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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).
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...)
[[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]]


== 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. 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.
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
|}
|}
[[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 (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)


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)
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 ==
== 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...]]


== Alternate sixths/seconds (related but not necessarily part of HT) ==
== 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 sounds like a tenth.
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|basically porcupine[7] melodic mode]]


=== Negri ===
=== Negri ===
A coooooooooooooool temperament?? I don't want to type these out anymore
A coooooooooooooool temperament?? I don't want to type these out anymore
[[File:"HT" (27-4)^(1-4).png|none|thumb|600x600px|abcdefg]]
[[File:"HT" (27-4)^(1-4).png|none|thumb|600x600px|basically negri[9] melodic mode]]


=== Blackwood ===
=== Blackwood ===
i love blackwood 10
i love blackwood
[[File:"HT" (243-16)^(1-5).png|none|thumb|600x600px|heyy this one's pretty neat it splits the octave into 5]]
[[File:"HT" (243-16)^(1-5).png|none|thumb|600x600px|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 ===
=== I don't even know what this one is but it has a good 11/7 lol ===
yea
...
[[File:"HT" (243-16)^(1-6).png|none|thumb|600x600px|i've had enough for now]]
[[File:"HT" (243-16)^(1-6).png|none|thumb|600x600px|idk if anyone is ever gonna use this]]
ok
...


== Alternate fourths/fifths ==
== Alternate fourths/fifths ==
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== Fourths-axial ==
== Fourths-axial ==
fourths instead of fifths on the axis todo
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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=== Ones that are part of analogous equivalence continua, all kinds of thirds ===
=== Ones that are part of analogous equivalence continua, all kinds of thirds ===
The "augmented" continuum:


==== Dicot/Mohajira ====
==== Dicot/Mohajira ====
[[File:"HT" fourths dicot.png|none|thumb|600x600px|adsdf]]
[[File:"HT" fourths dicot.png|none|thumb|600x600px|adsdf]]


==== Something with 425c gen ====
==== Squares ====
[[File:"HT" fourths 425cent.png|none|thumb|600x600px|asdasd]]
[[File:"HT" fourths 425cent.png|none|thumb|600x600px|asdasd]]


==== Something with 440c gen ====
==== Magi ====
[[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]]
[[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]]
table
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.
[[File:ProjectiveTuningSpace fourths augmented A.png|thumb|Image A]]
[[File:ProjectiveTuningSpace fourths augmented B.png|thumb|Image B]]
{| class="wikitable"
|+Continuation of fourths-axial augmented "HT" into various equivalence continua
!
!m
! colspan="3" |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 ====
==== Sixix/Amity ====
[[File:"HT" fourths amity.png|none|thumb|600x600px|sthdhfsfdha]]
[[File:"HT" fourths amity.png|none|thumb|600x600px|When using amity, the best approximation of 5-limit thirds is further away from the root]]


==== Lemba ====
==== Lemba ====
[[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
[[File:ProjectiveTuningSpace fourths diminished A.png|thumb|Image A]]
[[File:ProjectiveTuningSpace fourths diminished B.png|thumb|Image B]]
Same thing as with the above table.
{| class="wikitable"
|+Continuation of fourths-axial diminished "HT" into various equivalence continua
!
!m
! colspan="2" |temperament,
comma
|-
!approximated interval,
location in genchain
!
!6/5
1
!5/4
1
|-
!image
!
!A
!B
|-
!
!-2
|???
32/25
|
|-
!
!-1
|University
144/125
|
|-
!
!0
|Diminished
648/625
|
|-
!
!1
|Sixix
3125/2916
|???
9375/8192
|-
!
!2
|???
15625/13122
|Lemba
140625/131072
|-
!
!3
|
|Orson
[-21 3 7
|-
!
!4
|
|???
[25 -4 -8
|}


=== Alternative thirds outside the continua ===
=== Alternate thirds outside the continua ===
idk
The "blackwood" continuum:


alternate thirds sixths n stuff for this one too
==== 8/3 in 6 283c ====
[[File:"HT" fourths (8-3)^(1-6).png|none|thumb|600x600px|basically 17edo]]
 
==== Sensi, 32/9 in 7 314c ====
[[File:"HT" fourths (32-9)^(1-7).png|none|thumb|600x600px|possibly some merit to this one]]
 
==== Continuation of this "blackwood" pattern ====
{| class="wikitable"
|+((2/1)*(4/3)^m)^(1/(m+5))
!m
!interval
!parts
!cents of third
|-
|0
|2/1
|5
|240 (3rd?)
|-
|1
|8/3
|6
|283
|-
|2
|32/9
|7
|314
|-
|3
|128/27
|8
|todo
|-
|4
|512/81
|9
|
|-
|5
|2048/243
|10
|
|}
There are more continua but their octaves are already massive.
 
=== Alternate sixths/seconds ===
todo


== Hemifourths/fifths (???) ==
== Hemifourths/fifths (might be outside the scope) ==
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