Isomorphic Keyboards

One of the first obstacles an aspiring musician will face is trying to understand the relationships between the different notes they can make with their instrument of choice: a piano looks like it's missing every third or fourth black key; a guitar in standard tuning starts repeating pitches five frets along the next string, except for one string where it's only four; a saxophone plays different notes depending on which combinations of keys are held down; and fretless instruments like a violin or sliding instruments like a trombone offer practically no guidance to the desired positions on their own.

Chords make it all even more confusing; the same chord shape can get twisted into all sorts of arrangements depending on which note is the root. Playing a basic major chord on a piano can end up on all white keys, all black keys, or some combination of both. On a guitar, a major chord might involve all six strings or only five or four, and a player's fingers need to learn their own form of gymnastics to switch from one shape to another in time with the music.

Fortunately, there is a better way. Not only is it easier to learn, but it opens up a whole range of new possibilities unavailable on many traditional instruments.

This interface is called an isomorphic keyboard, so named because one hand shape plays the same type of chord no matter what the root pitch is. The pitches are arranged on a hexagonal grid in such a way that each direction represents movement by one certain interval. Such a layout is achieved by selecting any two intervals to define the pattern.

G		Ab		A		Bb		B
	E		F		Gb		G
C		Db		D		Eb		E
	A		Bb		B		C
F		Gb		G		Ab		A
	D		Eb		E		F
Bb		B		C		Db		D
	G		Ab		A		Bb
Eb		E		F		Gb		G
	C		Db		D		Eb
Ab		A		Bb		B		C
	F		Gb		G		Ab
Db		D		Eb		E		F

For example, in the above diagram, upward movement represents an ascending perfect fifth (C to G to D to A), so therefore downward movement is by a descending perfect fifth (C to F to Bb to Eb). Movement to the "northeast" is an ascending major third (C to E to Ab to C), so "southwest" is descending (C to Ab to E to C). Such motion is universal across the entire grid; starting on C and moving northeast to E and then moving upward from there will yield a perfect fifth above E, which is B. Given this arrangement, it is clear that motion along the "northwest"/"southeast" axis is by minor thirds.

G
	E
C

To play a basic major chord in such an arrangement, the shape is a triangle: a cluster of two hexes on the left bordering one hex on the right (such as C-major, C-E-G). This shape can be moved to any position on the grid and it will yield a major chord built on whatever pitch is in the lower-left hex; for example, look for B-Eb-Gb (B-major) or D-Gb-A (D-major). A minor chord is merely flipped around: the lone hex on the right is instead moved to the left (C-Eb-G).

Additionally, since the physical pattern between pitches does not change regardless of position, compositions can be played in a different key without any need to change the relative hand movements; exactly the same motions are made around a different tonic hex.

Again, an isomorphic arrangement is not limited to the one given in this example, but can instead be built from any two intervals, including those not available in standard Western twelve-tone tuning. An axis could be defined by the 7:4 harmonic (or "barbershop") seventh, or by the 352.9-cent neutral third of 17-EDO, or any other exotic interval your music might demand. As such, tunings that are completely unattainable on a traditional instrument can be accommodated with ease.

Is such a keyboard actually practical? Absolutely. Various isomorphic layouts have been in use for real instruments since the 1800s. Early on, these were unique, custom-built acoustic instruments, though by the late 20th century technology allowed the development of several electronic interfaces that could be automatically retuned to an entirely different layout at the whim of the performer. These specialized devices have typically been as costly as many other instruments, though there are now inexpensive apps for mobile touchscreen devices that make isomorphic keyboards easily accessible and widely available, and there are even free resources online that let visitors use their computer keyboards as isomorphic instruments!

19-EDO Guitar

Recently exhumed from my bedroom closet (along with a really nice tuner I didn't even know was in there) is an old electric guitar that has certainly seen better days. I don't believe it would be worth the cost to restore, but it's still basically functional. This makes it a prime candidate for some, shall we say, unorthodox modifications.

I want to rip out all the frets, refinish the neck, and install new frets with 19 to the octave instead of 12.

Why 19? Well, it's the next equally divided octave (EDO) up from 12 (the standard tuning of modern Western music) that works well for triadic harmony (the most basic chords). Not only does it sound good, it's actually better than normal tuning at approximating 5-limit just intonation. Any piece written for standard tuning can actually sound a little bit purer if it's played in 19-EDO. For example, in 12-EDO a major third is 13.7 cents sharp from the "sweet spot" of the just major third (5:4), but in 19-EDO it's only 7.4 cents flat. Even better, the 12-EDO minor third is 15.6 cents flat from just (6:5), but in 19-EDO it's less than 0.15 cents sharp! Also, as inversions of the thirds, these benefits apply to the major and minor sixths as well.

Of course, there's still more to gain: 19-EDO offers 7 more pitches per octave than standard tuning. The main benefit here is the introduction of some extremely useful new intervals: the traditional second, third, sixth, and seventh are all available in not only two varieties (major and minor) but four each: supermajor, major, minor, and subminor. Additionally, the tritone of 12-EDO is split into two distinct intervals, one for the augmented fourth (the real tritone, a stack of three whole tones) and the other for the diminished fifth. Utilizing these extra intervals opens up broad new expressive possibilities, both melodically and harmonically.

CENTS	INTERVAL
0.0	Unison
63.2	Subminor 2nd
126.3	Minor or Neutral 2nd
189.5	Major 2nd
252.6	Supermajor 2nd or Subminor 3rd
315.8	Minor 3rd
378.9	Major 3rd
442.1	Supermajor 3rd
505.3	Perfect 4th
568.4	Augmented 4th
631.6	Diminished 5th
694.7	Perfect 5th
757.9	Subminor 6th
821.1	Minor 6th
884.2	Major 6th
947.4	Supermajor 6th or Subminor 7th
1010.5	Minor 7th
1073.7	Neutral or Major 7th
1136.8	Supermajor 7th
1200.0	Octave

Now, given this list, it's clear that the minor second and its inversion, the major seventh, deviate from 12-EDO by 26.3 cents, which is significant enough they could be considered "neutral" (halfway) intervals. Fortunately, they both fare better as approximations of their respective just intervals (16:15 and 15:8), deviating only 14.6 cents; this is still a distinct difference, though it's slightly less terrible than the 15.6-cent deviations from just that the minor third and major sixth exhibit in 12-EDO. If those intervals are tolerated, certainly these can be as well.

With such a different system, you might think I'd need to relearn how to play every scale and every chord. In fact, the new frets that correspond to the equivalent old frets won't have moved far, so the same hand positions will sound the same scales and the same chords. Effectively, the standard frets will only have shifted up or down the neck a little to make room for the new arrivals. They will be different enough that they will sound "out of tune" if played alongside a 12-EDO instrument; however, 19-EDO is actually more "in tune" to pure intervals than the current standard!

Writing in Locrian

In modern Western music, the Locrian mode is built from the seventh degree of the major scale (at a piano, you could start on B and play only the white keys). In this mode, the tonic triad is diminished. This makes it very difficult (or even, according to traditional phrasing practice, impossible) to write in this mode without the result sounding like it's actually in a different key.

The tonic must be continually reasserted as the center, which is challenging because a diminished chord is quite dissonant and does not sound like a place of rest. This chord can be made more manageable, though less distinctly Locrian, by omitting its fifth degree and adding a minor seventh; another option is to "cheat" by raising the fifth degree chromatically only for the tonic chord. In either case, to define a Locrian identity, the modified tonic chord would require balance from the iii and V chords in proper diatonic form, where the true fifth degree of the mode can be voiced more comfortably.

If the fifth degree is raised permanently, the mode becomes Phrygian. This change makes the iii chord major, but more importantly, the V chord becomes diminished, which can be nearly as troublesome as a diminished tonic chord. In both Locrian and Phrygian, the second, third, sixth, and seventh degrees are all minor; they can sound even darker than the Aeolian mode, otherwise known as the natural minor scale, which has a major second. If the pattern of whole- and half-steps in the Locrian mode (HWWHWWW) is reversed, the result is the Lydian mode, which is completely major but features an augmented fourth.

Due to the difficulties inherent in working with the Locrian mode, it has always been extremely uncommon, though a few composers have made use of it. Claude Debussy exhibits the mode in three passages of his last orchestral work, Jeux. It can be done!

The Equally Divided Tetrachord

I've recently been studying tetrachords as well as equally divided octaves (EDOs). This morning, I was inspired to equally divide the just perfect fourth (4:3) into various sizes, starting with the tetrachord.

CENTS	ROLE
0	Hypate (Tonic)
166	Parhypate (Second)
332	Lichanos (Third)
498	Mese (Fourth)

The basic step size in the equally divided tetrachord (EDT) works out to 166 cents. This yields a second that is -34 cents from that of the standard 12-EDO major, and -37.9 cents from the just major tone (9:8), but only -1.5 cents from the 29-EDO neutral second. Likewise, the EDT also yields a third that is +32 cents from the 12-EDO minor, and +16.4 from the 5-limit minor (6:5) or +37.9 from the Pythagorean minor (32:27), but only +1 cent from the 29-EDO neutral third. As the perfect fourth of 29-EDO is only -1.4 cents from just (4:3), clearly a subset of 29-EDO makes an excellent approximation of the EDT. 36-EDO offers an even better approximation, and if its 500-cent perfect fourth is used as the basis instead, the EDT becomes an exact subset of 36-EDO.

Regardless of whether the EDT is played straight or tempered to an EDO, it features what can best be described as a neutral second that leans toward major and a neutral third that leans toward minor. Adding a disjunct EDT from the just perfect fifth (3:2) to round out an octave (additional pitches at 702, 868, 1034, and 1200 cents), the resulting pattern of the interval "leanings" (M2, m3, M6, m7) suggests the Dorian mode. Alternatively, adding a conjunct EDT from the fourth (additional pitches at 664, 830, 996, and 1200 cents), the fifth degree turns out very flat, -36 cents from 12-EDO and -37.9 from just (3:2), though still only +1.9 from the nearest degree of 29-EDO; the sixth is neutral leaning toward minor, +16.4 cents from just (8:5) and +2.4 cents from 29-EDO, while the seventh is the Pythagorean minor (16:9), +3 cents from 29-EDO.

This study has heightened my interest in 29-EDO, and I plan to explore it in greater detail. Also, of course, the perfect fourth can be divided equally into other step sizes as well; I've done the basic calculations for several of these, and a few look very interesting!