Schismic - Revision history

Vector at 02:04, 2 June 2026

2026-06-02T02:04:09Z

← Older revision		Revision as of 02:04, 2 June 2026
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	{{Infobox regtemp\|Mapping=1; 1 -8 -3\|Subgroup=2.3.5, 2.3.5.19\|Edo join 1=12\|Edo join 2=29\|Generators=3/2\|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)\|Odd limit 1=5\|Complexity 1=12\|Generators tuning=701.7\|Subgroups=2.3.5, 2.3.5.19\|Mistuning 1=0.217\|Optimization method=CWE\|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}		{{Infobox regtemp\|Mapping=1; 1 -8 -3\|Subgroup=2.3.5, 2.3.5.19\|Edo join 1=12\|Edo join 2=29\|Generators=3/2\|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)\|Odd limit 1=5\|Complexity 1=12\|Generators tuning=701.7\|Subgroups=2.3.5, 2.3.5.19\|Mistuning 1=0.217\|Optimization method=CWE\|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}
			[[File:Schismic17.png\|thumb\|Schismic 17-note MOS. This is a sharp tuning; the Pythagorean comma has been inflated to function as an [[aberrisma]].]]
	'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.		'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

Lériendil: /* Extensions */

2026-05-18T04:35:36Z

Extensions

← Older revision		Revision as of 04:35, 18 May 2026
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	=== Prime 7 ===		=== Prime 7 ===
	The ~~primary~~ 7-limit extension of Schismic is Garibaldi (also 12 & 29), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just) - that is, Garibaldi is not a microtemperament - but it is still more accurate than [[Meantone]] as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category. Garibaldi is an intuitive way of organizing just intonation as it tempers together the defining ~20-30c commas of the 7-limit.		The most important 7-limit extension of Schismic is Garibaldi (also 12 & 29, although 41 & 53 represents its tuning range better), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just) - that is, Garibaldi is not a microtemperament - but it is still more accurate than [[Meantone]] as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category. Garibaldi is an intuitive way of organizing just intonation as it tempers together the defining ~20-30c commas of the 7-limit.

	=== Primes 11 and 13 ===		==== Primes 11 and 13 ====
	A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.		A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.

	=== Prime 17 ===		=== Prime 17 ===
	Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 (preferring flatter schismic tunings) and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).		Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 (preferring flatter schismic tunings) and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).

	== Intervals ==		== Intervals ==

Lériendil at 04:19, 18 May 2026

2026-05-18T04:19:56Z

← Older revision		Revision as of 04:19, 18 May 2026
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	'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.		'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

	The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. 2.3.5.19 Schismic may be called ''Nestoria''.		The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out [[513/512]], which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma; [[8/5]] is split into two, with 24/19 and 19/15 being equated, tempering out 361/360. 2.3.5.19 Schismic may be called ''Nestoria''.

	While Schismic can be considered a 12-form temperament, the primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]]. Functionally reasonable tunings of Schismic[17] lie broadly between 29 and 53edo and include argent on the sharper end (e.g. 70c val) and Pythagorean tuning on the flatter end; [[aberrismic]] theory suggests that narrowing the Pythagorean comma further than 53edo would lead to it losing its functional property as a step of the scale.		While Schismic can be considered a 12-form temperament, the primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]]. Functionally reasonable tunings of Schismic[17] lie broadly between 29 and 53edo and include argent on the sharper end (e.g. 70c val) and Pythagorean tuning on the flatter end; [[aberrismic]] theory suggests that narrowing the Pythagorean comma further than 53edo would lead to it losing its functional property as a step of the scale.

Vector at 03:29, 18 May 2026

2026-05-18T03:29:53Z

← Older revision		Revision as of 03:29, 18 May 2026
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	The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. 2.3.5.19 Schismic may be called ''Nestoria''.		The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. 2.3.5.19 Schismic may be called ''Nestoria''.

	~~The~~ primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]].		While Schismic can be considered a 12-form temperament, the primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]]. Functionally reasonable tunings of Schismic[17] lie broadly between 29 and 53edo and include argent on the sharper end (e.g. 70c val) and Pythagorean tuning on the flatter end; [[aberrismic]] theory suggests that narrowing the Pythagorean comma further than 53edo would lead to it losing its functional property as a step of the scale.

	48edo and 58edo are the first two edos to observe the schisma - that is, to not support Schismic - while tuning the fifth within the aforementioned Schismic tuning range.		48edo and 58edo are the first two edos to observe the schisma - that is, to not support Schismic - while tuning the fifth within the aforementioned Schismic tuning range.

Inthar at 18:16, 13 April 2026

2026-04-13T18:16:28Z

← Older revision		Revision as of 18:16, 13 April 2026
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	{{Infobox regtemp\|Mapping=1; 1 -8 -3\|Subgroup=2.3.5, 2.3.5.19\|Edo join 1=12\|Edo join 2=29\|Generators=3/2\|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)\|Odd limit 1=5\|Complexity 1=12\|Generators tuning=701.7\|Subgroups=2.3.5, 2.3.5.19\|Mistuning 1=0.217\|Optimization method=CWE\|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}		{{Infobox regtemp\|Mapping=1; 1 -8 -3\|Subgroup=2.3.5, 2.3.5.19\|Edo join 1=12\|Edo join 2=29\|Generators=3/2\|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)\|Odd limit 1=5\|Complexity 1=12\|Generators tuning=701.7\|Subgroups=2.3.5, 2.3.5.19\|Mistuning 1=0.217\|Optimization method=CWE\|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}
	'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.		'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

	The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. 2.3.5.19 Schismic may be called ''Nestoria''.		The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. 2.3.5.19 Schismic may be called ''Nestoria''.

Inthar: /* Intervals */

2026-03-29T03:11:06Z

Intervals

← Older revision		Revision as of 03:11, 29 March 2026
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	\|905.87		\|905.87
	\|32/19, 27/16		\|32/19, 27/16
	\|3		\|'''3'''
	\|294.13		\|'''294.13'''
	\|19/16		\|'''19/16'''
	\|-		\|-
	\|4		\|4

Vector: nestoria canonical probably?

2026-03-29T02:24:03Z

nestoria canonical probably?

← Older revision		Revision as of 02:24, 29 March 2026
Line 2:		Line 2:
	'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.		'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

	The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. ~~Tempering out both 513/512 and 1216/1215 results in an extension~~ called ''Nestoria''.		The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. 2.3.5.19 Schismic may be called ''Nestoria''.

	The primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]].		The primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]].

Aura: Added more information on the 19-limit extension, Nestoria.

2026-03-29T00:13:25Z

Added more information on the 19-limit extension, Nestoria.

← Older revision		Revision as of 00:13, 29 March 2026
Line 2:		Line 2:
	'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.		'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

	The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third ~~(and consequently~~ 19/15 with the diatonic major third), tempering out ~~513~~/~~512~~, which is sometimes called ~~Boethius~~' comma.		The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. Tempering out both 513/512 and 1216/1215 results in an extension called ''Nestoria''.

	The primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]].		The primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]].

Vector at 23:32, 28 March 2026

2026-03-28T23:32:34Z

← Older revision		Revision as of 23:32, 28 March 2026
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	=== Prime 17 ===		=== Prime 17 ===
	Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).		Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 (preferring flatter schismic tunings) and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).

	== Intervals ==		== Intervals ==

Inthar at 21:44, 28 March 2026

2026-03-28T21:44:35Z

← Older revision		Revision as of 21:44, 28 March 2026
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	{{Infobox regtemp\|Mapping=1; 1 -8 -3\|Subgroup=2.3.5, 2.3.5.19\|Edo join 1=12\|Edo join 2=29\|Generators=3/2\|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)\|Odd limit 1=5\|Complexity 1=12\|Generators tuning=701.7\|Subgroups=2.3.5, 2.3.5.19\|Mistuning 1=0.217\|Optimization method=CWE\|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}		{{Infobox regtemp\|Mapping=1; 1 -8 -3\|Subgroup=2.3.5, 2.3.5.19\|Edo join 1=12\|Edo join 2=29\|Generators=3/2\|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)\|Odd limit 1=5\|Complexity 1=12\|Generators tuning=701.7\|Subgroups=2.3.5, 2.3.5.19\|Mistuning 1=0.217\|Optimization method=CWE\|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}
	'''Schismic''' or '''Schismatic'''[a], [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.		'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

	The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third (and consequently 19/15 with the diatonic major third), tempering out 513/512, which is sometimes called Boethius' comma.		The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third (and consequently 19/15 with the diatonic major third), tempering out 513/512, which is sometimes called Boethius' comma.
Line 243:		Line 243:
	\|40/39		\|40/39
	\|}		\|}
	[a] Despite ending in -isma, ''schisma'' is a [[Temperament naming#Comma declension categories\|3rd-declension]] comma name, therefore its temperaments are ''Schismatic'' and ''Schismic'' (which, as it is a 2.3.5 comma, refer to the same temperament).

	== Patent vals ==		== Patent vals ==
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	\|		\|
	\|}		\|}

			== Footnotes ==
			<sup>[a]</sup> Despite ending in -isma, ''schisma'' is a [[Temperament naming#Comma declension categories\|3rd-declension]] comma name, therefore its temperaments are ''Schismatic'' and ''Schismic'' (which, as it is a 2.3.5 comma, refer to the same temperament).
	{{Navbox regtemp}}		{{Navbox regtemp}}