Highly composite equal division: Difference between revisions
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Unlike highly composite edos, whose harmonic content tends to be random and usually contorted, highly composite EDFs often correspond to a useful edo. | Unlike highly composite edos, whose harmonic content tends to be random and usually contorted, highly composite EDFs often correspond to a useful edo. | ||
Highly composite EDFs have a possible usage in Georgian | Highly composite EDFs have a possible usage in Georgian-inspired music. Since [[Kartvelian scales]] are created by dividing the perfect fifth into an arbitrary number of steps, and complementing that with dividing 4/3 into an arbitrary number of steps, edos which correspond to highly composite EDFs have a high density of such scales per their size. | ||
=== Highly composite EDF-EDO correspondence === | === Highly composite EDF-EDO correspondence === | ||
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== Generalization == | == Generalization == | ||
=== Extensions === | |||
It is possible to define N-generalized highly composite numbers as a set of numbers, for which sum of powers of divisors relative to the number is greater than all the ones before it. This means that 0-generalized highly composite numbers are plain highly composite numbers, 1-generalized are superabundant numbers, etc. | |||
In terms of composition, this means that the equal division has the largest number of N-note combinations in its divisors relative to its size than all other equal divisions (of the same equave) before it. 0-generalized, highly composite equal divisions, have the largest amount of divisor scales, that is sub-EDOs. 1-generalized, superabundant equal divisions, have the largest amount of notes in those scales, etc. | |||
This also means that an N-generalized highly composite number has the largest amount of [[Regular temperament theory|rank-N temperaments]] that make use of sub-EDOs as generators relative to its size. For example, 2-generalized highly composite numbers therefore contain the most rank-2 temperaments that they both carry themselves as coprime to equal division's size, and also rank-2 temperaments of their subset equal divisons respectively. | |||
2-generalized highly composite numbers are the following ([[oeis:A208767|OEIS: A208767]]): | |||
{{EDOs|1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 360360, 720720}}. | |||
=== Superabundant equal division === | === Superabundant equal division === | ||
[[Wikipedia:Superabundant number|Superabundant numbers]] are slightly different from highly composite numbers, hence superabundant equal divisions can be considered separately from highly composite numbers whenever the properties of superabundant numbers are more relevant in context. '''Abundancy index''' is the ratio of the sum of divisors of the number to the number itself, and thus superabundant numbers are numbers where these indices increase to a record, and thus are the highest in proportion to the number itself. | [[Wikipedia:Superabundant number|Superabundant numbers]] are slightly different from highly composite numbers, hence superabundant equal divisions can be considered separately from highly composite numbers whenever the properties of superabundant numbers are more relevant in context. '''Abundancy index''' is the ratio of the sum of divisors of the number to the number itself, and thus superabundant numbers are numbers where these indices increase to a record, and thus are the highest in proportion to the number itself. | ||
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An example when they are not the same: [[50400edo|50400]] is the 27th highly composite number, that is not on the superabundant list. The count of divisors of 50400 is 108, which means it supports 106 symmetrical scales that aren't the chromatic and whole-octave scale. However, by the note count in all those scales, which is 102311 not counting 1 and 50400, the edo lags slightly behind [[27720edo]], with it's coefficient being 4.05 and 50400edo's coefficient being 4.03. This means that while 27720 is less composite than 50400, it carries a more impressive task in providing notes to compose with, if the composer is interested in smaller edos as subscales. And indeed it can be intuitively inferred from the fact that 27720 is divisible by 11, therefore contains [[11edo]] and thus introduces a new factor, while 50400 recycles edos from 1 to 10 multiple times. | An example when they are not the same: [[50400edo|50400]] is the 27th highly composite number, that is not on the superabundant list. The count of divisors of 50400 is 108, which means it supports 106 symmetrical scales that aren't the chromatic and whole-octave scale. However, by the note count in all those scales, which is 102311 not counting 1 and 50400, the edo lags slightly behind [[27720edo]], with it's coefficient being 4.05 and 50400edo's coefficient being 4.03. This means that while 27720 is less composite than 50400, it carries a more impressive task in providing notes to compose with, if the composer is interested in smaller edos as subscales. And indeed it can be intuitively inferred from the fact that 27720 is divisible by 11, therefore contains [[11edo]] and thus introduces a new factor, while 50400 recycles edos from 1 to 10 multiple times. | ||
=== Largely composite numbers === | === Largely composite numbers === | ||
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First few are (OEIS:A033833): {{EDOs|1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160}}. The sequence has a few notable members like 72edo and 96edo, alongside the 23-limit giant [[4320edo]] and a few already known highly composite numbers. | First few are (OEIS:A033833): {{EDOs|1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160}}. The sequence has a few notable members like 72edo and 96edo, alongside the 23-limit giant [[4320edo]] and a few already known highly composite numbers. | ||
== Prominence == | |||
Prominence (term proposed by Eliora) is the product of the number's count of divisors (highly composite) or its abundance index (superabundant) with its amount of distinct prime factors. Prominence serves as a good measure of different varieties of subsets that an equal division can provide, since a more prominent number is capable of approximating more distinct temperaments efficiently for its size. | Prominence (term proposed by Eliora) is the product of the number's count of divisors (highly composite) or its abundance index (superabundant) with its amount of distinct prime factors. Prominence serves as a good measure of different varieties of subsets that an equal division can provide, since a more prominent number is capable of approximating more distinct temperaments efficiently for its size. | ||
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[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||
{{todo|review|cleanup|comment=the way the page is laid out seems a bit chaotic and scattered, see if that can be rectified by switching up the heading structure}} | |||