User:Romeolz/Isomorphic layouts/Harmonic Table extensions: Difference between revisions
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== Canonical (12-based) extensions == | == 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. | 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) === | === w/ augmented octave (125/64 ~ 128/64 = 2/1) === | ||
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|high complexity low accuracy | |high complexity low accuracy | ||
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[[File:ProjectiveTuningSpace ht1.png|none|thumb|900x900px|The temperaments that support HT on PTS, major third based ones intersect at 3et, minor third ones at 4et]] | [[File:ProjectiveTuningSpace ht1.png|none|thumb|900x900px|The temperaments that support HT on PTS in 2.3.5-space, major third based ones intersect at 3et, minor third ones at 4et]] | ||
[[File:ProjectiveTuningSpace ht2.png|none|thumb|900x900px|Close-up]] | [[File:ProjectiveTuningSpace ht2.png|none|thumb|900x900px|Close-up (temperaments continue in the same pattern to infinity approaching the vertical dicot line on the right)]] | ||
= Harmonic Table-ish extensions = | = 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. | 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 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. Using meantone 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) | A prerequisite for a HT-ish layout is that a single octave is reachable using some combination of the two chosen harmonically close intervals. 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) | ||
== Alternate thirds == | == Alternate thirds == | ||
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[[File:"HT" (9-2)^(1-6).png|none|thumb|600x600px|Very close to 2.3.7 just intonation, but octaves are quite a reach.]] | [[File:"HT" (9-2)^(1-6).png|none|thumb|600x600px|Very close to 2.3.7 just intonation, but octaves are quite a reach.]] | ||
=== Orwell === | === 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. | 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... (alternate sixths etc., fourths, hemififths...) | [[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...) | ||
== Alternate sixths/seconds ( | == 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. | 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. | ||
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=== Semaphore/Barbados (again) === | === 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 | 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. | ||
[[File:"HT" (3-1)^(1-2).png|none|thumb|600x600px|The smallest possible octave! The range on this one is huge...]] | [[File:"HT" (3-1)^(1-2).png|none|thumb|600x600px|The smallest possible octave! The range on this one is huge...]] | ||
=== Porcupine === | === 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. | 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. | ||
[[File:"HT" (9-2)^(1-3).png|none|thumb|600x600px|aghsdjfkhdfgasd]] | [[File:"HT" (9-2)^(1-3).png|none|thumb|600x600px|aghsdjfkhdfgasd]] | ||
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== Fourths-axial == | == 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 === | === Ones that resemble pure HT === | ||
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[[File:"HT" fourths dicot.png|none|thumb|600x600px|adsdf]] | [[File:"HT" fourths dicot.png|none|thumb|600x600px|adsdf]] | ||
==== | ==== Squares ==== | ||
[[File:"HT" fourths 425cent.png|none|thumb|600x600px|asdasd]] | [[File:"HT" fourths 425cent.png|none|thumb|600x600px|asdasd]] | ||
==== | ==== Magi ==== | ||
[[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]] | [[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]] | ||
table | table | ||
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[[File:"HT" fourths lemba.png|none|thumb|600x600px|dffhddgfsfdsh]] | [[File:"HT" fourths lemba.png|none|thumb|600x600px|dffhddgfsfdsh]] | ||
==== Orwell/Orson ==== | ==== Orwell/Orson (fourths ver.) ==== | ||
[[File:"HT" fourths orwell.png|none|thumb|600x600px|hfhdgfsh]] | [[File:"HT" fourths orwell.png|none|thumb|600x600px|hfhdgfsh]] | ||
table | table | ||
=== Alternative thirds outside the continua === | === Alternative thirds outside the continua === | ||
i don't have the energy to search these now | |||
alternate thirds sixths n stuff for this one too | alternate thirds sixths n stuff for this one too | ||
== Hemifourths/fifths (???) == | == Hemifourths/fifths (???) == | ||