5040edo - Revision history

ArrowHead294 at 22:56, 20 February 2025

2025-02-20T22:56:08Z

← Older revision		Revision as of 22:56, 20 February 2025
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	== Scales ==		== Scales ==
	* Consecutive[43] - 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229		* Consecutive[43] – 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

	== References ==		== References ==

ArrowHead294 at 22:55, 20 February 2025

2025-02-20T22:55:59Z

Show changes

FloraC: -redundant categories. Mark todo

2024-08-07T11:48:55Z

-redundant categories. Mark todo

← Older revision		Revision as of 11:48, 7 August 2024
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	* https://mathworld.wolfram.com/PlatosNumbers.html		* https://mathworld.wolfram.com/PlatosNumbers.html

	~~[[Category:Equal divisions of the octave~~\|~~####]] <!-- 4-digit number -->~~		{{Todo\|review}}
	~~[[Category:Highly composite]]~~

Eliora: /* Theory */ i think that fact is best reserved for the 2520/2519 platonisma and not this page

2024-05-30T15:06:37Z

Theory: i think that fact is best reserved for the 2520/2519 platonisma and not this page

← Older revision		Revision as of 15:06, 30 May 2024
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	5040's divisors besides 1 and itself are {{EDOs\|2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520}} - which is a total of 58. This sequence does include some notable temperaments. For example, [[12edo]], which is the dominant tuning system in the world today, [[15edo]], notable for its usage by [[Easley Blackwood Jr.]], [[72edo]], notable for use in Gregorian chants and known for use by [[Joe Maneri]]. Other notable systems which are divisors of 5040 are all the EDOs from 1 to 10 inclusive, and also [[140edo]].		5040's divisors besides 1 and itself are {{EDOs\|2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520}} - which is a total of 58. This sequence does include some notable temperaments. For example, [[12edo]], which is the dominant tuning system in the world today, [[15edo]], notable for its usage by [[Easley Blackwood Jr.]], [[72edo]], notable for use in Gregorian chants and known for use by [[Joe Maneri]]. Other notable systems which are divisors of 5040 are all the EDOs from 1 to 10 inclusive, and also [[140edo]].

	5040 is the 19th superabundant and highly composite EDO, and it marks the end of the sequence where superabundant and highly composite numbers are the same - 7560 is the first highly composite that isn't superabundant. To provide some reference of how notable 5040 is, due to its profound divisibility the number has found some applied historical use. Ancient Greek philosopher Plato suggested that 5040 is the ideal number of people in a city, citing its high divisibility properties, and also the fact that the number that is two below 5040, 5038, is divisible by 11.		5040 is the 19th superabundant and highly composite EDO, and it marks the end of the sequence where superabundant and highly composite numbers are the same - 7560 is the first highly composite that isn't superabundant.

	5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.		5040 is also a sum of 43 consecutive primes, running from 23 to 229 inclusive.

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

Eliora: /* Regular temperament theory */

2023-11-06T11:41:06Z

Regular temperament theory

← Older revision		Revision as of 11:41, 6 November 2023
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	=== Regular temperament theory ===		=== Regular temperament theory ===
	From the regular temperament theory perspective, 5040edo is not as impressive, as for example zeta EDOs. ~~It does offer unique harmonies, although they are not simple. Since having errors less than~~ 25% ~~guarantees "naive consistency" to distance 1~~, ~~this means that~~ the best subgroup for 5040edo is 2.3.7.13.17.23.29.31.41.43.53.59.61.73.89.		From the regular temperament theory perspective, 5040edo is not as impressive, as for example zeta EDOs. Using the 25% error cutoff, the best subgroup for 5040edo is 2.3.7.13.17.23.29.31.41.43.53.59.61.73.89.

	There's an interesting property that arises in the subgroup, 2.7.13.17.29.31.41.47.61.67. In this subgroup, it makes a rank two temperament with 1111edo (1111 & 5040), which brings two notable number classes together - a repunit and a highly composite number.		There's an interesting property that arises in the subgroup, 2.7.13.17.29.31.41.47.61.67. In this subgroup, it makes a rank two temperament with 1111edo (1111 & 5040), which brings two notable number classes together - a repunit and a highly composite number.

Eliora: /* Regular temperament theory */ oops

2023-09-22T15:51:44Z

Regular temperament theory: oops

← Older revision		Revision as of 15:51, 22 September 2023
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	=== Regular temperament theory ===		=== Regular temperament theory ===
	From the regular temperament theory perspective, 5040edo is not as impressive, as for example zeta EDOs. It does offer unique harmonies, although they are not simple. Since having errors less than 25% guarantees "naive consistency" to distance 1, this means that the best subgroup for 5040edo is 2.3.7.13.17.23.29.31.41.47.61.67.		From the regular temperament theory perspective, 5040edo is not as impressive, as for example zeta EDOs. It does offer unique harmonies, although they are not simple. Since having errors less than 25% guarantees "naive consistency" to distance 1, this means that the best subgroup for 5040edo is 2.3.7.13.17.23.29.31.41.43.53.59.61.73.89.

	There's an interesting property that arises in ~~a stricter~~ subgroup, 2.7.13.17.29.31.41.47.61.67. In this subgroup, it makes a rank two temperament with 1111edo (1111 & 5040), which brings two notable number classes together - a repunit and a highly composite number.		There's an interesting property that arises in the subgroup, 2.7.13.17.29.31.41.47.61.67. In this subgroup, it makes a rank two temperament with 1111edo (1111 & 5040), which brings two notable number classes together - a repunit and a highly composite number.

	Using the patent val, 5040edo tempers out [[9801/9800]] in the 11-limit.		Using the patent val, 5040edo tempers out [[9801/9800]] in the 11-limit.

Eliora: /* Prime harmonics */ add continued as I talk on the page a lot about high prime limit

2023-09-22T10:49:39Z

Prime harmonics: add continued as I talk on the page a lot about high prime limit

← Older revision		Revision as of 10:49, 22 September 2023
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|5040}}		{{Harmonics in equal\|5040}}
			{{Harmonics in equal\|84\|columns=15\|start=12\|collapsed=1\|title=Approximation of prime harmonics in 5040edo (continued)}}

	=== Regular temperament theory ===		=== Regular temperament theory ===

BudjarnLambeth at 07:08, 9 July 2023

2023-07-09T07:08:21Z

← Older revision		Revision as of 07:08, 9 July 2023
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	~~{{novelty}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|5040}}		{{EDO intro\|5040}}

BudjarnLambeth at 04:23, 9 July 2023

2023-07-09T04:23:15Z

← Older revision		Revision as of 04:23, 9 July 2023
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	{{Infobox ET}}		{{novelty}}{{Infobox ET}}
	{{EDO intro\|5040}}		{{EDO intro\|5040}}

Eliora: /* Theory */ rewrite

2023-01-04T23:50:58Z

Theory: rewrite

← Older revision		Revision as of 23:50, 4 January 2023
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	== Theory ==		== Theory ==
	5040 is a factorial (7! = 1·2·3·4·5·6·7), superabundant, and a highly composite number. ~~Its divisors besides 1 and itself are {{EDOs\|~~2~~, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80,~~ 84, ~~90,~~ 105~~, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520}} - which is a total of 58~~.		5040 is a factorial (7! = 1·2·3·4·5·6·7), [[Highly composite equal division\|superabundant, and a highly composite]] number. The [[abundancy index]] of this number is about 2.84, or exactly 298/105.

	5040 is the ~~19th superabundant and highly composite EDO~~, and ~~it marks the end~~ of the ~~sequence where superabundant~~ and ~~highly composite numbers are the same - 7560 is the first highly composite that isn't superabundant~~.		5040's divisors besides 1 and itself are {{EDOs\|2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520}} - which is a total of 58. This sequence does include some notable temperaments. For example, [[12edo]], which is the dominant tuning system in the world today, [[15edo]], notable for its usage by [[Easley Blackwood Jr.]], [[72edo]], notable for use in Gregorian chants and known for use by [[Joe Maneri]]. Other notable systems which are divisors of 5040 are all the EDOs from 1 to 10 inclusive, and also [[140edo]].

	5040 is ~~both a~~ superabundant and a highly composite ~~number~~, ~~meaning its amount~~ of ~~symmetrical chords and subscales increases to a record,~~ and the ~~amount of notes which make up those scales, if stretched end-to~~-~~end, also~~ is ~~largest relative to~~ the ~~number~~'~~s size~~. ~~The abundance index~~ of 5040 is ~~about 2.84~~, ~~or exactly 298/105.~~		5040 is the 19th superabundant and highly composite EDO, and it marks the end of the sequence where superabundant and highly composite numbers are the same - 7560 is the first highly composite that isn't superabundant. To provide some reference of how notable 5040 is, due to its profound divisibility the number has found some applied historical use. Ancient Greek philosopher Plato suggested that 5040 is the ideal number of people in a city, citing its high divisibility properties, and also the fact that the number that is two below 5040, 5038, is divisible by 11.

	~~The~~ number has found some applied historical use. Ancient Greek philosopher Plato suggested that 5040 is the ideal number of people in a city, citing its high divisibility properties, and also the fact that the number that is two below 5040, 5038, is divisible by 11.

	5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.		5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.