
11/6 = 1049 cents
12/11 = 151 cents

All are very close to quartertones. However, seven-based intervals are just as out of tune in quartertone music as they are in 12-tone equal temperament.
The 36-tone equal temperament, or dividing the half-step in three, is better for capturing intervals based on the 7th harmonic, or 7-based intervals.

7/4 = 969 cents
7/6 = 267 cents
8/7 = 231 cents
9/7 = 435 cents

Each of these is approximately 33 cents above or below an equal-tempered pitch. Sims’s 72-pitch tuning combines these possibilities, allowing pitches both 33 and 50 cents away from the 12 standard ones, and also allows much closer approximations of standard major and minor thirds. The use of 72-tone equal temperament allows perfect transposibility in eleven-limit tuning, but at the price of tremendous inefficiency. Partch, after all, gets perfect eleven-limit tuning with only 43 pitches, and I’ve never succeeding in needing more than 31.

Other intervals are possible with equal temperaments not derived from the whole- or half-step, but from circles of fifths.

From BETWEEN U S: A HyperHistory of American Microtonalists

Certain divisions of the octave are natural because they represent the points at which multiples of the perfect fifth coincide with multiples of the octave. For example, 12.

12 perfect fifths = 12 x 701.955 cents = 8423.46 cents

7 octaves = 7 x 1200 cents = 8400 cents

8400 and 8423 are pretty close, so if you fudge the fifths a little, you can divide the octave into 12 steps and get both fifths and octaves.

Like wise, 19:

19 perfect fifths = 19 x 701.955 cents = 13337 cents
11 octaves = 11 x 1200 cents = 13200

13337 and 13200 are, proportionately pretty close, so 19 ends up being one of the natural divisions of the octave if you want perfect fifths. So do 31, 34, and, more spectacularly, 53.

The Colorado-based guitarist Neil Haverstick plays, and has recorded with, a 19-tone guitar, on his impressive discs, The Gate and Acoustic Stick. The influence of tuning on Haverstick’s blues playing is fun to listen to; he sometimes has to extend the rhythm of blues phrases to fit in all his chromatic pitches. He also plays a 34-tone-to-the-octave guitar. Fretted instruments such as guitars and lutes have the oldest history of playing equal temperaments, since you have to have equal temperament if the frets are going to go straight across the fretboard. Just intonation on guitar (or any other unequal temperament) requires jagged frets that shift up and down for each string. Since at least the 16th century it’s been considered easier just to tune guitars in equal.

The 16th-century theorist Nicola Vicentino invented a 31-tone-to-the-octave harpsichord, the keys divided between two manuals and with some of the black keys split. He claimed that with his 31-tone scale one could play melodies from the Hebrew and Arabic worlds and the Slavic and Germanic countries without distorting them into the Italian scale. He understood that tuning is ultimately a multicultural issue, and that standardized, invariant tuning was a means of oppression used against foreign musicians from allegedly inferior cultures. Incidentally, in 31-tone equal temperament the perfect fifth is a slightly flat 696.77 cents, almost exactly the same size as the meantone fifth Vincentino was used to. [Ed. Note: The 31-tone system has inspired a great many new music composers both in the Netherlands and the United States — American tricesimoprimalists include Joel Mandelbaum, and Jon Catler who in the 1980s led a rock band called J. C. and the Microtones.]

A 53-tone equal temperament has sometimes been held up as a dream tuning. In 53-tone, each pair of adjacent pitches is separated by 22.64 cents. The major third in this scale is playable as 384.9 cents (instead of an optimum 386.3); the minor third is 316.98 cents (instead of 315.6); and the perfect fifth is 701.886 cents (instead of 701.955). All of the (five-limit) intervals of European music can be played within a half of a percent accuracy in 53-tone equal temperament. Inspired by such realizations, an Englishman named T. Perronet Thompson built a 53-pitch organ in the 1850s, its keyboard a Dr. Seussian fantasy of split keys, curved keys, different colored-keys, and knobs sticking up through other keys. Estimated arrival time moving from a C major chord to a G major chord is probably four minutes, as the organist cogitates on where the right keys are, but it was a wonderful idea.

A couple of other modern experiments with equal temperaments should be mentioned. One is Easley Blackwood‘s remarkable series of 12 Microtonal Etudes, each written in a different equal temperament from 13 to 24 pitches to the octave. Blackwood invented his own different notation for each division of the octave; the score is published, and the CD is available on Cedille. The electronic sounds are a little cheesy, and it’s a little disappointing that, instead of treating you to the most unusual intervals, he concentrates wherever possible on intervals found in 12-pitch tuning; for instance, 15-pitch equal contains the same major thirds as 12-pitch. But the tunings themselves are all the weirder for not being grounded in any natural acoustic basis, and they’ll stretch your ears.

Wendy Carlos has also worked with equal-tempered scales not based on the octave, so that you get different pitches from octave to octave. For example, if you have a scale of 35-cent increments, you’ll have a pitch at 1190 cents and 1225, but not at 1200. I haven’t had an opportunity to hear her results, but her Web page – which seems to say virtually nothing about her tunings — has some brief samples of her music. Recently, at the last Festival of Microtonal Music that Johnny Reinhard organizes, I heard Skip LaPlante’s Music for Homemade Instruments group-sing a happy little tune in 13-tone equal temperament. And they really did it. There may be no natural acoustical basis for a 13- division of the octave, but it can be sung, and it blows your mind to hear it.

From BETWEEN U S: A HyperHistory of American Microtonalists

If 12-tone equal temperament is the Big Mac of tunings, then just intonation is the health food. Just intonation means that the pitches have been defined in terms of whole-number ratios between frequencies. For example, if we’re in the key of C and I refer to a 6/5 E flat, that means an E flat that vibrates at a frequency 6/5 as fast as C; in other words, if C above middle C vibrates at 500 cycles per second (cps), 6/5 E flat will vibrate at 600 cps. The number of potential pitches in a just intonation system is equal to the number of possible fractions: namely, infinite. Naturally, composers cannot deal creatively with a disordered infinity of pitches. We need schemes to limit and justify and order the world of potential pitches. In fact, I believe that good music can only issue from an elegant tuning, and the more elegant the tuning, the more fertile it will be as a generator of musics.

In just intonation, we use fractions to define pitches. To know what pitch a fraction represents, we need to know what key we’re in. If we’re in the key of C, then we define C as 1/1, and D is 9/8. That means that D is defined as the pitch that vibrates 9/8 as fast as C. 9/8 is also the name of an interval – in this case, a whole step. Normally, in talking about justly tuned pitches, we express fractions in terms within a single octave, or between 1 and 2. If 9/8 is D, then 9/2 and 9/16 are also D, but we tend to only use 9/8 because it’s in the octave between 1 and 2. We’re used to calling pitches in different octaves all Cs, or all B flats, but it can be difficult for people to get used to the notion that 7/8 = 7/4 = 7/2 = 14/1. The pitches denoted by those fractions are all octaves of each other, because multiplication or division by 2 only changes octaves.

One of the ways we differentiate between different just intonation systems is by what prime numbers are employed factors in the tuning’s fractions. For instance, five-limit tuning is tuning in which all fractions can be expressed as powers or multiples of the numbers 2, 3, and 5 (not 1 because 1 is merely identity, and not 4 because 4 is merely an octave of 2). In seven-limit tuning, the list is expanded to 2, 3, 5, and 7. Eleven-limit tuning goes up to 2, 3, 5, 7, and 11, and so on. Finally we’ll address the possibility of 13-limit and higher tunings.

If the arithmetic here confuses you, you’ll find a fuller, more gradual explanation on my tuning page. If this sparks your interest, you’ll find all sorts of just-intonation resources at the Just Intonation Network Web page, which will lead you to an encyclopedic array of tuning sites. Or if this just scares you, you can go back to the tuning page.

From BETWEEN U S: A HyperHistory of American Microtonalists

You may have never been aware of it, but you’ve been listening to music in five-limit tuning all your life – or rather, an equal-tempered approximation of it. European music is based on the desire to get two intervals in tune: the perfect fifth (3/2) and the major third (5/4), as well as the minor third (6/5) between those two intervals. The basic problem of tuning is that any two prime-numbered intervals are incommensurate. That is, there is no number of major thirds that will equal any other number of perfect fifths. Major thirds are 386.3 cents wide, perfect fifths are 701.955 cents wide, and there just aren’t any reasonably small numbers those will both divide into evenly. [Another way to say it: Major thirds are based on 5, perfect fifths on 3, and no power of 5 (5, 25, 125, 625…) will ever equal a power of 3 (3, 9, 27, 81, 243…).] Therefore we have to make decisions about which pitches to have fifths on and which to have thirds on.

Let’s start by taking four pitches and tuning them to perfect fifths: F, C, G, and D, with C as our tonic, defined as 1/1.

F 4/3 = 498 cents
C 1/1 = 0 cents
G 3/2 = 702 cents
D 9/8 = 204 cents

If we continue tuning the circle of fifths to these perfect 3/2 fifths, we’ll end up with a 3-limit tuning known as Pythagorean, because it is limited to the intervals Pythagoras is alleged to have discovered. That gives us rather harsh major thirds of 81/64 (408 cents), though, and right now we’re looking for pure major thirds of 5/4 (386.3 cents). So let’s build a pure major third above and below each of our four established pitches. This gives us a pretty evenly-spaced 12-pitch scale:

C 1/1 = 0 cents
Db 16/15= 112 cents
D 9/8 = 204 cents
Eb 6/5= 316 cents
E 5/4 = 386 cents
F 4/3 = 498 cents
F# 45/32 = 590 cents
G 3/2 = 702 cents
Ab 8/5 = 814 cents
A 5/3 = 884 cents
Bb 9/5 = 1018 cents
B 15/8 = 1088 cents

This is a fine, perfectly in-tune scale with 12 pitches. Its only drawback is that it is only in tune for the key of C. For example, if you want to play a D chord, the interval between D 9/8 and A 5/3 isn’t 3/2, as a perfect fifth should be, but 40/27 (5/3 divided by 9/8 = 5/3 x 8/9 = 40/27). And at 680 cents instead of 702, that “wolf fifth” between D and A is going to howl. You can retune A to be in tune with D, but then your F chord is no longer in tune.

There is a fascinating recording of a piano work in five limit tuning: Terry Riley‘s The Harp of New Albion (Celestial Harmonies 14018). The work employs a piano tuned to the above scale on C#, except that the G is 64/45 instead of 45/32. And the movements of the piece run through several keys, including D and Bb, so that you get a powerful sense of what happens when you intentionally modulate within a limited five-limit system.

This is the basic problem with tuning Western music, with its need for fifths and thirds and its limitation of only 12 pitches. If we allowed ourselves more than 12 pitches per octave, the problem would have many easier solutions. In fact, there were experiments in 16th century Italy with constructing harpsichords with octaves of 19 and 31 pitches to the octave just to avoid this dilemma. The academics of their day, as academics always will, prevented such innovations from catching on. But all of our historical European tunings, from meantone to well temperament and even our present accursed equal temperament, are approximations of this five limit tuning. So you can explore those historical European solutions, or you can go on to the composers who, after being stalled for 330 years, finally plunged ahead into seven-limit tuning.

From BETWEEN U S: A HyperHistory of American Microtonalists

Back in the 16th century, lots of forward-looking musicians wanted to expand the range of European tuning to include intervals based not only on 3 (perfect fifths) and 5 (major and minor thirds), but also 7. After all, Indian and Arabic musicians had gaily been using 7-based intervals for centuries. But they were infidels, and one way the Christians wanted to distance them from the heathens was by insisting that no one could really perceive such tiny pitch differences. So the Italian academics won just after 1600 (over the continuing objections of the seminal mathematician Marin Mersenne), and closed off the wonderful number 7 to the Western world for more than 300 years, until Harry Partch rediscovered it and began using it in the 1920s.

Adding the number 7 and its octaves (14, 28) to our original stable of numbers (2, 3, 5) gives us a very interesting array of new intervals:

15/14 = 119.443 cents
8/7 = 231.174 cents
7/6 = 266.871 cents
9/7 = 435.084 cents
7/5 = 582.512 cents
10/7 = 617.488 cents
14/9 = 764.916 cents
12/7 = 933.129 cents
7/4 = 968.826 cents
28/15 = 1080.557 cents

Now keep in mind that the equal-tempered intervals we’re used to are all sizes divisible by 100. An equal tempered whole step is 200 cents, a perfect fourth is 500 cents, and so on. Seven-limit intervals often create intervals a third of a half-step away from their equal temperament semi-equivalents. The 8/7 “major second” is 31 cents “sharp.” The 7/6 “minor third” is 23 cents “flat.” This specific difference creates a certain flavor for seven-limit tuning, oddly off from the tuning we’re used to and sometimes bitterly flat, yet strangely consonant. In addition, the 7/5 tritone offers a much more consonant tritone than anything we’re used to in European tuning.

Add these 10 simple seven-based intervals to the basic five-limit intervals, and you get a scale that some would consider unwieldy. The most perplexing compositional problem of working in just intonation is, once you open up the field to seven, how do you choose which pitches to use?

One of the most brilliant solutions is the one La Monte Young adopted in his six-hour piano masterpiece The Well-Tuned Piano. He eliminated all factors of the number 5, so that he was only working with multiples of 2, 3, and 7. And he arrived at the eccentric yet very beautiful 12-pitch scale, suitable for piano tuning:

Eb 1/1 = 0 cents
E 567/512 = 177 cents
F 9/8 = 204 cents
F# 147/128 = 240 cents
G 21/16 = 471 cents
G# 1323/1024 = 444 cents
A 189/128 = 675 cents
Bb 3/2 = 702 cents
B 49/32 = 738 cents
C 7/4 = 969 cents
C# 441/256 = 942 cents
D 63/32 = 1173 cents

Note that the scale doesn’t uniformly ascend: G# is lower than G, and C# is lower than C. The scale is basically a five-pitch pentatonic scale around 0, 200, 450, 700, and 950 cents, with slightly different versions available for each pitch. Young kept the tuning secret for 27 years until I tuned my synthesizer to it and published it in an article (with his permission) in Perspectives of New Music, Winter 1993, Volume 31 Number 1. There’s a lot more to say about this scale, and I say some of it on my La Monte Young web page. Unfortunately, the Gramavision recording of The Well- Tuned Piano is out of print and nearly impossible to obtain. Please don’t ask me how to get a copy, because I can’t tell you. You can find out more about La Monte, though, at the Mela Foundation Web Page.

Michael Harrison, Young’s protege and piano tuner, also writes piano music in seven-limit just intonation. He has a CD available on New Albion records.

Another important masterpiece in seven-limit tuning is Ben Johnston‘s String Quartet No. 4, “Amazing Grace.” This lushly emotive 1973 work, a series of variations on the old hymn “Amazing Grace,” begins in a simple pentatonic scale and keeps adding new pitches with each variation until it runs through a glorious 22-pitch, seven-limit scale in the final variation. It’s Ben Johnston’s most popular work, and an instant favorite for everyone who hears it. The best recording is an old one by the Fine Arts Quartet on Gasparo records – unfortunately still only on vinyl. The Kronos Quartet has made a perfectly acceptable recording on Nonesuch, but their attention to tuning isn’t as meticulous.

From seven-limit tuning, the next logical step is eleven-limit tuning.

From BETWEEN U S: A HyperHistory of American Microtonalists

Eleven-limit tuning means that we define intervals and pitches by multiplying and dividing the numbers 2, 3, 5, 7, and 11, with no prime numbers larger than 11 used as factors. Adding 11 into the mix gives us a very interesting range of new intervals:

12/11 = 150.637 cents

11/10 = 165.004 cents
11/9 = 347.408 cents
14/11 = 417.508 cents
15/11 = 536.951 cents
11/8 = 551.318 cents
16/11 = 648.682 cents
22/15= 663.049 cents
11/7 = 782.492 cents
18/11 = 852.592 cents
20/11 = 1034.996 cents
11/6 = 1049.363 cents

Note that of these intervals, nine are within 15 cents of a quartertone (50 cents between two equal-tempered steps in 12-tone equal temperament), and six of those – 12/11, 11/9, 11/8, 16/11, 18/11, 11/6 – are within three cents. Eleven-limit tuning produces many of the pitches that we think of as quartertones. The peculiar quality of eleven-limit tuning is to smooth out the scale by giving us mediating pitches half-way in-between the pitches we’re used to.

The great champion of eleven-limit tuning, of course, is Harry Partch. His 43-tone scale, 43 non-equal steps to the octave, uses no prime factors larger than 11. I won’t give his scale here, because you can easily find it in his book Genesis of a Music and other places. There is loads of Harry Partch information on the Internet, the best sites being Corporeal Meadows and a British Harry Partch web page. In addition, all of Harry Partch’s major and minor works are being released on the innova and CRI labels.

The other person I can mention as having written a significant-sized output in eleven-limit tuning is myself, Kyle Gann. Eleven-limit tuning appeals to me for all those in-between notes, those quarter-tones that slide so easily between the pitches we’re used to. My own attraction to microtonality is the potential for extreme chromaticism and a minimalist approach to voice-leading in which lines can remain almost motionless while the harmony changes key wildly. You can hear this effect in my 1997 piece How Miraculous Things Happen, of which you can find an audio excerpt here. The piece uses 24 pitches to the octave (but very unevenly distributed, not quartertone), and capitalizes on the interval of 11/9 (347 cents) to slide smoothly between the minor (316 cents) and major (386) thirds. My largest just-intonation work so far is a 35-minute, one-man opera, Custer and Sitting Bull. The four movements of this work expand from 20 to 31 pitches, each movement based on a different tuning principle. You can also find audio samples of this piece at the same place, and program notes and tuning charts at my Custer and Sitting Bull web page.

Partch charmingly expressed his reasons for not proceeding past 11 to the 13-limit: “When a hungry man has a large table of aromatic and unusual viands spread before him he is unlikely to go tramping along the seashore and in the woods for still other exotic fare. And however skeptical he is of the many warnings regarding the unwholesomeness of his fare – like the ‘poison’ of the ‘love-apple’ tomato of a comparatively few generations ago – he has no desire to provoke further alarums.” Personally, I’ve never been able to get comfortable enough with my perception of the 13th harmonic as a consonance to pass that limit myself. But if you’re hungrier than Partch, and more attuned than myself, you’ll want to go on to the thirteen-limit and beyond.

From BETWEEN U S: A HyperHistory of American Microtonalists

There are an infinity of numbers, and an infinity of prime numbers. There are an infinity of fractions. There are, correspondingly, an infinity of pitches within any octave. Studies have suggested that, depending on circumstances of timbre, register, volume, and so on, the human ear and brain can distinguish about 250 pitches per octave. I myself have found, as a composer, that two pitches only 5 or 6 cents apart turn out to be impractically close, and I can’t meaningfully distinguish them within the context of a piece of music. Other composers, with other, more acoustically pure musical aims, may well find a vast gulf of difference between 200 and 205 cents. To be comfortable, I need at least 16 to 20 cents between the pitches in my scales, but Partch‘s scale contains pitches only 14.4 cents apart. There are, refreshingly, no rules here, and no limitations besides the composer’s own personal idiosyncrasies.

The number 13 opens up still further territories. The 13th harmonic is 840.53 cents above an octave of its fundamental, and some simple 13-based intervals include:

13/12 = 139 cents
13/11 = 289 cents
16/13 = 359 cents
13/10 = 454 cents
18/13 = 563 cents
13/9 = 637 cents
20/13 = 746 cents
13/8 = 841 cents
13/7 = 1072 cents

Note a preponderance of pitches about 40 cents away from our equal-tempered pitches.
The only composer I know of working consistently in 13-limit tuning is Mayumi Reinhard, and she swears that’s where the action is.

The 17th and 19th harmonics come too close to equal temperament to sound very exotic in most contexts; 17/16 is 105 cents, and 19/16 is 297.5 cents, both nearly divisible by 100. Ben Johnston‘s Suite for Microtonal Piano (1977), though, is a fine example of a piece in 19-limit tuning. The 12 pitches of the piano are tuned to the 16th, 17th, 18th, 19th, 20th, 21st, 22nd, 24th, 26th, 27th, 28th, and 30th harmonics of C. (This is still only 19-limit because all the other numbers factor down to prime numbers smaller than 19: 21 = 3 x 7, 22 = 2 x 11, and so on.) I don’t know of a better work for demonstrating the fascinating possibilities just intonation holds for modulation. The first and fifth movements are in the key of C, the second movement is in D, and the fourth is in E, meaning you get some pretty strange scales over D and E. The third movement is dodecaphonic. Johnston is possibly the only major composer who’s written dodecaphonic music in just tunings. The Suite for Microtonal Piano is recorded by Philip Bush on the Koch label, along with Johnston’s Sonata for Microtonal Piano (1964), a complexly prickly work in a highly extended five-limit tuning.

An equally fascinating work in 31-limit tuning is Johnston’s String Quartet No. 9, recorded by the Stanford Quartet (now renamed the Ives Quartet). Johnston has his intrepid string players play in a harmonic series scale from the 16th to the 32nd harmonic, including transpositions and inversions of the scale. The 31st harmonic is basically a quartertone between the 30th and 32nd, and when the strings cadence from dominant to tonic with that 31st harmonic, that cadence sounds nailed down for good. The work is mostly sweetly neoclassical, providing an unexpectedly normal context for odd harmonic events, and it’s very well performed. (The original 1964 tuning of La Monte Young‘s The Well-Tuned Piano was also in 31-limit tuning, not seven-limit as it eventually ended up.)

Johnston has more recently gone up to the 43rd harmonic (43-limit tuning) in recent works that aren’t recorded yet. The only person I know of to go higher than that in just intonation is La Monte Young in his sine-tone installations. For 20 years, Young has explored in his scintillating sound sculptures the harmonics between the higher octaves of the 7th and 9th harmonics. His current installation at the Mela Foundation is The Base 9:7:4 Symmetry in Prime Time When Centered Above and Below the Lowest Term Primes in the Range 288 to 224 with the Addition of 279 and 261…. The complete title is many times longer), which you can visit at 275 Church Street in New York City (call 212-925-5098 for times). It includes the 1072nd, 1096th, 2096th, and 2224th harmonics over its base drone, as well as other, lower tone complexes. All of these are octaves of prime-numbered harmonics: 1072 = 67 x 16, 1096 = 137 x 8, and so on. In more recent works Young has gone up above the 5000th harmonic. My sketchy introduction here can’t begin to do his sine-tone installations justice, but you can read more about them in my article “The Outer Edge of Consonance: The Development of La Monte Young’s Tuning Systems,” in Sound and Light: La Monte Young and Marian Zazeela (Lewisburg, PA: Bucknell University Press, 1996, pp. 152-190).

Until some madman surpasses Young, this takes us as far as we can go in discussion of new just-intonation dimensions.

From BETWEEN U S: A HyperHistory of American Microtonalists

Many musicians think that to attack today’s equal-tempered tuning is to attack the European classical music tradition itself. Not at all true, in fact, quite the opposite. Through most of European history – all except the last 100 years, in fact – tuning was an art, not a science, and the differences between different keys in the old keyboard tunings had an enormous, if subtle, influence in the way the great composers wrote their music. Ever wonder why F major is considered a calm, pastoral key, or why there are no Vivaldi concerti in F# major? Tuning is the answer. And the old tunings need not be dead, for they still have much to offer the modern composer. I keep my own pianos tuned to 18th-century well temperament, and compose for the wonderful variety of intervals it provides.

For simplicity’s sake I’m going to present only two phases of European tuning, meantone and well temperament.

From BETWEEN U S: A HyperHistory of American Microtonalists

Meantone is the name of one of the most elegant tunings in the history of European music, a beautiful tuning that provided near-perfect consonance in a variety of keys. The tuning was first explicitly defined in 1523 by an Italian theorist name Pietro Aaron, though it is suspected that some rough form of meantone had been in use through much of the 15th century. Meantone tuning dominated European music until the early 18th century, and continued being used in certain backwaters, especially England and especially among organ tuners, through the late 19th century. Having lasted some 250 to 400 years, depending on the area, meantone has been the most durable tuning in European history so far. [Ed. Note: It is also, most likely, the tuning that the European colonial settlers brought to America, and its compromises of good and bad triads undoubtedly informed the compositional choices of William Billings, Francis Hopkinson and other early American composers.]

Meantone is basically a 12-pitch keyboard tuning, a workable compromise that allows eight usable major triads and eight minor triads, the other four of each being real howlers. The premise of keyboard tuning in general is that you can have your major thirds in tune or your perfect fifths in tune, but both cannot be in tune, and some compromise is always necessary. Let’s look at why:

A well-tuned perfect fifth = 702 cents.
A well-tuned major third = 386.3 cents.
An octave = 1200 cents.

If we tune our perfect fifths in tune, we’ll have C to G, G to D, D to A, and A to E all 702 cents wide. 4 x 702 = 2808 cents. Therefore the two octaves and a major third from C to E (C G D A E) will be 2808 cents, and, subtracting two octaves or 2400 cents, the major third C to E will be 408 cents. 408 cents is an awfully wide and harsh major third, not really tunable by ear, bad for singing, and inharmonious.

Meantone’s solution is two squeeze down the perfect fifths until the major thirds are perfect. What we want is C to E at 386.3 cents. Therefore, two octaves and a major third will be 2786.3 cents, and each perfect fifth will be 1/4th of that amount, 696.575 cents. (The ratio between pitches of a meantone fifth is actually the fourth root of 5, since if you take the fourth root of 5 to the fourth power, you get 5, which is the ratio of two octaves and a pure major third.) (Don’t worry if you didn’t follow that. Not necessary.) The perfect fifth in meantone is just over 5 cents flat. But acoustically, the ear is less disturbed by out-of-tune fifths than by out-of-tune thirds, since with fifths the out-of-tune harmonics are higher up in register and further away and less obvious.

So meantone strives to give us as many perfect 5-to-4 major thirds as possible, which, when limited to 12 keys per octave on a keyboard, is 8. C, D, E, A, and G are tuned to slightly narrow fifths and slightly broad fourths, and then the rest of the pitches are tuned to pure major thirds: E-G#, F-A, A-C#, Eb-G, G-B, Bb-D, and D-F#. The result, notated in cents above C, is the following scale:

C 0
C# 76.0
D 193.2
Eb 310.3
E 386.3
F 503.4
F# 579.5
G 696.6
G# 772.6
A 889.7
Bb 1006.8
B 1082.9

And, if you’ve tuned your first five pitches right, C-E is a pure major third as well. The other four major thirds are 427 cents wide and sound terrible: G#-C, F#-Bb, C#-F, and B-Eb. In fact, as notated, those aren’t major thirds at all, but diminished fourths. In meantone, there is no such note as Db, but only C#. There is no D#, but only Eb. Unless, that is, you redo the tuning slightly to center it around some key other than C, as was sometimes done.

And so, in meantone, you simply can’t use triads with those unavailable major thirds. During the meantone period, you can’t really use keys with more than three sharps or flats in the key signature. Look through music of the 16th and 17th centuries, and you will find no pieces in Ab major, F# major, or Bb minor. Such keys need pitches that don’t exist in meantone tuning.

BUT – and this is the great advantage, the eight major and eight minor keys you can use sound so much sweeter than they do in our music. Those thirds sound so lovely, and thus all European music from the mid-15th to mid-18th centuries (and beyond) was based on the primacy of thirds. It became excusable to omit the fifth from a triad, but not the third, because the third was in tune and the fifth wasn’t. (I highly recommend Orlando Gibbons‘s Lord Salisbury Pavane and Galliard as a sterling example of exploration of meantone tuning. This late-16th-century work, a masterpiece of early keyboard music, meanders through every possible chord in meantone plus one dissonant B-major triad as a passing chord. Play through it in equal temperament and it sounds OK. Then play it in meantone, and its colors suddenly come alive, and you hear the work’s luscious beauty as Gibbons’s original audience did. Then play it in equal temperament again, and it collapses disappointingly back into black and white, just like Dorothy coming back to Kansas.)

Historically, there are different kinds of meantone, based on their division of the syntonic comma. The syntonic comma is the discrepancy between four perfect fifths and two octaves and a major third, about 21.5 cents. The classic Pietro Aaron meantone I’ve outlined above is called 1/4th-comma meantone, because 1/4 of the comma was subtracted from each perfect fifth. There are less extreme meantones such as 1/5th-comma, 1/6th-comma, even 5/18ths-comma. The less subtracted from each fifth, the more out-of-tune the thirds will be. 1/11th-comma meantone is actually identical to equal temperament. [The classic tuning book from which all this material is drawn is J. Murray Barbour’s Tuning and Temperament (New York: Da Capo Press, 1972).]
I wish I could recommend specific recordings in meantone tuning. I suspect that many exist, but early music groups, even when they are attentive to authentic tunings, are not often in the habit of specifying what keyboard tunings they use. Anyone can contact me at [email protected] with information about specific meantone recordings, I will add them to my historical tuning web page.

From BETWEEN U S: A HyperHistory of American Microtonalists

In 1893, the august Grove Dictionary of Music and Musicians stated that Johann Sebastian Bach wrote The Well-Tempered Clavier to demonstrate equal temperament. That was a misconception. And it has filtered down into hundred of music history texts, a lie so pervasively believed that it will take generations to correct.
Bach, after all, did not write The Equal-Tempered Clavier. But in his day, they did refer to a temperament in which all keys were usable as an “equal” temperament. But the truth was (and all this material comes from a wonderful book, Tuning: Containing the Perfection of Eighteenth-Century Temperament, the Lost Art of Nineteenth-Century Temperament, and the Science of Equal Temperament, East Lansing: Michigan State University Press, 1991, by piano technician Owen Jorgensen) that, back then, they had no way to truly divide the octave equally on a piano or harpsichord. They didn’t have oscillators or electronic tuners. And while, by Bach’s day, composers wanted to have all keys available, no one was interested in making them all the same. Each key had its own interval pattern and its own different color, so that one kind of music sounded better in E flat minor, while another came off much better in F major. There were reasons for choosing a specific key, and Bach wrote his preludes and fugues to illustrate those differences, not to suggest that they didn’t exist.

I do an experiment with my students: in a blindfold situation, I play the Bach preludes each in several different keys, and let them guess which key the prelude was written for. They usually get it right. The E flat minor Prelude, for example, sounds really flabby in E minor, and the C major prelude takes on a weird, too-bright quality in C#.

The tunings that dominated the 18th and most of the 19th centuries are now referred to as well temperaments. There were many different varieties. Tuning, for most of musical history, was an art, not a science, and each piano tuner has leeway to use his own taste. Some of these temperaments have reasonably familiar names: Werckmeister III, Kirnberger III, Vallotti-Young. (That’s Thomas Young, not La Monte – he wasn’t quite in the picture yet, or Beethoven’s sonatas might have sounded very different.) Some of these tunings you can find more information about on my Historical Tuning web page, but to save space I’ll only describe one here: Thomas Young’s well temperament of 1799.

The principle of well temperament is this: Imagine the circle of fifths. Now imagine that you squeeze the fifths nearest C slightly to make the thirds (C-E, G-B, F-A) a little flatter and more in tune. That means that you’ll have larger fifths and thirds on the opposite side of the circle of fifths, around F#. In most well temperament, and especially this one, the keys most closely related to C major/A minor have the flattest fifths and the most consonant thirds. Black-note keys like F# major and Eb minor have purely in-tune fifths, but their thirds are a little harsh and bitter. Therefore, when you want to write something like a Funeral March in an especially bitter, tragic key, you write it in B flat Minor – which is what Chopin did with his Funeral March Sonata. When you want a kind of majestic spookiness, as Beethoven did in the adagio of the Hammerklavier, you write it in F# minor, where the thirds aren’t bad and the fifths are especially stately and in tune.

A 0 cents
Bb 102 cents
B 196 cents
C 306 cents
C# 396 cents
D 502 cents
Eb 600 cents
E 698 cents
F 804 cents
F# 894 cents
G 1004 cents
G# 1098 cents

As you can see, no pitch here is more than 6 cents away from equal temperament, but the differences, if subtle, are still striking. The major third C-E is 392 cents, the major third F#-A# 408 cents, much harsher. All the black-key perfect fifths are perfectly 702 cents, while C-G is only 698 cents; the black-key fifths sound much purer. C# major and F# major are really active keys, bristling with overtones. C and F major are sweet, mild keys, and E flat minor is pungent. This is the tuning in which Beethoven heard his music (before he went deaf), and it clearly influenced his choices of keys, as it did for every other composer before the mid-19th century.

The first recording of Beethoven’s music in the original temperament appeared a couple of years ago: Beethoven in the Temperaments, with pianist Enid Katahn and piano tuner Edward Foote (Gasparo). The disc includes the Moonlight Sonata, the Waldstein, and the Pathetique in a late-18th-century well temperament that brings out subtle color differences among the keys.

Well temperament is hardly a dead issue even for composers today. The Californian composer Lou Harrison loves to use well temperament, and in fact wrote his entire Piano Concerto, recorded by New World Records, in Kirnberger III, which is fairly similar to Young’s tuning above. (Check out the Lou Harrison Web Page.) The mystic New York composer Elodie Lauten often writes music in well temperament, often combining it with equal temperament at the same time for a scintillating, slightly out-of-tune effect.

I have both my pianos tuned to Young’s 1799 temperament, my Steinway grand at home and the Disklavier in my Bard College office. I basically write for keyboard in well temperament, and I see no reason to go back to bland equal temperament. I know of no keyboard music that doesn’t sound better and more interesting in well temperament. The only music that equal temperament supports, as Lou has often put it, is 12-tone music; in 12-tone music, all the major thirds are theoretically equal, all the major 7ths, and so on, so I suppose one might want equal temperament to play 12-tone music authentically. But I see no other advantages. And how many of us play a repertoire dominated by 12-tone music?

Chronologically, this brings us back to 12-tone equal temperament – and if you can’t say anything good about a tuning, you shouldn’t say anything at all, I suppose, which is why I’ll direct you to the just intonation page or back to the tuning page.

From BETWEEN U S: A HyperHistory of American Microtonalists

A tremendous amount of work remains to be done in the realm of determining the tunings of non-Western musics. This is my academically acceptable way of professing ignorance. I have made some attempts on my own to match tunings of various non-European musics on my synthesizer, and to analyze pitch structures in terms of tuning. Not only is this sometimes difficult (especially when the music is fast or more than one pitch is sounded at a time), but the appropriate methodology varies from culture to culture. Many non-Western musics do not hold sustained pitches in place the way European music does, but slide and glissando and even yelp from one word to the next. An entire generation of musicologists could devote themselves to this problem without exhausting it. However – I’ll tell you what I know, and what I’ve found.

At Bard College we have a gamelan. Its official slendro scale is as follows:

Ab 16/15 = 112 cents
G 1/1 = 0 cents
Eb 8/5 = 814 cents
D 3/2 = 702 cents
C# 11/8 = 551 cents

These cent measurements arrived with the gamelan – the ratios I supplied myself. The gamelan instruments are in pairs, tuned about 30 cents apart to create the characteristic shimmering of gamelan music, but this tuning applies separately to each half of the pairs. I once wrote about gamelan music in terms of ratio tunings, and received an irate letter from some gamelan maven who informed me that gamelan musicians don’t tune in terms of ratios, and that I was imposing a foreign notion. And yet I’ve heard Lou Harrison talk about gamelans in terms of ratio tuning, and the cent-sequence 551, 702, 814, 0, 112 can only be interpreted as intending the ratios above. Not my field, but this particular bit of evidence leaves little room for interpretation.
Among Americans, Lou Harrison, Barbara Benary, Evan Ziporyn, and Jarad Powell, and many others, have written music for gamelan, and inspired by gamelan. With more than 200 Indonesian gamelans operating in the U.S., gamelan must really be considered a major current in American music.

Arabic music is well known for its use of 11-based intervals which sound, to our ears, like fairly exact quarter-tones. Treatises on Arabic music make reference within a scale on, say, G, to pitches halfway between Bb and B (11/9, or 347 cents) and halfway between F and F# (11/6, or 1049 cents). Modern transcriptions of Arabic songs and violin music sometimes use fairly standard quarter-tone notation for these pitches of the scale, and the quarter-tones are easily audible and very distinct on recordings.
That Indian ragas use a scale of 22 pitches is well known, but the exact tuning seems to be in doubt. The Hindu specialist Alain Danielou lists a tuning for the 22 pitches with some authority, but a student of mine (Jane Gilvin) researched Danielou and couldn’t locate any such tunings in the Sanskrit treatises he supposedly derived them from. Danielou, in his Music and the Power of Sound, interprets the Indian scale as pairs of notes, some five limit and others drawn from a cycle of 53 perfect fifths, and thus within three-limit tuning:

1/1 = 0 cents
256/243 = 90

16/15 = 112
10/9 = 182
9/8 = 204
32/27 = 294
6/5 = 316
5/4 = 386
81/64 = 408
4/3 = 498
27/20 = 520
45/32 = 590
64/45 = 610
3/2= 702
128/81 = 792
8/5 = 814
5/3 = 884
27/16 = 906
16/9 = 996
9/5 = 1018
15/8 = 1088
243/128 = 1110

Several of the paired notes are separated by the syntonic comma of 21.5 cents, or 81/80. It is unclear what authority Danielou asserts for these ratios, but he is a fascinating figure nonetheless.

Finally, many musicians believe that the “blue” notes that jazz singers and sax players play – bending the third, sixth, and seventh steps of the minor scale downward a touch – is a return to a seven-limit scale inherited from Africa. I have analyzed a few recordings by Billie Holiday with inconclusive results. There are certainly points at which she distinctly sings about a third of a half-step flat on those scale steps, with reference to the piano – and other places at which she sings in tune with the piano. My impression, drawn from the most modest evidence, is that sometimes she bends the third, sixth, and seventh scale steps downward for expressive purposes, and that she may find there “notches” representing the ratios 7/6 (267 cents), 14/9 (765 cents), and 7/4 (969 cents). Ben Johnston has written some jazz arrangements couched in such specific “mis-tunings.”

Through the spread of pop music, the Euramerican 12-pitch equal-tempered scale is in danger of wiping out indigenous tunings in much of the world, especially India and Southeast Asia, even in Arabic countries. At the same time, however, more and more American musicians are studying Eastern cultures, becoming experts in Indian or Indonesian performance, and absorbing new tunings. It is clear that if we want American music to join the rest of the world rather than squashing it, we need to get out of our 12-pitch equal-tempered RUT.

With that depressing thought you can go to the just intonation page, the historical European temperaments page, the equal temperaments page, or back to the tuning page.

From BETWEEN U S: A HyperHistory of American Microtonalists