Evolutionary Harmony with a Schismatic-Fourth Keyboard Mapping

This page explains how a retuned conventional keyboard can be used to add harmonic spice to conventional music in the western tradition. You can add nicely tuned microtones to standard triads. You don't need any specialist equipment and you should be able to sneak the microtones into a context with other musicians playing the usual notes on their usual instruments.

There's a certain amount of speculation here. I've made a fair bit of microtonal music, but I do it in an antisocial way by retuning everything unconventional harmonic structures that mean normal instruments will sound out of tune and normal musicians won't know what's going on. So I'm not talking about what I've done, but what I would do if I were working with other people.

I make vague statements about "normal musical instruments" and "conventional this or that" because there's a lot of variation in the tuning flexibility of existing instruments, from pianos that have a fixed tuning an there's not much you can do about it during a performance to violins and trombones, and even the human voice if Auto-Tune breaks, that can play any pitch you like. So the worst case is that you'll be playing along with instruments tuned to 12 tone equal temperament (12-equal) but you're still likely to find some instruments following the keyboard and some musicians will already be used to adaptive just/Pythagorean intonation. These ideas make a lot of sense for a keyboard used for ear training or rehearsing string quartets and choirs, but I don't think anybody's doing that. That must be because I didn't explain it very well so I'm trying again here.

My intended audience is somebody with a good knowledge of conventional harmony and an interest in exploring microtonality but no current knowledge of it. Then again, the conventional parts of the chords are fairly simple, and I'll throw in obscure microtonal terminology, so whoever you are see what you can take from it.

Quick Start

This tuning doesn't require any specialist microtonal equipment, so you can dive right in and hear what it sounds like. You'll need a MIDI keyboard, a cable to plug it into your computer, and a soft synth that supports full-keyboard tuning tables. If you don't have a MIDI keyboard, well, there's not much that you can do right now, but at least you can read about the cool things you'll be able to do when you get one. If you don't have the cable to plug the keyboard into your computer, you probably want a MIDI DIN to USB cable, and music shops will have them and you can probably find one online. If you're not sure about the soft synth, have a look at ZynAddSubFX.

To apply the tuning, download the scala file or a variant giving pure octaves. You might notice that I call this "cassava" as a mispronounced abbreviation of "cassandra fourth". Your synthesizer will hopefully be able to load this file. You can also get a ZynAddSubFX tuning file with the reference pitch consistent with this page.

Get everything set up and you can marvel at the strange sounds you've twisted your keyboard into producing and how stupidly difficult you've made it to play normal tunes. You might be able to find some exciting microtonal tunes that you've never heard before. You can certainly look for exciting chords as well, but there are a lot to search through, so it's good to have some theory to guide you. So I'll carry on talking about it.

The Keyboard

The fundamental idea is to take a standard keyboard (the layout some of us call "Halberstadt") and map it so that the repeating pattern of twelve keys covers twelve pitches to a perfect fourth instead of an octave. This adds up to twenty-nine distinct pitches to the octave, which opens up a lot more subtlety of pitch, as well as stretching tunes and chords out in an uncomfortable way. The choice of a perfect fourth isn't arbitrary. I explained it some time ago, but didn't do a very good job, so here I am trying again.

The principle that makes the harmonic landscape a bit more familiar is that there are twelve black notes to the octave. They aren't tuned the same way as a standard piano, but they're close enough that if you only play on the black notes. It should all make sense given standard notation and so on. They're closer still to a Pythagorean scale: most perfect fifths are very tuned, but one in each octave, called the "wolf", is very different. The white notes are used to get clean chords or microtonal melodic affects as you see fit.

This table shows the pattern of large and small steps making up a perfect fourth:

Pitch C C♯ D E♭ E
Keyboard key name C D E F G A B C
Intervals π r π π r π r π π r π r

The first row shows what you'll hear when you press a key. The second row shows the pitches assigned to the white keys in conventional tunings. There's a problem here in that what you see is different to what you expect to hear, and there's a danger of confusing the names of pitches with keys. I'll try to resolve this confusion by, for example, calling the thing you press "the C key" and the sound you hear "C". They happen to agree here, but if you press the E key you'll hear D♭. I hope this makes sense.

There are two intervals between consecutive keys on the keyboard. I call them "π" and "r". You might know π as the Pythagorean comma (or an approximation of it depending on the tuning). Two pitches that differ py π aren't distinguished in conventional notation. You might call r "the 5-limit chromatic semitone". It corresponds to the difference between a minor and major third. It's a bit smaller than a common, equally tempered semitone, and a fair bit bigger than π, but small enough that it sometimes works like a quartertone.

I can assign names to other intervals on the keyboard, defined as algebraic combinations of π and r:

π + rlimma (Pythagorean diatonic semitone)
2π + r diatonic semitone (apotome or Pythagorean chromatic semitone)
2π + 2rminor tone
3π + 2rmajor tone
5π + 3rclean minor third
5π + 4rclean major third
7π + 5rperfect fourth

This is starting to look like a mathematical text, and somebody out there's convinced they need a Ph.D. in something or other to understand it. But the pattern of intervals fully apparent if you look at the keyboard. The steps between black keys are all semitones. Where only one white key separates a pair of black keys, you have a limma. Where two white keys separate them, you have a diatonic semitone, which as its name suggests is the semitone you'll naturally find in a 5-limit diatonic scale. The diatonic semitone is larger than the chromatic semitone, and it looks larger.

Two semitones make a whole tone. There two different sizes of whole tone on the keyboard. The smaller one, between the F♯ and B♭ keys, is the minor tone, and the larger one, between C♯ and F♯ and so on, is the major tone. The major tone is historically more important as it's in the Pythagorean scale between the fundamental intervals of the fourth and fifth. Go up a major tone and find the key that looks like it's an octave above where you started: this is a perfect fifth from where you started. Go up what looks like another octave, and what you have sounds like an octave from the starting point.

The minor tone and diatonic semitone are used to get 5-limit intervals: a major tone plus a minor tone gives you a 5-limit major triad as occurring in the harmonic series. A major tone plus a diatonic semitone makes up a 5-limit minor third, the other component of a 5-limit triad.

I assume you're going to use a five-octave keyboard (that is, five octaves of conventional tuning) mapped like this:

Pitch G A♭ A B♭ B
Pitch C C♯ D E♭ E
Pitch F F♯ G A♭ A
Pitch B♭ B C C♯ D
Pitch E♭ E F F♯ G
Keyboard key name C D E F G A B C
Intervals π r π π r π r π π r π r r

There's a visual pattern to these assignments. If you look at the spacing of most keyboards, the black keys tend to belong to a white key. Or, looked at in a way more relevant to this mapping, each white key tends to belong to a black key. C belongs to C♯, E belongs to E♭, and so on, in a way that I highlight with the choice of key names. The G and A keys both belong to the same black key, that I'm calling G♯ today. The "belongs to" concept translates as "is separated by a comma from". The D key is ambiguous: it could belong to either C♯ or E♭. Today it belongs to E♭.

The implication of the tuning pattern is that there's a single chain of fifth or octaves (they're fourth-equivalent, and also equivalent to a major tone). A perfect fifth is 17 steps on the keyboard, and 11 of the 12 intervals of 17 steps are perfect fifths. The bad fifth (or bad major tone) is between the A and D keys, which mirrors the wolf fifth of a just major scale. (I originally put the wolf tone between D and G, but changed it for the sake of the 11-limit chords that I'll show later.) An octave is 29 steps on the keyboard, an only 11 out of 12 intervals of 29 steps are octaves because the pattern doesn't repeat every octave. (You could make it do so if you really wanted.)

The black keys on the keyboard also follow a single chain of fifths (or tones or octaves). There's one obvious way of doing this, and it's like a pentatonic scale centered on the G♯ key. This leads to wolf fifths and octaves: intervals that look like fifths or octaves but are mistuned because the black keys here are not tuned to 12-equal. I assigned pitch names so these wolves end up in the middle of a C major scale. The wolf fifth of this scale is between the D and A pitches. This corresponds to the "syntonon diatonic" scale that gives just F, C, and G major chords but might be confusing because moving by major fifths takes you off the scale. But it doesn't take you off the keyboard. The chain of fifths above C takes you to an A pitched on the B key a comma above the other A pitched on the B♭ key. An equal tempered fifth would leave you between the B and B♭ keys (but much closer to the B key).

Cassandra Tuning

The tuning is based on Erv Wilson's Cassandra. I've optimized it for 11-limit harmony but it also works in the 13-limit. Here's a Scala file to cut and paste:

Schismatic fourth tuned to 11-limit Cassandra

You can also download the scala file but you probably already did.

Cassandra belongs to the schismatic family of temperaments. They're structured so that all pitches are reachable on a single spiral of fifths, which matches the intervals I described on the keyboard before. That's a good thing as an evolutionary step from music that tends to relate chords by perfect fifths. It becomes a problem when you play alongside instruments fixed to 12-equal because the fifths have to be tuned differently. That means the best agreement with with equally tempered instruments is over a few chords with roots related by fifths.

This particular tuning is optimized for the 11-limit, meaning you can get some interesting pitches that are roughly quartertones relative to the twelve standard pitches. A consequence of this is that π (the smallest interval on the keyboard) is a bit bigger than it would be in Pythagorean intonation, and easier to hear as a melodic step. It happens that the comma π is about half the size of the chromatic semitone r, so the tuning is roughly a subset of 41 tone equal temperament.

A drawback is that the fourths and fifths are further from equal temperament than other tunings like 53 tone equal temperament. This means that the further you get on the spiral of fifths from your reference pitch, the further your retuned keyboard will get from any conventionally tuned instruments you're trying to play with. The discrepancy is about 2.4 cents per fifth, so six fifths away an you'll be more than 14 cents away from 12-equal. This will probably be enough that people will start to notice that you're out of tune in such a way that wasn't the point of shoving 29 pitches into the octave. The idea is that the retuned keyboard is used as the pitch reference for other instruments. Musicians tend to think in terms of Pythagorean fifths (and string players tune to them) so the Cassandra tuning is closer to those expectations than 12-equal is. With a little more mistuning relative to 12-equal, but in the other direction, you get meantone temperament, which smooths out all the commas. While this has its advantages, the fifths will tend to stand out as flat because they're further from Pythagorean than the schismatic family.

Roots chained by fifths will drift away from their equal tempered positions, and thirds will generally sound out of tune relative to equal temperament. If you're playing against a bass guitar, get it to concentrate on the roots of chords and nearby fifths will sound fine and there'll be a light/dark effect on the opposite side of the spiral of fifths. If you're playing with guitars or normally tuned keyboards that insist on playing chords, there'll be clashes. Either accept a bit of discordance, or arrange so that the equally tempered instruments hold back when you take your spicy chord solos and return the compliment during 12-equal sections.

If you want to tune a bit closer to 12-equal, you can use a Scala file for 53 tone equal temperament. This is a fine tuning that gets intervals close to their theoretically ideal tunings until you get to the 11-limit quartertone, which you might not even care about. Because the commas are smaller, the steps on the keyboard are more uneven, and this might be a problem with some microtonal melodic tricks. There are probably irregular tunings that do a better job of most fifths by breaking a few of them but I haven't worked them out.

11-limit Hexads

Here are the fingerings for some complete 11-limit chords, voiced as 8:9:10:11:12:14 (except E♭, which only fits on with a strange voicing, so I'll say no more about it). I chose the voicing so that it sounds good, and you can play it on a normally sized keyboard with the two hands most of us have.

The root pitch The keys you press
E♭C♯ F AE♭C♯ F♯ B♭
B♭C♯ F♯ B♭D F♯ C
FC♯ F♯ B♭D F♯C
CC♯ F♯ B♭D F♯C
GC♯ F♯ B♭D F♯C

Using that chart (and a dash of enharmonic equivalence) you can play out of tune with any major-based chord in a conventional piece of music. Let's look at each pitch that I numbered 8:9:10:11:12:14.

8 is the root of the chord. It should be in tune with the roots other instruments are playing. The assumption here is that it won't be, so you set the reference pitch so that the most important chords are tuned well enough.

9 is a major tone above the root, which makes it the ninth degree of the chord. It's tuned close to twelve tone equal temperament, so it'll agree about as well as the root with other instruments playing the ninth. That also means it's not very exciting so it might be safer to leave it out and not draw attention to whatever mistuning there is.

10 is a major third above the root. It's about 20 cents away from 12-equal, which is enough to be clearly noticeable, but most of that's because 12-equal's wrong. So there's a dilemma here. If you play it your keyboard will sound out of tune with everything else. So you could leave it out, but then you won't be telling everybody else that they're wrong. You decide. From what we know about the way thirds are sung, you can get away with quite a lot, but be careful.

11 is where things get properly microtonal. You can think of it as the eleventh degree of the chord, and it's significantly flat of the usual eleventh. Relative to the root, it's between a perfect fourth and a tritone. If you like quartertones, you can take pride in this pitch being a good and proper quartertone away from wherever the instruments in twelve tone equal temperament think it should be, although not as good a quartertone as it would be in just intonation. There's good theory to say that it's in tune with the other pitches in the chord, although it might take a bit of getting used to. Use it where you like the sound of it.

12 is the fifth of the chord. It's not very exciting.

14 is the seventh degree of the chord. It's tuned about a third of a semitone flat of 12-equal. In barbershop harmony, and maybe other contexts, it can be tuned a lot like this. In other contexts, it's safest to pounce on chords where other instruments aren't playing the seventh.

Tritone Substitutions

Tritone substitutions are where a chord replaces a chord with a root a tritone away. With tunings like Cassandra, that tends to mean replacing a 5-limit triad with a 7-limit tetrad, and it's a way of sneaking 7-limit chords into a 5-limit context. Here are the tritone substitutions of chords with roots on the black keys of the retuned keyboard.

The root pitch The substituted root The keys you press
GD♭ (G♯ key)F♯E♭ BF
DA♭ (G♯ key)F♯E♭ BF
AE♭ (G♯ key)F♯E♭ BF
AE♭ (C♯ key)BG♯E B♭
EB♭ (C♯ key)BG♯E B♭
BF (C♯ key)BG♯E B♭
G♭C (F♯ key)EC♯ AE♭
D♭G (F♯ key)EC♯ AE♭
A♭D (F♯ key)EC♯ AE♭
A♭D (B♭ key)G♯FC♯ G
E♭A (B♭ key)G♯FC♯ G
B♭E (B♭ key)G♯FC♯ G
B♭E (E♭ key)C♯ B♭F♯ C
FB (E♭ key)C♯ B♭F♯C
CF♯ (E♭ key)C♯ B♭F♯C
GC♯ (E♭ key)C♯ B♭F♯C

And a few more that subsitute for pitches on white keys.

The root pitch The substituted root The keys you press
G♭C (F key)E♭C G♯D
D♭G (F key)E♭C G♯D
A♭D (F key)E♭C G♯D

15-limit Scales

Now I'll show you 8:9:10:11:12:13:14:15:16 chords. There big enough that they qualify as scales, and they're voiced in a single octave (as heard) so you might be able to play them as chords. (I can, but not comfortably).

The root pitch The keys you press
G♭E AC♯ F AC E♭ F# A
D♭E AC♯ F AC E♭ F# A
A♭E AC♯ F AC E♭ F# A

The second A in that chart is missing its 15 and 16 because they fall off the end of the keyboard.

There are quite a few chromatic degrees that don't have a root in here. If a different section of the spiral of fifths would suit your music better, you can change the reference pitch that matches the tuning to your keyboard (however you set that up on your synthesizer).

A quick roundup of the new notes here: 13 is a kind of thirteenth degree that's offset by a quartertone from 12-equal. It isn't particularly well tuned by the 11-limit Cassandra, but I don't expect anybody will notice. 15 is a major seventh, so you can use the chart to get a major seventh chord, or a minor triad. 16 is the octave above the root.

Xenharmonic Triads

Here are some 7:11:13:14 triads. They include intervals we've seen before, but they're useful as triads because there are a lot of them and they're a long way from 12-equal, so you can use them to elaborate harmony played by conventional instruments. You don't need to worry about sounding out of tune because the point is to add xenharmonic spice.

The missing otonal root pitch The lowest sounding pitch The keys you press
A♭G♭E♭B♭F G♯
E♭D♭E♭B♭F G♯
B♭A♭E♭B♭F G♯
(G♭)EBF♯C♯ E
(D♭)BBF♯C♯ E
(A♭)G♭EBF♯ A
(E♭)D♭EBF♯ A
(B♭)A♭EBF♯ A
(F)E♭G♯E♭ B♭C♯
(C)B♭G♯E♭ B♭C♯
(G)FC♯G♯E♭ F♯
(D)CC♯G♯E♭ F♯
(A)GC♯G♯E♭ F♯
(A)GF♯C♯ G♯ B
(E)DF♯C♯ G♯ B
(B)AF♯C♯ G♯ B

The first column shows you the pitch of chords you can elaborate by pressing the keys on the right. Sometimes these pitches are in parentheses (or brackets) which means they don't correspond to a key on this keyboard. That also means they're likely to be out of tune with equal temperament, but if you're creative you might find something to do with them. The second column shows the lowest pitch of the chord that sounds on this keyboard.

Minor Chords

If you give minor triads the same intervals as major triads, but in the opposite direction, there are as many of them on the keyboard as there are major triads. There are so many that you'd have to be insane to produce a list like this of all the root position minor triads:

The root pitch The keys you press
B♭C♯ AF♯
F C♯ AF♯
C FC♯ B♭
G FC♯ B♭
F♯ D B♭G
F♯ E♭ BG♯
C♯E♭ BG♯
G♯E♭ BG♯

With most of these, you can see that they come in pairs of chords on neighboring keys and they get listed with the same root pitch. These chords are a comma apart and conventional notation doesn't notate commas. Which one you use depends on whether you want the minor triad's root to be in tune with the spiral of thirds that the major triads' roots follow or you want the minor triad to share pitches with its relative major.

These 5-limit tunings fit really well with 16th century harmony. These days there are other ways to do it. On of those is Pythagorean intonation: everything based on simple steps along the cycle of fifths. It's easy to play such chords on this keyboard, and if you want to you can work it out yourself. What I will show are 9-limit 6:7:9 triads. These have a minor third smaller than the 5-limit or Pythagorean tunings, and further from 12-equal. They have a nice sound and I'll show them here with an added major sixth to get 6:7:9:10.

The root pitch The keys you press
B♭C♯ GF♯ B♭
FC♯ GF♯ B♭
CC♯ GF♯ B♭
GC♯ GF♯ B♭

There's some overlap, but these chords tend to work on roots where the 5-limit minor triads don't.

Melodic Tricks

You can play tunes by mixing up pitches from chords with pitches in between the pitches from chords, and whatever you like. There are some special things you can do with more notes, though, and the currently fashionable one is dividing a minor third into four roughly equal steps. This is covered in a recent Jacob Collier interview ("Microtonal Voice Leading" about 10 minutes in). On our keyboard, you can use the keys B♭, C, D, E, and F♯. The step between the D and E keys is only ⅔ as big as the others, and sounds a lot like a quartertone, so this doesn't fully comply with Mr Collier's requirements, but it's still a lot better than a naïve quartertone implementation. There are some harmonic relationships that make this a special trick beyond the celebrity endorsement. We can describe it with the ratios 20:21:22:23:24. 20:22 simplifies to 10:11 so if the B♭ and F♯ keys are the third and fifth degrees of an 11-limit hexad, the D key is the 11th degree. 21:24 simplifies to 7:8, which means that if the F♯ key is the root (or octave) of a 7-limit tetrad (an 11-limit hexad without the 9 and 11) (like G7 with a barbershop seventh), the C key is in tune with the seventh. There are also good melodic reasons to like this pattern: every step is a small semitone, and small semitones are good melodic steps. You can also extend this pattern to a major third by adding the G# key.

Another trick is an equal division of a perfect fourth as 6:7:8 to get two intervals sometimes called "semi-fourths". On the keyboard, this looks like dividing an octave into two tritones. This division is part of the 7-limit tetrad. You can go further and divide a fourth into four large semitones or six small semitones (like the ones in the previous paragraph).

Finally, how about dividing a whole tone into three equal parts? You can do this with the keys F♯, G, A, and B♭ for a nearly equal (much closer than it looks on the keyboard). The steps here are the melodically-good small semitones again. By the rule of major and minor tones being roughly the same, you can roughly equally divide a major tone as well (e.g. the C♯, D, E, and F# keys) giving one large semitone. Put them together, and you have six roughly equal divisions of a major third, and if six works so does three (e.g. the C♯, E, G, and G♭ keys).

Further Explorations

This keyboard works quite well for a piece of hardware being used in a way it was never intended. It's far from ideal, though. It's a big stretch for even normal chords, and a lot of voicings don't work at all. Here are some possible improvements:

Tuning documents