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).

Legend

dim oct = octave derived from diminished temperament, 2/1 ~ (6/5)^4, NOT A LITERAL DIMINISHED OCTAVE (ex. C4-Cb5)
aug oct = octave derived from augmented temperament, 2/1 ~ (5/4)^3, NOT A LITERAL AUGMENTED OCTAVE (ex. C4-C#5)
when one of these is crossed out, it means that octave mapping is no longer there and maps to another interval, and the arrow signifies where it would have been
magic twelfth = twelfth derived from magic temperament (and so on...)

12edo HT for reference

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. This is where we get 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.

HT with diminished octave, \| = 80/81, m3 /| = A2 \| \| \| = 6/5 (unmirrored)

HT with diminished octave, \| = 80/81, m3 /| = A2 \| \| \| = 6/5 (mirrored)

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.

HT with magic twelfth, \| = 80/81, M3 \| = DD4 /| /| /| /| = 5/4

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.

HT with hanson twelfth, \| = 80/81, m3 /| = AA2 \| \| \| \| \| = 6/5

More pure HT extensions using equivalence continua

The octave mappings are each 25/24 apart

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

The temperaments that support HT on PTS in 2.3.5-space, major third based ones intersect at 3et, minor third ones at 4et

Close-up (temperaments continue in the same pattern to infinity approaching the vertical dicot line on the right)

Harmonic Table-ish extensions

These are all the extensions outside of "pure HT", the way HT works in 12edo. We can replace the fifths and thirds with other intervals that are closely related harmonically.

"Why such few temperaments? Where is meantone? Why do some temperaments not work?"

A prerequisite for a HT-ish layout is that a single octave is reachable using some combination of the two chosen harmonically close intervals (usually 3/2 and 5/4). Using meantone's fifths and thirds we can only reach the double-octave. (this probably has something to do with the monzos of the intervals and commas but I don't know how yet)

It is always possible to reach the octave in any ET where the interval sizes are coprime, so the ET doesn't have to support any of these temperaments. Using a HT-like layout like this is highly impractical.

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.

An even smaller octave than the augmented layout!

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.

Very close to 2.3.7 just intonation, but octaves are quite a reach.

Orwell (fifths ver.)

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.

A powerful temperament for even 11-limit just intonation, but the layout is quite spread apart...

Alternate sixths/seconds (this is where it starts to get weird)

Using sixths instead of thirds has the effect that thirds are shifted towards lower octaves. These can resemble layouts akin to Lumatone's "Melodic Mode", with a clear albitonic scale.

Every sixth has a second of the same quality one fifth below. The sixth-layouts can be sheared along the fifths-axis to get the secundal equivalent.

Semaphore/Barbados (again)

This one is strange, the 951 cent interval derived from splitting 3/1 in half can be interpreted as 7/4, 12/7, 26/15 and many others. It's bordering on being a seventh, but it works because the interval one fifth up from it still behaves like a tenth.

The smallest possible octave! The range on this one is huge...

Porcupine

Porcupine is a powerful 11-limit system with a distinctive sound. Here, with a pure 3/1, the sixth is quite flat at 868 cents. The 3/1 can be sharpened to improve other intervals.

basically porcupine[7] melodic mode

Negri

A coooooooooooooool temperament?? I don't want to type these out anymore

basically negri[9] melodic mode

Blackwood

i love blackwood

heyy this one's pretty neat it splits the octave into 5

I don't even know what this one is but it has a good 11/7 lol

...

idk if anyone is ever gonna use this

...

Alternate fourths/fifths

todo

Fourths-axial

Instead of having a fifths-axis, the fifths are replaced by fourths. This effectively flips the harmony upside down when ignoring octaves. Fourths-axial layouts may be harder to conceptualize than fifths-axial ones.

Ones that resemble pure HT

Augmented

asdfghjkl

Diminished

adsfgh

Ones that are part of analogous equivalence continua, all kinds of thirds

The "augmented" continuum:

Dicot/Mohajira

adsdf

Squares

asdasd

Magi

fgdfjhfgds

Coming up with an equivalence continuum is tough for this one because the generator varies so much. If one were to choose a single JI interval all these should be approximating, the simple temperaments would be unrecognizable.

Image A

Image B

Continuation of fourths-axial augmented "HT" into various equivalence continua
	m	temperament, comma
approximated interval, location in genchain		5/4 1	9/7 1	5/3 2
image		A	B	todo
	-2	Yo 10/9	??? 8/7
	-1	Dicot 25/24	Ruru 54/49
	0	Augmented 128/125	Triru 729/686	Father 16/15
	1	Symbolic 2048/1875	Squares 19683/19208	Dicot 25/24
	2	??? 32768/28125	Magi 537824/531441	Magic 3125/3072
	3		Satribizo [7 -15 6	Augmented 128/125
	4		??? [9 -18 7	??? [18 -1 -7
	5

And the "diminished" continuum:

Sixix/Amity

When using amity, the best approximation of 5-limit thirds is further away from the root

Lemba

dffhddgfsfdsh

Orwell/Orson (fourths ver.)

hfhdgfsh