Diaschismic - Revision history

FloraC: Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

2026-06-06T12:39:48Z

Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

← Older revision		Revision as of 12:39, 6 June 2026
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	[[Category:Diaschismic family]]		[[Category:Diaschismic family]]
	[[Category:Starling temperaments]]		[[Category:Starling temperaments]]
	[[Category:~~Hemifamity~~ temperaments]]		[[Category:Aberschismic temperaments]]

FloraC at 10:18, 9 February 2026

2026-02-09T10:18:51Z

← Older revision		Revision as of 10:18, 9 February 2026
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	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Diaschismic		\| Title = Diaschismic
	\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17		\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
	\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)		\| Comma basis = [[2048/2025]] (5-limit); <br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)
	\| Edo join 1 = 46 \| Edo join 2 = 58		\| Edo join 1 = 46 \| Edo join 2 = 58
			\| Mapping = 2; 1 -2 -8 -12 -15 1
	\| Generators = 3/2 \| Generators tuning = 703.9 \| Optimization method = CWE		\| Generators = 3/2 \| Generators tuning = 703.9 \| Optimization method = CWE
	\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]		\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
	~~\| Mapping = 2; 1 -2 -8 -12 -15 1~~
	\| Pergen = (P8/2, P5)		\| Pergen = (P8/2, P5)
	\| Color name = Saguguti		\| Color name = Saguguti

FloraC: Cleanup on infobox

2026-02-09T09:06:00Z

Cleanup on infobox

← Older revision		Revision as of 09:06, 9 February 2026
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	\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)		\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)
	\| Edo join 1 = 46 \| Edo join 2 = 58		\| Edo join 1 = 46 \| Edo join 2 = 58
	\| ~~Generator~~ = 3/2 \| ~~Generator~~ tuning = 703.9 \| Optimization method = CWE		\| Generators = 3/2 \| Generators tuning = 703.9 \| Optimization method = CWE
	\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]		\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
	\| Mapping = 2; 1 -2 -8 -12 -15 1		\| Mapping = 2; 1 -2 -8 -12 -15 1
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	\| Odd limit 2 = 17-limit 21 \| Mistuning 2 = ??? \| Complexity 2 = 46		\| Odd limit 2 = 17-limit 21 \| Mistuning 2 = ??? \| Complexity 2 = 46
	}}		}}

	'''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament\|temperament]] [[generator\|generated]] by a [[3/2\|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out\|tempered out]], and we also have a whole tone plus a period represent [[8/5]].		'''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament\|temperament]] [[generator\|generated]] by a [[3/2\|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out\|tempered out]], and we also have a whole tone plus a period represent [[8/5]].

Overthink: /* As a detemperament of 12et */ Avoid overlap

2026-01-11T20:15:07Z

As a detemperament of 12et: Avoid overlap

← Older revision		Revision as of 20:15, 11 January 2026
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	Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.		Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.
			{{Clear}}

	== Chords and harmony ==		== Chords and harmony ==

Overthink: no longer in infobox

2026-01-11T20:13:33Z

no longer in infobox

← Older revision		Revision as of 20:13, 11 January 2026
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	{{Infobox Regtemp		{{Infobox Regtemp
	\| Title = Diaschismic~~; srutal archagall~~		\| Title = Diaschismic
	\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17		\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
	\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)		\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)

FloraC: /* Tunings */ + more edo tunings

2026-01-11T10:32:04Z

Tunings: + more edo tunings

← Older revision		Revision as of 10:32, 11 January 2026
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	\|-		\|-
	\|		\|
	\| 12/7		\| 7/6
	\| 703.681		\| 703.681
	\|		\|
			\|-
			\| [[162edo\|95\162]]
			\|
			\| 703.704
			\| 162cef val
	\|-		\|-
	\|		\|
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	\| 703.965		\| 703.965
	\|		\|
			\|-
			\| [[150edo\|88\150]]
			\|
			\| 704.000
			\| 150c val
	\|-		\|-
	\|		\|

FloraC: /* Chords */

2026-01-11T10:27:08Z

Chords

← Older revision		Revision as of 10:27, 11 January 2026
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	Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.		Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.

	== Chords ==		== Chords and harmony ==
	{{~~Main~~\| Chords of diaschismic }}		{{See also\| Chords of diaschismic }}

	== Tunings ==		== Tunings ==

FloraC: Reduce duplication. Rework on infobox (this is a 17-limit temp!). Expand on 7-limit

2026-01-11T10:26:38Z

Reduce duplication. Rework on infobox (this is a 17-limit temp!). Expand on 7-limit

← Older revision		Revision as of 10:26, 11 January 2026
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	{{Infobox Regtemp		{{Infobox Regtemp
	\| Title = Diaschismic; ~~Srutal~~ archagall		\| Title = Diaschismic; srutal archagall
	\| Subgroups = 2.3.5, 2.3.5.17		\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11.13.17
	\| Comma basis = [[2048/2025]] (~~2.3.~~5); <br /> [[136/135]], [[256/255]] (~~2.3.5.~~17)		\| Comma basis = [[2048/2025]] (5-limit);<br>[[126/125]], [[2048/2025]] (7-limit); <br>[[126/125]], [[136/135]], [[176/175]], <br>[[196/195]], [[256/255]] (17-limit)
	\| Edo join 1 = 12 \| Edo join 2 = 22		\| Edo join 1 = 46 \| Edo join 2 = 58
	\| Generator = 16/15 \| Generator tuning = ~~104~~.~~898~~ \| Optimization method = ~~POTE~~		\| Generator = 3/2 \| Generator tuning = 703.9 \| Optimization method = CWE
	\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]		\| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
	\| Mapping = 2; 1 -2 1		\| Mapping = 2; 1 -2 -8 -12 -15 1
	\| Pergen = (P8/2, P5)		\| Pergen = (P8/2, P5)
	\| Color name = Saguguti		\| Color name = Saguguti
	\| Odd limit 1 = 5 \| Mistuning 1 = 3.259 \| Complexity 1 = 12		\| Odd limit 1 = 5 \| Mistuning 1 = 3.259 \| Complexity 1 = 12
	\| Odd limit 2 = ~~(2.3.5.~~17~~) 25~~ \| Mistuning 2 = ??? \| Complexity 2 = 22		\| Odd limit 2 = 17-limit 21 \| Mistuning 2 = ??? \| Complexity 2 = 46
	}}		}}

	'''Diaschismic''', sometimes known as [[srutal ~~vs diaschismic\|srutal]]~~ in the [[5-limit]], is a half-octave [[regular temperament\|temperament]] [[generator\|generated]] by a [[3/2\|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out\|tempered out]], and we also have a whole tone plus a period represent [[8/5]]. 9/8 splits in two very naturally into [[17/16]] × [[18/17]], and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out [[S-expression\|S16]] = [[256/255]], S17 = [[289/288]], and their product [[136/135]], leading to a 2.3.5.17 [[subgroup]] extension called '''srutal archagall'''.		'''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament\|temperament]] [[generator\|generated]] by a [[3/2\|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out\|tempered out]], and we also have a whole tone plus a period represent [[8/5]].

	The canonical [[extension]] to the [[7-limit]] lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]], as well as the argent comma, [[5120/5103]].		The canonical [[extension]] to the [[7-limit]] lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]], as well as the argent comma, [[5120/5103]]. This equates the [[64/63\|septimal comma]] with the [[81/80\|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals.

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. The mappings of primes [[11/1\|11]] and [[13/1\|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]], leading to the same mapping of prime 17 as in srutal archagall. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]].		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. The mappings of primes [[11/1\|11]] and [[13/1\|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp.

	~~An alternative extension~~ to ~~the full~~ 17~~-limit is~~ [[~~srutal~~]], ~~which has~~ a ~~more complex mapping of prime 7 at +15 fifths~~, tempering out [[~~4375~~/~~4374~~]]~~. The mappings for primes 11 and 13 follow through parapyth~~, and ~~17 is mapped as in srutal archagall~~. This works best ~~for a sharper tuning,~~ between [[~~34edo~~]] (~~with the 34d~~ [[~~val~~]]) and [[~~46edo~~]].		Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S\|16}}), [[289/288]] ({{S\|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup\|2.3.5.17-]][[subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.

	Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.		An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.

	See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic]] for technical data.		See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic\|#Septimal diaschismic]] for technical data.

	== Interval chain ==		== Interval chain ==
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	[[File: Diaschismic 12et Detempering.png\|thumb\|Diaschismic as a 58-tone 12et detempering]]		[[File: Diaschismic 12et Detempering.png\|thumb\|Diaschismic as a 58-tone 12et detempering]]

	Diaschismic is naturally considered as a [[detemperament]] of the [[12edo\|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 generator steps, which represent the ~~syntonic~~ comma. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.		Diaschismic is naturally considered as a [[detemperament]] of the [[12edo\|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by 6 generator steps, which represent the generic comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.

	The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.		The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.

	Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the ~~syntonic~~ comma in 58edo. 104edo tunes it to one half the size of the ~~syntonic~~ comma, which may be seen as a good compromise.		Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. 104edo tunes it to one half the size of the comma step, which may be seen as a good compromise.

	== Chords ==		== Chords ==

Overthink: expand on extensions

2026-01-10T20:47:25Z

expand on extensions

← Older revision		Revision as of 20:47, 10 January 2026
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	'''Diaschismic''', sometimes known as [[srutal vs diaschismic\|srutal]] in the [[5-limit]], is a half-octave [[regular temperament\|temperament]] [[generator\|generated]] by a [[3/2\|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out\|tempered out]], and we also have a whole tone plus a period represent [[8/5]]. 9/8 splits in two very naturally into [[17/16]] × [[18/17]], and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out [[S-expression\|S16]] = [[256/255]], S17 = [[289/288]], and their product [[136/135]], leading to a 2.3.5.17 [[subgroup]] extension called '''srutal archagall'''.		'''Diaschismic''', sometimes known as [[srutal vs diaschismic\|srutal]] in the [[5-limit]], is a half-octave [[regular temperament\|temperament]] [[generator\|generated]] by a [[3/2\|perfect fifth]] or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]], is [[tempering out\|tempered out]], and we also have a whole tone plus a period represent [[8/5]]. 9/8 splits in two very naturally into [[17/16]] × [[18/17]], and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out [[S-expression\|S16]] = [[256/255]], S17 = [[289/288]], and their product [[136/135]], leading to a 2.3.5.17 [[subgroup]] extension called '''srutal archagall'''.

	The canonical [[extension]] to the [[7-limit]] lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]], as well as the ~~hemifamity~~ comma, [[5120/5103]].		The canonical [[extension]] to the [[7-limit]] lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]], as well as the argent comma, [[5120/5103]].

	A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]]. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255.		A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)\|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. The mappings of primes [[11/1\|11]] and [[13/1\|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]], leading to the same mapping of prime 17 as in srutal archagall. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]].

	Another option for extension is [[pajara]], which inherits the inaccuracy of [[archy]] while providing a much simpler representation of the 7-limit.		An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]].

			Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.

	See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic]] for technical data.		See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic]] for technical data.

Overthink: added instead of space

2025-11-29T21:15:52Z

added instead of space

← Older revision		Revision as of 21:15, 29 November 2025
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	{{Infobox Regtemp		{{Infobox Regtemp
	\| Title = Diaschismic; ~~srutal~~ archagall		\| Title = Diaschismic; Srutal archagall
	\| Subgroups = 2.3.5, 2.3.5.17		\| Subgroups = 2.3.5, 2.3.5.17
	\| Comma basis = [[2048/2025]] (2.3.5); <br /> [[136/135]], [[256/255]] (2.3.5.17)		\| Comma basis = [[2048/2025]] (2.3.5); <br /> [[136/135]], [[256/255]] (2.3.5.17)