388edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:32Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
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	\| [[Berkelium]]		\| [[Berkelium]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	[[Category:Cuthbert]]		[[Category:Cuthbert]]

Overthink: delete stuff beyond 113-limit

2025-12-25T09:19:00Z

delete stuff beyond 113-limit

← Older revision		Revision as of 09:19, 25 December 2025
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	{{Harmonics in equal\|388\|columns=12}}		{{Harmonics in equal\|388\|columns=12}}
	{{Harmonics in equal\|388\|start=13\|columns=12\|collapsed=1}}		{{Harmonics in equal\|388\|start=13\|columns=12\|collapsed=1}}
	~~{{Harmonics in equal\|388\|start=25\|columns=12\|collapsed=1}}~~

	=== Subsets and supersets ===		=== Subsets and supersets ===
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	388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.		388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.

	311edo also deals better with composite harmonics than 388edo. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113~~, 139, 149, and 151~~. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113~~, 131, 137, 139, and 149~~. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]] or [[494edo]]. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle.		311edo also deals better with composite harmonics than 388edo. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, and 113. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, and 113. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]] or [[494edo]]. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle.

	Another system notable in high limits around this size is [[422edo]], using the sharp-tending 422[[wart\|l]] [[val]] for prime 37.		Another system notable in high limits around this size is [[422edo]], using the sharp-tending 422[[wart\|l]] [[val]] for prime 37.

Overthink: /* Approximation to JI */ + more notes

2025-12-12T23:29:59Z

Approximation to JI: + more notes

← Older revision		Revision as of 23:29, 12 December 2025
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	388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.		388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.

	311edo also deals better with composite harmonics than ~~388~~. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 47 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]]. Note that ~~311~~ has generally higher absolute errors than ~~388~~ due to its smaller size, but smaller size also means the system is easier to handle.		311edo also deals better with composite harmonics than 388edo. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]] or [[494edo]]. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle.

	Another system notable in high limits around this size is [[422edo]].		Another system notable in high limits around this size is [[422edo]], using the sharp-tending 422[[wart\|l]] [[val]] for prime 37.

			=== 37-odd-limit intervals ===

	{{Q-odd-limit intervals\|388\|limit=37}}		{{Q-odd-limit intervals\|388\|limit=37}}

FloraC: Sectioning

2025-10-20T10:41:34Z

Sectioning

← Older revision		Revision as of 10:41, 20 October 2025
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	{{Harmonics in equal\|388\|start=25\|columns=12\|collapsed=1}}		{{Harmonics in equal\|388\|start=25\|columns=12\|collapsed=1}}

	=== Approximation to JI ===		=== Subsets and supersets ===
	This ~~EDO~~ has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.		Since 388 factors into primes as {{nowrap\| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs\| 2, 4, 97, and 194 }}.

			== Approximation to JI ==
			This edo has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.

	388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.		388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.
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	{{Q-odd-limit intervals\|388\|limit=37}}		{{Q-odd-limit intervals\|388\|limit=37}}

	~~=== Subsets and supersets ===~~
	~~Since 388 factors into {{nowrap\| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs\| 2, 4, 97, and 194 }}.~~

	== Regular temperament properties ==		== Regular temperament properties ==

Overthink: /* Approximation to JI */

2025-10-20T00:47:45Z

Approximation to JI

← Older revision		Revision as of 00:47, 20 October 2025
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	388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.		388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.

	311edo also deals better with composite harmonics than 388. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22, 41/28, 41/22, and their octave complements. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]]. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.		311edo also deals better with composite harmonics than 388. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 47 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]]. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.

	Another system notable in high limits around this size is [[422edo]].		Another system notable in high limits around this size is [[422edo]].

			{{Q-odd-limit intervals\|388\|limit=37}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

Overthink: /* Theory */

2025-10-20T00:43:54Z

Theory

← Older revision		Revision as of 00:43, 20 October 2025
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	It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].		It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

	~~{{Todo\|inline=1\| expand \| comment=Explain its higher limit properties.}}~~

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|388\|columns=12}}		{{Harmonics in equal\|388\|columns=12}}
			{{Harmonics in equal\|388\|start=13\|columns=12\|collapsed=1}}
			{{Harmonics in equal\|388\|start=25\|columns=12\|collapsed=1}}

			=== Approximation to JI ===
			This EDO has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.

			388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1\|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.

			311edo also deals better with composite harmonics than 388. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22, 41/28, 41/22, and their octave complements. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]]. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.

			Another system notable in high limits around this size is [[422edo]].

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 388 factors into {{nowrap\| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs\| 2, 4, 97, and 194 }}.		Since 388 factors into {{nowrap\| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs\| 2, 4, 97, and 194 }}.

	== Regular temperament properties ==		== Regular temperament properties ==

Overthink at 04:44, 15 September 2025

2025-09-15T04:44:47Z

← Older revision		Revision as of 04:44, 15 September 2025
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	It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].		It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

			{{Todo\|inline=1\| expand \| comment=Explain its higher limit properties.}}

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: Cleanup and update

2025-03-16T15:07:23Z

Cleanup and update

← Older revision		Revision as of 15:07, 16 March 2025
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	388edo is the first edo that is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]]; it is also consistent through the [[37-odd-limit]].		388edo is the first edo that is [[consistency\|distinctly consistent]] through to the [[27-odd-limit]]; it is also consistent through the [[37-odd-limit]].

	The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[~~tricot~~ comma]], {{monzo\| 39 -29 3 }}, the [[minortone comma]], {{monzo\| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider\|raider comma]], {{monzo\| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.		The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[alphatricot comma]], {{monzo\| 39 -29 3 }}, the [[minortone comma]], {{monzo\| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider\|raider comma]], {{monzo\| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.

	It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\|190 &~~amp;~~ 198}} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].		It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 388 factors into {{~~factorization~~\|~~388~~}}, 388edo has subset edos {{EDOs\| 2, 4, 97, and 194 }}.		Since 388 factors into {{nowrap\| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs\| 2, 4, 97, and 194 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
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	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
Line 28:		Line 28:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 615 -388 }}		\| {{Monzo\| 615 -388 }}
	\| {{~~mapping~~\| 388 615 }}		\| {{Mapping\| 388 615 }}
	\| +0.0337		\| +0.0337
	\| 0.0337		\| 0.0337
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	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 23 6 -14 }}, {{monzo\| 39 -29 3 }}		\| {{Monzo\| 23 6 -14 }}, {{monzo\| 39 -29 3 }}
	\| {{~~mapping~~\| 388 615 901 }}		\| {{Mapping\| 388 615 901 }}
	\| −0.0633		\| −0.0633
	\| 0.0501		\| 0.0501
Line 43:		Line 43:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 235298/234375, 2100875/2097152		\| 4375/4374, 235298/234375, 2100875/2097152
	\| {{~~mapping~~\| 388 615 901 1089 }}		\| {{Mapping\| 388 615 901 1089 }}
	\| +0.0224		\| +0.0224
	\| 0.1546		\| 0.1546
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 5632/5625, 235298/234375		\| 3025/3024, 4375/4374, 5632/5625, 235298/234375
	\| {{~~mapping~~\| 388 615 901 1089 1342 }}		\| {{Mapping\| 388 615 901 1089 1342 }}
	\| +0.0643		\| +0.0643
	\| 0.1617		\| 0.1617
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374		\| 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374
	\| {{~~mapping~~\| 388 615 901 1089 1342 1436 }}		\| {{Mapping\| 388 615 901 1089 1342 1436 }}
	\| +0.0216		\| +0.0216
	\| 0.1758		\| 0.1758
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700		\| 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700
	\| {{~~mapping~~\| 388 615 901 1089 1342 1436 1586 }}		\| {{Mapping\| 388 615 901 1089 1342 1436 1586 }}
	\| +0.0116		\| +0.0116
	\| 0.1646		\| 0.1646
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330		\| 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330
	\| {{~~mapping~~\| 388 615 901 1089 1342 1436 1586 1648 }}		\| {{Mapping\| 388 615 901 1089 1342 1436 1586 1648 }}
	\| +0.0280		\| +0.0280
	\| 0.1600		\| 0.1600
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	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| 565.97		\| 565.97
	\| 75/52		\| 75/52
	\| [[~~Trillium~~]] / [[pseudotrillium]]		\| [[Alphatrillium]] / [[pseudotrillium]]
	\|-		\|-
	\| 2		\| 2
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	\|-		\|-
	\| 4		\| 4
	\| 123\388<br />(26\388)		\| 123\388<br>(26\388)
	\| 380.41<br />(80.41)		\| 380.41<br>(80.41)
	\| 81/65<br />(22/21)		\| 81/65<br>(22/21)
	\| [[Quasithird]]		\| [[Quasithird]]
	\|-		\|-
	\| 97		\| 97
	\| 161\388<br />(1\388)		\| 161\388<br>(1\388)
	\| 497.938<br />(3.09)		\| 497.938<br>(3.09)
	\| 4/3<br />(?)		\| 4/3<br>(?)
	\| [[Berkelium]]		\| [[Berkelium]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:Cuthbert]]		[[Category:Cuthbert]]

ArrowHead294 at 14:49, 20 February 2025

2025-02-20T14:49:16Z

← Older revision		Revision as of 14:49, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|388~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:44, 15 January 2025

2025-01-15T19:44:45Z

← Older revision		Revision as of 19:44, 15 January 2025
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	The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[tricot comma]], {{monzo\| 39 -29 3 }}, the [[minortone comma]], {{monzo\| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider\|raider comma]], {{monzo\| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.		The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }}, the [[tricot comma]], {{monzo\| 39 -29 3 }}, the [[minortone comma]], {{monzo\| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider\|raider comma]], {{monzo\| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.

	It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the 190 & 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].		It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap\|190 & 198}} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	\| {{monzo\| 23 6 -14 }}, {{monzo\| 39 -29 3 }}		\| {{monzo\| 23 6 -14 }}, {{monzo\| 39 -29 3 }}
	\| {{mapping\| 388 615 901 }}		\| {{mapping\| 388 615 901 }}
	\| ~~−0~~.0633		\| −0.0633
	\| 0.0501		\| 0.0501
	\| 1.62		\| 1.62
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	\| [[Berkelium]]		\| [[Berkelium]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:Cuthbert]]		[[Category:Cuthbert]]