NED Talk 3.0: An Introduction to Xenharmonics

01.18.2013 § 1 Comment

This post is designed to be supplemental listening & learning to the talk I gave in San Francisco on February 29th, 2013, “An Introduction to Xenharmonics.” If you weren’t there in person, you can check out this video equivalent of it here:

Robert Craft: “Is any musical element still susceptible to radical exploitation and development?”

Igor Stravinsky: “Yes, pitch. I even risk a prediction that pitch will comprise the main difference between the ‘music of the future’ and our music, and I consider that the most important aspect of electronic music is the fact that it can manufacture pitch. Our mid-twentieth century situation, in regard to pitch, might perhaps be compared to that of the mid-sixteenth century, when, after Willaert and others had proved the necessity of equal temperament, the great pitch experiments began—Zarlino’s quarter-tone instrument, Vincentino’s thirty-nine-tones-to-the-octave archicembalo, and others. These instruments failed, of course, and the well-tempered clavier was established (though at least three hundred years before Bach), but our ears are more ready for such experiments now—mine are at any rate. I had been watching the Kuramatengu play in Osaka one afternoon recently and had become accustomed to the Noh flute. Later, in a restaurant, I suddenly heard an ordinary flute playing ordinary (well-tempered) music. I was shocked, music apart—I think I could keep the music apart anyway—by the expressive poverty of the tuning.”

Memories And Commentaries, 1970

Xenharmonics is a word coined by Ivor Darreg (who is pretty much the man) to refer to the design and practice of harmonies which are foreign to the norms, whether those be of the western canon or any other tradition. There is a lot out there for you to find about xenharmonics, and I’m writing this supplement to my talk all in a flurry. Basically all I hope for is this post to give you a starter kit to explore it.


Starting from scratch. It is the set of all ratios. Hence, every ratio between pitches you can imagine is found here.

In Voyage into the Golden Screen, Per Nørgård uses two harmonic series, offset very slightly.


You may notice that pitches get closer and closer together as they get higher. That’s because pitch space is logarithmic, specifically base 2. That is, every time you multiply by 2, you’ve moved the same distance, no matter where you were.


Just Intonation is the use of pitches which are in pure ratios to each other. This gives them a certain fundamental beauty in their purity. Compare the sound of pure Just Intonation intervals with tempered ones here; Jon Catler has designed a guitar which contains frets for both:


Another major approach to harmony is to use pitches which are all a single-sized step away from each other. There is a different sort of beauty in the consistency of the intervals between the pitches, and it can often be easier to use a system like this.

This is one of my favorite songs in an equal division of the octave other than our standard 12. This one is 15ed2, that is, it divides the pitch ratio 2:1 into fifteen equal steps. Within that tuning it uses a Blackwood style Moment of Symmetry scale. Moment of Symmetry scales are an important idea popularized by Erv Wilson, which describe most historical tunings as iterations of an generator looping around an interval of equivalence until no more than two different step sizes remain. There is much more to be said about that, which you can read about here: But in terms of the results it means it just alternates between moving by 1 and 2. So it would go 1-3-4-6-7-9-10-12-13-14-15/1.

Protagonism in Violet



You can read Igliashon Jones’s thoughts behind this tuning here:

They relate interestingly to Rothenberg Propriety, and the Stern-Brocot tree and Farey Sequence.


Ivor Darreg was very fond of looking at equal divisions without focusing on the underlying JI intervals which shine though / they imply. But many people choose their equal division to best approximate whichever set of JI ratios which they most care about at that time. For example, our common tuning system uses 12 steps to an octave in order to closely approximate 3- and 5- odd limit harmony, in other words, all the ratios for which 3 and 5 are the highest odd numbers in them. This tuning does a pretty bad job at approximating

This piece is written in 31ed2, which has much better approximations of 7-limit harmony than 12.


JC & The Microtones – Cow People

One way we may look at a temperament is tempering out a “comma.”


As I said in my talk, it is possible to choose a smaller and smaller atomic unit of pitch to approach better approximations of your underlying JI intervals. However, the resulting system gets increasingly difficult to work with, as you must memorize more and more steps.

Georg Friedrich Haas took an interesting approach here with his piece Limited Approximations. He chose 72-tet, which divides the space between notes in our standard 12-tet by six. Since most 5-limit and 7-limit intervals are off by 1/6ths or 1/3rds of the space between notes, this results in incredibly close approximations of these intervals. Plus, dividing the notes up this small is very near the limit of most people’s ability to distinguish pitches, so it is very near pure glissando. And Haas achieves this by tuning each of six pianos a sixth step apart, so each pianist can play individually without much strain, and the orchestra just takes their tuning cues from them.

Haas also works in pitch circularity, while not truly xenharmonic, is another fun thing you can do with pitch. Check out the Shepard-Risset Glissando to get the gist. Then check out his epic “in vain”, which contrasts harmonic series tuning with common tuning:


The octave is the most important interval (2:1). It is found in every human musical tradition. We consider pitches with this relation to be the same note, with the same letter. We call it “pitch class,” or “octave equivalence.”

Well, in Bohlen-Pierce, this most important of intervals is thrown out. You are left with 3:1 as the new interval of equivalence, known as the tritive. Equal temperament is 13 steps. There is still room for development in the BP modes, but you can read Elaine Walker’s research here:

You can listen to a fantastic song she wrote in it here:

ZIA – Stick Men


Wendy Carlos: you may know her for Switched-On Bach, or the original Tron soundtrack. But the most important work of this electronic music pioneer is her xenharmonic masterpiece, Beauty in the Beast. Her most famous innovations here are her equal divisions of smaller-than-the-octave intervals.


Carlos’s harmonic tuning from this album is also interesting. She uses many of the pitches in the 6th octave of the harmonic series to create a scale. One can take this to an extreme and look at the 8th octave of the harmonic series, which contains 128 notes to choose from. If you use any subset of these notes, since they all share a fundamental, they have a wonderful harmonic synergy. And since if these pitches are within human hearing range then their fundamental is so low that it’s below human hearing range, it is merely psychoacoustic, which is awesome and mysterious.

Check out this piece by La Monte Young which uses this tuning:


You can flip the harmonic series upside down, and open a whole ‘nother world of tonality.

Undertone Blues by Willie McBlind


This is another great invention by Erv Wilson. Here you just take a set of numbers, like 3,5,7,11, say and take all of their unique products, and lay them out on a lattice. The beauty is that the scale is symmetrical insofar as it contains an equal number of harmonic and subharmonic ratios.


Kraig Grady’s site has all sorts of other wonderful things on it.

This is another method like CPSs of working with both harmonics and subharmonics. Just lay out pitches so that columns have the same denominator and rows have the same numerator. You’ll have one diagonal line of all 1/1 home tones.

Listen to this piece by Harry Partch, the master of it:


Uses third and fourth steps, more on this later.


The Gamelan is a style of instrument / orchestra historically developed in Bali. It had a surge of popularity in California in the 70’s. It uses tunings close to 5ed2 and a subset of 9ed2. Interestingly, pieces of the orchestra are stylistically left out of tune with each other. The resulting difference tones, which are very low in pitch (that is, if you play 401Hz and 400Hz simultaneously, their difference tone is 1Hz), create an effect we call “beating,” which for many is considered unpleasant. To them, however, these beats are considered meditative stepping stones to God.

Here’s a clip of some gamelan music:


Harmonic entropy is an idea put forth by Paul Erlich, and it is a complicated idea that I still struggle with, honestly. I understand that basically it is a measure of dissonance of a chord, and that one way to think of it is as how low in pitch the shared fundamental would be of the pitches included in your chord. For example, a justly tuned major chord will have low harmonic entropy, because its three pitches are in the ratio 4:5:6 to each other; their shared fundamental is only two octaves below the root. However, if you picked three random pitches which were in really dissonant relation to each other, their shared fundamental will tend to be extremely low, and they their overtone numbers will be really high, like they are the 19675th, 349893rd, and 2897893rd overtones of it or whatever.




Sagittal notation:

logo to learn about Benedetti height, vals & monzos, wedgies, etc. home of XPSC, Xenharmonic Praxis Summer Camp, among many other wonderful things for some cool Max Patches, “Hex” and “2032”




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Ron Sword custom guitars:


Genesis of a Music by Harry Partch

Tuning Timbre Spectrum Scale by Bill Sethares:

New Musical Resources by Henry Cowell


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