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intro 12based 19based and others started
 
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more extensions, tables, 19edo
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= 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.
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.
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.


== Canonical (12-based) extensions ==
== Canonical (12-based) extensions ==
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== 19-based extensions ==
== 19-based extensions ==
sadsdasdsad
In 19edo, it just so happens that there is another 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) ===
dfghjk
This one is more akin to the familiar augmented layout. The octaves are a bit far apart but still reachable.


=== w/ hanson twelfth (46656/15625 ~ 46875/15625 = 3/1) ===
=== w/ hanson twelfth (46656/15625 ~ 46875/15625 = 3/1) ===
dhsdadsdf
This one is like the diminished layout, as its octaves run diagonally, or horizontally when mirrored. The octave starts to be a bit tough to reach, especially multiple octaves. Hanson temperament can get the closest to 5-limit just intonation out of all the pure HT temperaments.
 
== More pure HT extensions using equivalence continua ==
The major third extensions can be visualized in the layouts like so: every time m increments, the octave mapping moves up by one 25/24, which is the difference between 5/4 and 6/5. From this we get the equation* (5/4)^3 * (25/24)^m ~ 2/1 ⇒ (125/128) * (25/24)^m ~ 1 ⇒ (25/24)^m ~ (128/125).


== More extensions using equivalence continua ==
This is essentially the same thing as the Father–3 equivalence continuum.
[7 0 -3> [-10 -1 5> [-13 -2 7>
{| class="wikitable"
|+Major thirds (25/24)^m ~ (128/125)
!m
!comma
!monzo
!temperament name
!
|-
!-1
|16/15
|[4 -1 -1⟩
|father
|exotemperament
|-
!0
|128/125
|[7 0 -3⟩
|augmented
|
|-
!1
|3125/3072
|[-10 -1 5⟩
|magic
|
|-
!2
|78125/73728
|[-13 -2 7⟩
|wesley
|
|-
!3
|1953125/1769472
|[-16 -3 9⟩
|(3 & 33c)
|high complexity low accuracy
|}
The minor third extensions can be visualized in the layouts like so: every time m increments, the octave mapping moves down by one 25/24, which is the difference between 5/4 and 6/5. From this we get the equation* (6/5)^4 * (24/25)^m ~ 2/1 ⇒ (648/625) * (24/25)^m ~ 1 ⇒ (25/24)^m ~ (648/625).


[3 4 -4> [-6 -5 6> [-9 -6 8>
I couldn't find any established way of conceptualizing this continuum.
{| class="wikitable"
|+Minor thirds (25/24)^m ~ (648/625)
!m
!comma
!monzo
!temperament name
!
|-
!-1
|27/25
|[0 3 -2⟩
|bug
|exotemperament
|-
!0
|648/625
|[3 4 -4⟩
|diminished
|
|-
!1
|15625/15552
|[-6 -5 6⟩
|hanson
|nearly just
|-
!2
|390625/373248
|[-9 -6 8⟩
|doublewide
|
|-
!3
|9765625/8957952
|[-12 -7 10⟩
|(4 & 33c)
|high complexity low accuracy
|}
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