User:Romeolz/Isomorphic layouts/Harmonic Table extensions - Revision history

Romeolz at 19:54, 3 November 2025

2025-11-03T19:54:51Z

← Older revision		Revision as of 19:54, 3 November 2025
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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 ==

Romeolz: pts images for fourths continua

2025-11-03T18:52:41Z

pts images for fourths continua

← Older revision		Revision as of 18:52, 3 November 2025
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	==== 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
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	!5/3		!5/3
	2		2
			\|-
			!image
			!
			!A
			!B
			!todo
	\|-		\|-
	!		!
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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]]
	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
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	!5/4		!5/4
	1		1
			\|-
			!image
			!
			!A
			!B
	\|-		\|-
	!		!
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	=== Alternate sixths/seconds ===		=== Alternate sixths/seconds ===
	~~asdfhjköä~~		todo

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

Romeolz: fourths, blackwood continuum and table math

2025-11-02T20:40:38Z

fourths, blackwood continuum and table math

← Older revision		Revision as of 20:40, 2 November 2025
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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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	\|		\|
	\|}		\|}
			And the "diminished" continuum:

	==== Sixix/Amity ====		==== Sixix/Amity ====
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	\|Diminished		\|Diminished
	648/625		648/625
	\|~~???~~		\|
	~~625/512~~
	\|-		\|-
	!		!
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	\|Sixix		\|Sixix
	3125/2916		3125/2916
	\|~~Lemba~~		\|???
	~~140625~~/~~131072~~		9375/8192
	\|-		\|-
	!		!
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	\|???		\|???
	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~~		asdfhjköä

	~~alternate thirds sixths n stuff for this one too~~

	== Hemifourths/fifths (???) ==		== Hemifourths/fifths (???) ==

Romeolz: table

2025-10-30T17:02:10Z

table

← Older revision		Revision as of 17:02, 30 October 2025
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	!		!
	!1		!1
	\|~~todo~~		\|Sixix
			3125/2916
	\|Lemba		\|Lemba
	140625/131072		140625/131072
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	!		!
	!2		!2
	\|		\|???
			15625/13122
	\|Orson		\|Orson
	[-21 3 7		[-21 3 7
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	!3		!3
	\|		\|
	\|		\|todo
	\|-		\|-
	!		!

Romeolz: tables

2025-10-29T19:01:16Z

tables

Show changes

Romeolz: grammar n stuff

2025-10-28T16:28:58Z

grammar n stuff

← Older revision		Revision as of 16:28, 28 October 2025
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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. ~~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. 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 (~~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\|aghsdjfkhdfgasd]]

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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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	[[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		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 ===
	~~idk~~		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 (???) ==

Romeolz: /* Ones that are a part of an analogous equivalence comtinuum */

2025-10-27T19:07:15Z

Ones that are a part of an analogous equivalence comtinuum

← Older revision		Revision as of 19:07, 27 October 2025
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	[[File:"HT" fourths diminished.png\|none\|thumb\|600x600px\|adsfgh]]		[[File:"HT" fourths diminished.png\|none\|thumb\|600x600px\|adsfgh]]

	=== Ones that are a part of an analogous equivalence ~~comtinuum~~ ===		=== Ones that are part of analogous equivalence continua, all kinds of thirds ===

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

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	==== Something with 440c gen ====		==== Something with 440c gen ====
	[[File:"HT" fourths 440cent.png\|none\|thumb\|600x600px\|fgdfjhfgds]]		[[File:"HT" fourths 440cent.png\|none\|thumb\|600x600px\|fgdfjhfgds]]
			table

			==== Sixix/Amity ====
			[[File:"HT" fourths amity.png\|none\|thumb\|600x600px\|sthdhfsfdha]]

			==== Lemba ====
			[[File:"HT" fourths lemba.png\|none\|thumb\|600x600px\|dffhddgfsfdsh]]

			==== Orwell/Orson ====
			[[File:"HT" fourths orwell.png\|none\|thumb\|600x600px\|hfhdgfsh]]
			table

			=== Alternative thirds outside the continua ===
			idk

	alternate thirds sixths n stuff for this one too		alternate thirds sixths n stuff for this one too

	== Hemifourths/fifths (???) ==		== Hemifourths/fifths (???) ==

Romeolz: /* Fourths-axial */

2025-10-27T18:12:44Z

Fourths-axial

@@ Line 172: / Line 172: @@
 fourths instead of fifths on the axis todo
 alternate thirds sixths n stuff for this one too
 == Hemifourths/fifths (???) ==

Romeolz: /* Alternate sixths (related but not necessarily part of HT) */ idk

2025-10-26T19:50:59Z

Alternate sixths (related but not necessarily part of HT): idk

← Older revision		Revision as of 19:50, 26 October 2025
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	[[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 (related but not necessarily part of HT) ==		== Alternate sixths/seconds (related but not necessarily part of HT) ==
	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.

			Every sixth has a second of the same quality one fifth below. The sixth-layouts can be sheared along the fifths-axis to get the secundal equivalent.

	=== Semaphore/Barbados (again) ===		=== Semaphore/Barbados (again) ===
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	[[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\|i've had enough for now]]
	ok		ok

			== Alternate fourths/fifths ==
			todo

			== Fourths-axial ==
			fourths instead of fifths on the axis todo

			alternate thirds sixths n stuff for this one too

			== Hemifourths/fifths (???) ==

Romeolz: pts

2025-10-25T21:00:25Z

pts

← Older revision		Revision as of 21:00, 25 October 2025
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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 ht2.png\|none\|thumb\|900x900px\|Close-up]]

	= Harmonic Table-ish extensions =		= Harmonic Table-ish extensions =