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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]]


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"Why such few temperaments? Where is meantone? Why do some temperaments not work?"
"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'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 (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 ==
== Alternate thirds ==
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=== Orwell (fifths ver.) ===
=== 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 (this is where it starts to get weird) ==
== Alternate sixths/seconds (this is where it starts to get weird) ==
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=== 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. The 3/1 can be sharpened to improve other intervals.
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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=== 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 ====
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==== Magi ====
==== 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 ====
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==== Orwell/Orson (fourths ver.) ====
==== 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 ===
i don't have the energy to search these now
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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