User:Romeolz/Isomorphic layouts/Harmonic Table extensions: Difference between revisions
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= Harmonic Table extensions = | = Harmonic Table extensions = | ||
The Harmonic Table (HT), aka | 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. | 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). | ||
[[File:12edo harmonic table augmented diminished octave and unison.png|none|thumb|900x900px|12edo HT for reference]] | [[File:12edo harmonic table augmented diminished octave and unison.png|none|thumb|900x900px|12edo HT for reference]] | ||
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== 19-based extensions == | == 19-based extensions == | ||
In 19edo, it just so happens that there is | 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) === | === w/ magic twelfth (3125/1024 ~ 3072/1024 = 3/1) === | ||
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=== Orwell === | === 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 7/6. | 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. | ||
[[File:"HT" (3-1)^(1-7).png|none|thumb|600x600px|A powerful temperament for even 11-limit just intonation, but the layout is quite spread apart...]]To be continued... | [[File:"HT" (3-1)^(1-7).png|none|thumb|600x600px|A powerful temperament for even 11-limit just intonation, but the layout is quite spread apart...]]To be continued... (alternate sixths etc., fourths, hemififths...) | ||