User:Romeolz/Isomorphic layouts/Harmonic Table extensions

Harmonic Table extensions

The Harmonic Table (HT), aka (names) is an isomorphic layout designed to work with 12edo. It exploits the fact that 12 is divisible by 3 and 4, by mapping 1{backslash}3 and 1{backslash}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.

Canonical (12-based) extensions

In the 12edo HT, the octave can be reached using augmented temperament (1{backslash}3) horizontally, or using diminished temperament (1{backslash}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.

images

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.

images

19-based extensions

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

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

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More extensions using equivalence continua

[7 0 -3> [-10 -1 5> [-13 -2 7>

[3 4 -4> [-6 -5 6> [-9 -6 8>