Xenharmonic Wiki

User:Romeolz/Isomorphic layouts/Harmonic Table extensions: Difference between revisions

← Older edit
VisualWikitext
Romeolz (talk | contribs)
88 edits
m table
Romeolz (talk | contribs)
88 edits
mNo edit summary
 
(2 intermediate revisions by the same user not shown)
Line 3: Line 3:


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]]


Line 123: Line 131:
"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 ==
Line 181: Line 191:


=== 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 ====
Line 190: Line 201:
==== Magi ====
==== Magi ====
[[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]]
[[File:"HT" fourths 440cent.png|none|thumb|600x600px|fgdfjhfgds]]
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.
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"
{| class="wikitable"
|+Continuation of fourths-axial augmented "HT" into various equivalence continua
|+Continuation of fourths-axial augmented "HT" into various equivalence continua
Line 207: Line 220:
!5/3
!5/3
2
2
|-
!image
!
!A
!B
!todo
|-
|-
!
!
Line 273: Line 292:
|
|
|}
|}
And the "diminished" continuum:


==== Sixix/Amity ====
==== Sixix/Amity ====
Line 282: Line 302:
==== 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]]
Same thing as with the above 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"
{| class="wikitable"
|+
|+Continuation of fourths-axial diminished "HT" into various equivalence continua
!
!
!m
!m
Line 297: Line 319:
!5/4
!5/4
1
1
|-
!image
!
!A
!B
|-
|-
!
!
Line 314: Line 341:
|Diminished
|Diminished
648/625
648/625
|???
|
625/512
|-
|-
!
!
Line 321: Line 347:
|Sixix
|Sixix
3125/2916
3125/2916
|Lemba
|???
140625/131072
9375/8192
|-
|-
!
!
Line 328: Line 354:
|???
|???
15625/13122
15625/13122
|Lemba
140625/131072
|-
!
!3
|
|Orson
|Orson
[-21 3 7
[-21 3 7
|-
|-
!
!
!3
!4
|
|
|???
[25 -4 -8
|}
=== Alternate thirds outside the continua ===
The "blackwood" continuum:
==== 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
|todo
|-
|-
!
|4
!4
|512/81
|9
|
|
|-
|5
|2048/243
|10
|
|
|}
|}
There are more continua but their octaves are already massive.


=== Alternative thirds outside the continua ===
=== Alternate sixths/seconds ===
i don't have the energy to search these now
todo
 
alternate thirds sixths n stuff for this one too


== Hemifourths/fifths (???) ==
== Hemifourths/fifths (might be outside the scope) ==
Retrieved from "https://en.xen.wiki/w/User:Romeolz/Isomorphic_layouts/Harmonic_Table_extensions"