Kleismic - Revision history

Vector at 01:57, 2 June 2026

2026-06-02T01:57:57Z

← Older revision		Revision as of 01:57, 2 June 2026
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	\| Odd limit 2 = 2.3.5.13 15 \| Mistuning 2 = 2.35 \| Complexity 2 = 15		\| Odd limit 2 = 2.3.5.13 15 \| Mistuning 2 = 2.35 \| Complexity 2 = 15
	}}		}}
			[[File:Kleismic11.png\|thumb\|Kleismic 11-note MOS]]
	'''Kleismic''', [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a [[canonical extension\|structurally induced]] extension to 2.3.5.13, it equates three 6/5's to one semitwelfth [[26/15]] and equates 25/24 to 26/25 and 27/26.		'''Kleismic''', [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a [[canonical extension\|structurally induced]] extension to 2.3.5.13, it equates three 6/5's to one semitwelfth [[26/15]] and equates 25/24 to 26/25 and 27/26.

Inthar at 02:30, 21 April 2026

2026-04-21T02:30:51Z

← Older revision		Revision as of 02:30, 21 April 2026
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	Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.		Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.

	Kleismic contains [[interordinal]] intervals~~; this is easy to see~~ because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.		Kleismic contains [[interordinal]] intervals, because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics 1–15 are labeled in '''bold'''.		In the following table, odd harmonics 1–15 are labeled in '''bold'''.

Inthar at 02:42, 19 April 2026

2026-04-19T02:42:11Z

← Older revision		Revision as of 02:42, 19 April 2026
Line 19:		Line 19:
	Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.		Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.

	Kleismic contains interordinal intervals; this is easy to see because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.		Kleismic contains [[interordinal]] intervals; this is easy to see because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics 1–15 are labeled in '''bold'''.		In the following table, odd harmonics 1–15 are labeled in '''bold'''.

Inthar at 02:41, 19 April 2026

2026-04-19T02:41:46Z

← Older revision		Revision as of 02:41, 19 April 2026
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	Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.		Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.

	Kleismic can be seen as a counterpart to [[~~diaschismic~~]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - ~~diaschismic~~ splits 9/8 into two; ~~kleismic~~ splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.		Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.

	Kleismic contains interordinal intervals. Because it equates 9/8 to three 25/24s, ~~this is a necessary consequence, as~~ the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.		Kleismic contains interordinal intervals; this is easy to see because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics 1–15 are labeled in '''bold'''.		In the following table, odd harmonics 1–15 are labeled in '''bold'''.

Vector at 02:18, 19 April 2026

2026-04-19T02:18:34Z

← Older revision		Revision as of 02:18, 19 April 2026
Line 19:		Line 19:
	Kleismic can be seen as a counterpart to [[diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - diaschismic splits 9/8 into two; kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.		Kleismic can be seen as a counterpart to [[diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - diaschismic splits 9/8 into two; kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.

	Kleismic contains interordinal intervals. Because it equates 9/8 to three 25/24s, this is a necessary consequence, as the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that augmenting or diminishing a ~~major~~ or ~~minor third~~ respectively by 25/24 results in an interordinal.		Kleismic contains interordinal intervals. Because it equates 9/8 to three 25/24s, this is a necessary consequence, as the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics 1–15 are labeled in '''bold'''.		In the following table, odd harmonics 1–15 are labeled in '''bold'''.

Vector at 02:18, 19 April 2026

2026-04-19T02:18:00Z

← Older revision		Revision as of 02:18, 19 April 2026
Line 16:		Line 16:

	Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.		Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.

			Kleismic can be seen as a counterpart to [[diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - diaschismic splits 9/8 into two; kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.

			Kleismic contains interordinal intervals. Because it equates 9/8 to three 25/24s, this is a necessary consequence, as the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that augmenting or diminishing a major or minor third respectively by 25/24 results in an interordinal.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics 1–15 are labeled in '''bold'''.		In the following table, odd harmonics 1–15 are labeled in '''bold'''.

Inthar: /* List of patent vals */

2026-03-14T17:00:13Z

List of patent vals

@@ Line 107: / Line 107: @@
 == List of patent vals ==
-The following patent vals support 2.3.5.13 Kleismic. Contorted vals are not included.
+:''Main article: [[Kleismic/Patent vals]]''
 {{Navbox regtemp}}
 {{cat|Temperaments}}

Inthar: /* List of patent vals */

2026-03-14T00:43:37Z

List of patent vals

← Older revision		Revision as of 00:43, 14 March 2026
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	The following patent vals support 2.3.5.13 Kleismic. Contorted vals are not included.		The following patent vals support 2.3.5.13 Kleismic. Contorted vals are not included.

	{\| class="wikitable"		{\| class="wikitable sortable"
	!\|Edo !! Generator tuning !! Fifth tuning		!\|Edo !! Generator tuning !! Fifth tuning
	\|-		\|-

Inthar: /* List of patent vals */

2026-03-14T00:40:53Z

List of patent vals

@@ Line 107: / Line 107: @@
 == List of patent vals ==
 {{Navbox regtemp}}
 {{cat|Temperaments}}

Inthar: /* Interval chain */

2026-03-14T00:26:31Z

Interval chain

← Older revision		Revision as of 00:26, 14 March 2026
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	<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced		<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced

			== List of patent vals ==
	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{cat\|Temperaments}}		{{cat\|Temperaments}}