136/135 - Revision history

FloraC: Move fiventeen to subgroup temps

2026-03-21T12:08:39Z

Move fiventeen to subgroup temps

@@ Line 9: / Line 9: @@
 [[Tempering out]] this comma in the full 17-limit results in the rank-6 '''diatismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''diatic''' temperament.
-Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth.
+Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S|16}}) and [[289/288]] ({{S|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called [[fiventeen]], which is a rank-2 temperament generated by an octave and a perfect fifth.
 === Diatic ===

FloraC: /* Temperaments */ update data

2026-03-21T11:59:17Z

Temperaments: update data

← Older revision		Revision as of 11:59, 21 March 2026
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[Tp tuning\|~~subgroup~~]] [[~~CEE~~]]: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = ~~705~~.~~440~~{{c}}		* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
	* [[Tp tuning\|~~subgroup~~]] [[~~CTE~~]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.~~1088~~{{c}}		* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}
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	: mapping generators: ~2, ~3, ~5		: mapping generators: ~2, ~3, ~5

	[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.~~1088~~{{c}}, ~5/4 = ~~387~~.~~8544~~{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}

	{{Optimal ET sequence\|legend=1\| 10, 12, 22, 34, 80, 114, 194bc }}		{{Optimal ET sequence\|legend=1\| 10, 12, 22, 34, 80, 114, 194bc }}

			[[Badness]] (Sintel): 0.139

	=== Diatismic ===		=== Diatismic ===
	The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering 136/135 is [[46edo]], which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then [[80edo]] makes a good and more accurate tuning ~~that extends to the [[23-limit]]~~. Alternatively, if you ~~don't~~ care (as much) about prime 11, [[68edo]] makes a great tuning ~~in the no-11's [[19-limit]] and no-11's no-29's [[31-limit]]~~.		The only edo tuning that has less than 25% [[relative error]] for all primes in the [[17-limit]] tempering out 136/135 is [[46edo]], which also tunes 20/17 with less than 25% relative error and 51/40 even more accurately. If you allow 7/4 to be sharper than 25% then [[80edo]] makes for a good and more accurate tuning. Alternatively, if you do not care as much about prime 11, [[68edo]] makes for a great tuning.

	[[Subgroup]]: 2.3.5.7.11.13.17		[[Subgroup]]: 2.3.5.7.11.13.17
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	: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13		: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

	[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.~~1088~~{{c}}, ~5/4 = ~~387~~.~~8544~~{{c}}, ~7/4, ~11/8, ~13/8		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}, ~5/4 = 389.0228{{c}}, ~7/4 = 970.2512{{c}}, ~11/8 = 553.4578{{c}}, ~13/8 = 842.6669{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}, ~5/4 = 388.6162{{c}}, ~7/4 = 969.9161{{c}}, ~11/8 = 552.6614{{c}}, ~13/8 = 841.9647{{c}}

	{{Optimal ET sequence\|legend=1\| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*		{{Optimal ET sequence\|legend=1\| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }} *

	<nowiki>*</nowiki> [[optimal patent val]]: [[177edo\|177]]		<nowiki>*</nowiki> [[optimal patent val]]: [[177edo\|177]]

			[[Badness]] (Sintel): 1.15

	== Etymology ==		== Etymology ==

FloraC: Intro to the temps

2026-03-21T10:57:42Z

Intro to the temps

← Older revision		Revision as of 10:57, 21 March 2026
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	== Temperaments ==		== Temperaments ==
			[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''diatismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''diatic''' temperament.

			Since 136/135 = ([[225/224]])⋅([[256/255]]), it would make sense to temper out both [[256/255]] ({{S\|16}}) and [[289/288]] ({{S\|17}}), thereby tempering diatic to [[srutal archagall]], which is equivalently described as "[[charic]] [[semitonic]]". This can be further restricted to the 2.3.17/5-subgroup {136/135}, called fiventeen, which is a rank-2 temperament generated by an octave and a perfect fifth.

	=== Fiventeen ===		=== Fiventeen ===
	[[~~17edo~~]] ~~makes a good tuning (especially for its size) for the 2.3.17~~/~~5-subgroup {136/135} rank 2 temperament which~~ implies a [[supersoft]] [[pentic]] pentad of [[~]]30:34:40:45:51~~:60 (because as aforementioned~~ [[~~17/15~~]] ~~is equated with~~ [[~~9/8~~]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and ~~80 + 17 =~~ [[97edo]] ~~and 97~~ + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then ~~34 + 80 =~~ [[114edo]] and ~~amazingly~~ even ~~114 + 80 =~~ [[194edo\|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and ~~63 + 80 =~~ [[143edo]] tunings are found in the optimal ET sequence for fiventeen.		In fiventeen, [[17/15]] is equated with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo\|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

	[[Subgroup]]: 2.3.17/5		[[Subgroup]]: 2.3.17/5
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	<nowiki>*</nowiki> [[optimal patent val]]: [[177edo\|177]]		<nowiki>*</nowiki> [[optimal patent val]]: [[177edo\|177]]

	~~=== Srutal archagall ===~~
	[[Srutal archagall]] is an efficient rank-2 temperament tempering out both [[256/255\|S16]] and [[289/288\|S17]], which is equivalently described as [[charic]] [[semitonic]] due to the fact that {S16⋅S17, [[24576/24565\|S16/S17]]} = {[[256/255\|S16]], [[289/288\|S17]]}

	== Etymology ==		== Etymology ==

FloraC: Rework

2026-03-21T10:39:28Z

Rework

← Older revision		Revision as of 10:39, 21 March 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}
	'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[~~small~~ comma]]. It is the interval that separates [[17/10]] and [[27/16]] (or their octave complements [[20/17]] and [[32/27]]) and that separates [[30/17]] and [[16/9]] (or their octave complements [[17/15]] and [[9/8]]). It is also the difference between [[16/15]] and [[18/17]] with an [[S-expression]] of [[256/255\|S16]] × [[289/288\|S17]] or ((16/15)(17/16))/((17/16)(18/17)).		'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[small comma\|small]] [[17-limit]] [[comma]]. It is the interval that separates [[17/10]] and [[27/16]] (or their octave complements [[20/17]] and [[32/27]]) and that separates [[30/17]] and [[16/9]] (or their octave complements [[17/15]] and [[9/8]]). It is also the difference between [[16/15]] and [[18/17]] with an [[S-expression]] of [[256/255\|S16]]⋅[[289/288\|S17]] or ((16/15)⋅(17/16))/((17/16)⋅(18/17)).

	== Temperaments ==		== Temperaments ==
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	{{Mapping\|legend=2\| 1 0 -3 \| 0 1 3 }}		{{Mapping\|legend=2\| 1 0 -3 \| 0 1 3 }}
			: mapping generators: ~2, ~3
	: ~~sval~~ mapping generators: ~2, ~3

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[Tp tuning\|subgroup]] [[CEE]]: 2 = ~~1\1~~, ~3/2 = 705.440		* [[Tp tuning\|subgroup]] [[CEE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.440{{c}}
	* [[Tp tuning\|subgroup]] [[CTE]]: ~2 = ~~1\1~~, ~3/2 = 704.1088		* [[Tp tuning\|subgroup]] [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}

	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}
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	{{Mapping\|legend=2\| 1 0 0 -3 \| 0 1 0 3 \| 0 0 1 1 }}		{{Mapping\|legend=2\| 1 0 0 -3 \| 0 1 0 3 \| 0 0 1 1 }}
			: mapping generators: ~2, ~3, ~5

	~~: sval mapping generators: ~2, ~3, ~5~~		[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}, ~5/4 = 387.8544{{c}}

	[[Optimal tuning]] (~~[[Tp tuning\|subgroup]]~~ [[CTE]]): ~2 = ~~1\1~~, ~3/2 = 704.1088, ~5/4 = 387.8544

	{{Optimal ET sequence\|legend=1\| 10, 12, 22, 34, 80, 114, 194bc }}		{{Optimal ET sequence\|legend=1\| 10, 12, 22, 34, 80, 114, 194bc }}
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| ]]
	\|}		\|}
	: ~~sval~~ mapping generators: ~2, ~3, ~5, ~7, ~11, ~13		: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

	[[Optimal tuning]] (~~[[Tp tuning\|subgroup]]~~ [[CTE]]): ~2 = ~~1\1~~, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8		[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1088{{c}}, ~5/4 = 387.8544{{c}}, ~7/4, ~11/8, ~13/8

	{{Optimal ET sequence\|legend=1\| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*		{{Optimal ET sequence\|legend=1\| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*
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	=== Srutal archagall ===		=== Srutal archagall ===
	[[Srutal archagall]] is an efficient rank-2 temperament tempering out both [[256/255\|S16]] and [[289/288\|S17]], which is equivalently described as [[charic]] [[semitonic]] due to the fact that {~~S16 × S17~~ , [[24576/24565\|S16/S17]]} = {[[256/255\|S16]], [[289/288\|S17]]}		[[Srutal archagall]] is an efficient rank-2 temperament tempering out both [[256/255\|S16]] and [[289/288\|S17]], which is equivalently described as [[charic]] [[semitonic]] due to the fact that {S16⋅S17, [[24576/24565\|S16/S17]]} = {[[256/255\|S16]], [[289/288\|S17]]}

	== Etymology ==		== Etymology ==
	The name was formerly ''diatonisma'', suggested by [[~~User:~~Xenllium]] in 2023, but this name ~~has [[comma naming\|strong reasons]] against it due to implying an ambiguously-named~~ "diatonic" subgroup temperament. Therefore ''fiventeenisma'' and ''~~diatisma~~'' were proposed. ~~However, due~~ to ~~the need for a separate name for~~ the rank 2 2.3.17/5 subgroup temperament ~~and due to its relation to the chord (see [[Talk:136/135]])~~, ~~the name "fiventeen" was given to the temperament~~ and hence ~~due to the lack of a need for "-ic/-ismic/-isma" (as that can apply to~~ the ~~already-short~~ name of ''~~diatisma~~''~~, itself a rename & shortenage of~~ ''~~diatonisma~~''~~) the name was shortened to just "fiventeen"~~.		The name of this comma was formerly ''diatonisma'', suggested by [[Xenllium]] in 2023, but this name would imply a problematic "diatonic" subgroup temperament. Therefore ''diatisma'', a shortenage of ''diatonisma'', and ''fiventeenisma'' a portmanteau of ''five'' and ''seventeen'' for its relation to a chord involving primes 5 and 17, were proposed by [[Godtone]] in 2024. The name ''fiventeen'' was soon given to the rank-2 2.3.17/5-subgroup temperament, and hence the name ''fiventeenisma'' became just ''fiventeen comma''.

	== See also ==		== See also ==
	* [[Small comma]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]

	[[Category:Commas ~~with unknown etymology~~]]		[[Category:Diatismic]]
			[[Category:Commas named for their regular temperament properties]]

	~~{{todo\|improve readability\|inline=1\|comment=Rewrite the etymology section to be easier to parse and less vague.}}~~

BudjarnLambeth at 00:30, 16 November 2024

2024-11-16T00:30:00Z

← Older revision		Revision as of 00:30, 16 November 2024
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	* [[Small comma]]		* [[Small comma]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]

			[[Category:Commas with unknown etymology]]


			{{todo\|improve readability\|inline=1\|comment=Rewrite the etymology section to be easier to parse and less vague.}}

Godtone: /* Fiventeen */

2024-11-11T14:49:22Z

Fiventeen

← Older revision		Revision as of 14:49, 11 November 2024
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	== Temperaments ==		== Temperaments ==
	=== Fiventeen ===		=== Fiventeen ===
	[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and 80 + 17 = [[97edo]] and 97 + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then 34 + 80 = [[114edo]] and amazingly even 114 + 80 = [[194edo\|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and 63 + 80 = [[143edo]] tunings are found in the optimal ET sequence for fiventeen.		[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of [[~]]30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and 80 + 17 = [[97edo]] and 97 + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then 34 + 80 = [[114edo]] and amazingly even 114 + 80 = [[194edo\|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and 63 + 80 = [[143edo]] tunings are found in the optimal ET sequence for fiventeen.

	[[Subgroup]]: 2.3.17/5		[[Subgroup]]: 2.3.17/5

FloraC: You're not supposed to "recommend" one optimization method over another. Ideally we wanna keep CTE and CWE here but CEE could stay for now

2024-10-02T17:51:33Z

You're not supposed to "recommend" one optimization method over another. Ideally we wanna keep CTE and CWE here but CEE could stay for now

← Older revision		Revision as of 17:51, 2 October 2024
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	== Temperaments ==		== Temperaments ==
	=== Fiventeen ===		=== Fiventeen ===
	[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and 80 + 17 = [[97edo]] and 97 + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank 3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, ~~as it correctly finds~~ [[34edo]], then [[80edo]], then 34 + 80 = [[114edo]] and amazingly even 114 + 80 = [[194edo\|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and 63 + 80 = [[143edo]] tunings are found in the optimal ET sequence for fiventeen.		[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and 80 + 17 = [[97edo]] and 97 + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then 34 + 80 = [[114edo]] and amazingly even 114 + 80 = [[194edo\|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and 63 + 80 = [[143edo]] tunings are found in the optimal ET sequence for fiventeen.

	[[Subgroup]]: 2.3.17/5		[[Subgroup]]: 2.3.17/5
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* subgroup CE: 2 = 1\1, ~3/2 = 705.440 ~~(recommended)~~		* [[Tp tuning\|subgroup]] [[CEE]]: 2 = 1\1, ~3/2 = 705.440
	* [[Tp tuning\|subgroup CTE]]: ~2 = 1\1, ~3/2 = 704.1088 ~~(flat of 46edo)~~		* [[Tp tuning\|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 704.1088

	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}
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	: sval mapping generators: ~2, ~3, ~5		: sval mapping generators: ~2, ~3, ~5

	[[Optimal tuning]] ([[Tp tuning\|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544		[[Optimal tuning]] ([[Tp tuning\|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544

	{{Optimal ET sequence\|legend=1\| 10, 12, 22, 34, 80, 114, 194bc }}		{{Optimal ET sequence\|legend=1\| 10, 12, 22, 34, 80, 114, 194bc }}
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	: sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13		: sval mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

	[[Optimal tuning]] ([[Tp tuning\|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8		[[Optimal tuning]] ([[Tp tuning\|subgroup]] [[CTE]]): ~2 = 1\1, ~3/2 = 704.1088, ~5/4 = 387.8544, ~7/4, ~11/8, ~13/8

	{{Optimal ET sequence\|legend=1\| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*		{{Optimal ET sequence\|legend=1\| 22, 27eg, 29g, 34d, 39dfg, 41g, 46, 58, 80, 104c, 114e, 126(f), 136ef, 148d, 167g, 216bdef }}*

Godtone: /* Fiventeen */ the CTE tuning being flat of 46edo is honestly insane/shows it's not the right thing to use here; i would much prefer to only have CE gives a much more reasonable result (between 17 EDO and 80 EDO) but in case anyone is interested in the CTE tuning i wil leave it here

2024-10-02T04:06:58Z

Fiventeen: the CTE tuning being flat of 46edo is honestly insane/shows it's not the right thing to use here; i would much prefer to only have CE gives a much more reasonable result (between 17 EDO and 80 EDO) but in case anyone is interested in the CTE tuning i wil leave it here

← Older revision		Revision as of 04:06, 2 October 2024
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	== Temperaments ==		== Temperaments ==
	=== Fiventeen ===		=== Fiventeen ===
	[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]) although [[80edo]] might be preferred for ~~a more accurate~~ [[51/40]] to optimize plausibility slightly more, and it and [[~~46edo~~]] ~~might be preferred for~~ more accurate ~~fifths~~. The same is true of the related rank 3 temperament diatic, described below.		[[17edo]] makes a good tuning (especially for its size) for the 2.3.17/5-subgroup {136/135} rank 2 temperament which implies a [[supersoft]] [[pentic]] pentad of 30:34:40:45:51:60 (because as aforementioned [[17/15]] is equated with [[9/8]]), corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and 80 + 17 = [[97edo]] and 97 + 17 = [[114edo]] do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank 3 temperament diatic, described below, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, as it correctly finds [[34edo]], then [[80edo]], then 34 + 80 = [[114edo]] and amazingly even 114 + 80 = [[194edo\|194bc-edo]], though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and 63 + 80 = [[143edo]] tunings are found in the optimal ET sequence for fiventeen.

	[[Subgroup]]: 2.3.17/5		[[Subgroup]]: 2.3.17/5
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	: sval mapping generators: ~2, ~3		: sval mapping generators: ~2, ~3

	[[Optimal tuning]] ([[Tp tuning\|subgroup CTE]]): ~2 = 1\1, ~3/2 = 704.1088		[[Optimal tuning]]s:
			* subgroup CE: 2 = 1\1, ~3/2 = 705.440 (recommended)
			* [[Tp tuning\|subgroup CTE]]: ~2 = 1\1, ~3/2 = 704.1088 (flat of 46edo)

	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 46, 63, 143 }}

FloraC: Improve readability by preferring verbal descriptions to math expressions. Remove words like "deduce", "therefore" (which deduces which?) in favor of a focus on presenting facts.

2024-06-01T13:36:51Z

Improve readability by preferring verbal descriptions to math expressions. Remove words like "deduce", "therefore" (which deduces which?) in favor of a focus on presenting facts.

← Older revision		Revision as of 13:36, 1 June 2024
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[small comma]]. It is ~~equal to (~~[[32/27]])/([[20/17]]) and ~~also (~~[[51/40]])/([[81/64]]). It is also the difference between [[16/15]] and [[18/17]] ~~(which is easily deduced from its~~ S-expression of [[256/255\|S16]] * [[289/288\|S17]])~~, and is therefore the difference between~~ ([[17/16]])~~([[16/15]]~~) ~~= [[17~~/~~15]] and~~ (~~[[18/17]])~~([[17/16]]) ~~= [[9~~/~~8]] as the two~~ 17~~/16's cancel~~.		'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[small comma]]. It is the interval that separates [[17/10]] and [[27/16]] (or their octave complements [[20/17]] and [[32/27]]) and that separates [[30/17]] and [[16/9]] (or their octave complements [[17/15]] and [[9/8]]). It is also the difference between [[16/15]] and [[18/17]] with an [[S-expression]] of [[256/255\|S16]] × [[289/288\|S17]] or ((16/15)(17/16))/((17/16)(18/17)).

	== Temperaments ==		== Temperaments ==

Godtone: better style

2024-05-27T12:10:07Z

better style

← Older revision		Revision as of 12:10, 27 May 2024
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[small comma]]. It is equal to ([[32/27]])/([[20/17]]) and also ([[51/40]])/([[81/64]]). It is also the difference between [[16/15]] and [[18/17]] which is easily deduced from its S-expression of [[256/255\|S16]] * [[289/288\|S17]], and is therefore the difference between ([[17/16]])([[16/15]]) = [[17/15]] and ([[18/17]])([[17/16]]) = [[9/8]] as the two 17/16's cancel.		'''136/135''', the '''diatisma''', '''diatic comma''' or '''fiventeen comma''', is a [[17-limit]] [[small comma]]. It is equal to ([[32/27]])/([[20/17]]) and also ([[51/40]])/([[81/64]]). It is also the difference between [[16/15]] and [[18/17]] (which is easily deduced from its S-expression of [[256/255\|S16]] * [[289/288\|S17]]), and is therefore the difference between ([[17/16]])([[16/15]]) = [[17/15]] and ([[18/17]])([[17/16]]) = [[9/8]] as the two 17/16's cancel.

	== Temperaments ==		== Temperaments ==