Harmonic Table extensions

The Harmonic Table (HT), aka Sonome, Tonnetz or 5-L lattice, is an isomorphic layout designed to work with 12edo. It exploits the fact that 12 is divisible by 3 and 4, by mapping 1\3 and 1\4 right next to the origin. It can seem that the Harmonic Table can only be used for 12edo in that case, but it can be used for other tunings by making some changes.

I will refer to "pure HT" when talking exclusively about layouts that map 5/4, 6/5 and 3/2 to the same location as the 12edo HT (including reflections and rotations).

Canonical (12-based) extensions

In the 12edo HT, the octave can be reached using augmented temperament (1\3) horizontally, or using diminished temperament (1\4) diagonally. 12edo is the unique intersection of these two temperaments. From this we can derive the first two temperaments that support versions of the Harmonic Table by themselves. This adds flexibility to the choice of tuning.

w/ augmented octave (125/64 ~ 128/64 = 2/1)

This is seemingly the most intuitive of all the versions, as the octave is reachable and nearly horizontal without modification.

HT with augmented octave, \| = 80/81, M3 \| = D4 /|/| = 5/4 (idk why the Sagittal wiki template isn't working)

w/ diminished octave (1296/625 ~ 1250/625 = 2/1)

This is much more obscure than the augmented version. The diminished octave mapping sees much less use in the 12edo HT. It runs diagonally, so the playing direction is altered, or alternatively, the layout can be mirrored along the fifth-axis, swapping the places of the major and minor third.

19-based extensions

In 19edo, it just so happens that there is a different interval, the twelfth, that can be reached by stacking both major thirds and minor thirds. The twelfth can be reached with 5 major thirds or 6 minor thirds, compared to the 3 major thirds or 4 minor thirds of 12edo. The octave mappings aren't as obvious, because they aren't located on an offset axis.

w/ magic twelfth (3125/1024 ~ 3072/1024 = 3/1)

This one is more akin to the familiar augmented layout. The octaves are a bit far apart but still reachable.

w/ hanson twelfth (46656/15625 ~ 46875/15625 = 3/1)

This one is like the diminished layout, as its octaves run diagonally, or horizontally when mirrored. The octave starts to be a bit tough to reach, especially multiple octaves. Hanson temperament can get the closest to 5-limit just intonation out of all the pure HT temperaments.

More pure HT extensions using equivalence continua

The major third extensions can be visualized in the layouts like so: every time m increments, the octave mapping moves up by one 25/24, which is the difference between 5/4 and 6/5. From this we get the equation* (5/4)^3 * (25/24)^m ~ 2/1 ⇒ (125/128) * (25/24)^m ~ 1 ⇒ (25/24)^m ~ (128/125).

This is essentially the same thing as the Father–3 equivalence continuum.

Major thirds (25/24)^m ~ (128/125)
m	comma	monzo	temperament name
-1	16/15	[4 -1 -1⟩	father	exotemperament
0	128/125	[7 0 -3⟩	augmented
1	3125/3072	[-10 -1 5⟩	magic
2	78125/73728	[-13 -2 7⟩	wesley
3	1953125/1769472	[-16 -3 9⟩	(3 & 33c)	high complexity low accuracy

The minor third extensions can be visualized in the layouts like so: every time m increments, the octave mapping moves down by one 25/24, which is the difference between 5/4 and 6/5. From this we get the equation* (6/5)^4 * (24/25)^m ~ 2/1 ⇒ (648/625) * (24/25)^m ~ 1 ⇒ (25/24)^m ~ (648/625).

I couldn't find any established way of conceptualizing this continuum.

Minor thirds (25/24)^m ~ (648/625)
m	comma	monzo	temperament name
-1	27/25	[0 3 -2⟩	bug	exotemperament
0	648/625	[3 4 -4⟩	diminished
1	15625/15552	[-6 -5 6⟩	hanson	nearly just
2	390625/373248	[-9 -6 8⟩	doublewide
3	9765625/8957952	[-12 -7 10⟩	(4 & 33c)	high complexity low accuracy

Harmonic Table-ish extensions

These are all the extensions outside of "pure HT", the way HT works in 12edo.

Alternate thirds

This encompasses all HT extensions where 5/4 and 6/5 are replaced with some other type of "third", like 9/7 and 7/6, or 16/13 and 39/32. The definition of a "third" is nebulous, so I'm using it to refer to intervals between 240 and 460 cents.

Semaphore/Barbados

By splitting the fourth in half we get an interval of about 249 cents. This can be interpreted in many ways, as 7/6, or 15/13, or many others. Using this unlocks more tunings for use, like 24, which are otherwise incompatible with HT.

Stearnsmic

Splitting 9/2 into 6 equal parts yields an interval very close to 9/7 at about 434 cents. The octave is now already larger than in the hanson layout.

Orwell

Orwell offers good approximations of even 11-limit intervals, using its generator of about 272 cents to split 3/1 into seven. The generator can be interpreted as 7/6.

To be continued... (alternate sixths etc., fourths, hemififths...)

User:Romeolz/Isomorphic layouts/Harmonic Table extensions

Contents