Highly composite EDO: Difference between revisions
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Contrast this with zeta EDOs, that approximate just intonation well, being highly harmonic EDOs. | Contrast this with zeta EDOs, that approximate just intonation well, being highly harmonic EDOs. | ||
== | == Highly composite numbers in EDOs == | ||
The defining feature of a SA-EDO is that the EDO has the largest number of notes within sub-EDOs that it contains, in proportion to the EDO's number. This means that: | The defining feature of a SA-EDO is that the EDO has the largest number of notes within sub-EDOs that it contains, in proportion to the EDO's number. This means that: | ||
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The factor of being relative to the number's size is important. For example, [[36edo|36]] has a sum of divisors of 91, while 130 has a sum of divisors of 252. If all the sub-scales were stretched end-to-end, this means 36 has 91 notes, and 130 has 252. However, 91/36 = 2.527, while 252/130 = 1.938, meaning that 36 carries a more impressive task in replicating sub-EDOs and symmetrical chords relative to it's size. | The factor of being relative to the number's size is important. For example, [[36edo|36]] has a sum of divisors of 91, while 130 has a sum of divisors of 252. If all the sub-scales were stretched end-to-end, this means 36 has 91 notes, and 130 has 252. However, 91/36 = 2.527, while 252/130 = 1.938, meaning that 36 carries a more impressive task in replicating sub-EDOs and symmetrical chords relative to it's size. | ||
== | === Superabundant EDOs and the difference with highly composite === | ||
The difference between SA and HC is the following: the highly composite numbers count the amount of divisors, that is sub-EDOs, going to a record., while superabundant EDOs count the amount of note in those divisors if they were stretched end-to-end. | The difference between SA and HC is the following: the highly composite numbers count the amount of divisors, that is sub-EDOs, going to a record., while superabundant EDOs count the amount of note in those divisors if they were stretched end-to-end. | ||
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And indeed it's somewhat obvious - 27720 is divisible by 11, therefore contains [[11edo]], while 50400 recycles EDOs from 1 to 10 multiple times. | And indeed it's somewhat obvious - 27720 is divisible by 11, therefore contains [[11edo]], while 50400 recycles EDOs from 1 to 10 multiple times. | ||
== First highly | == First highly composite EDOs == | ||
First | <code>First highly composite EDOs: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160.</code> | ||
Superabundant EDOs that are also highly composite, excluding the first 19: 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 367567200, 698377680, 735134400. | Superabundant EDOs that are also highly composite, excluding the first 19: 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 367567200, 698377680, 735134400. | ||
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The sequence is finite and has 430 terms starting with 10080. | The sequence is finite and has 430 terms starting with 10080. | ||
12edo is the predominantly used tuning in the world today, and in addition it is the only known so far highly composite EDO that's also a zeta edo. Others have not been found yet, and given the lack of such EDOs until hundreds of thousands it's likely if another one is found, it would be of any harmonic use since it's amount of steps would be astronomical. | [[12edo]] is the predominantly used tuning in the world today, and in addition it is the only known so far highly composite EDO that's also a zeta edo, and the only one with a step size above [[just noticeable difference]]. Others have not been found yet, and given the lack of such EDOs until hundreds of thousands it's likely if another one is found, it would not be of any harmonic use since it's amount of steps would be astronomical. | ||
== Variations upon the theme == | == Variations upon the theme == | ||
=== Extensions === | === Extensions === | ||
It is possible to define N-generalized | 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 EDO has the largest number of N-note cobinations in its divisors relative to its size than all other EDOs before it. 0-generalized, highly composite numbers, have the largest amount of divisor scales in EDO. 1-generalized have the largest amount of notes in those scales, and etc. | In terms of composition, this means that the EDO has the largest number of N-note cobinations in its divisors relative to its size than all other EDOs before it. 0-generalized, highly composite numbers, have the largest amount of divisor scales in EDO. 1-generalized have the largest amount of notes in those scales, and etc. | ||
=== | === Largely composite numbers === | ||
Largely composite numbers are a variant of a sequence of numbers where the divisor count is nondecreasing, as opposed to strictly increasing | Largely composite numbers are a variant of a sequence of numbers where the divisor count is nondecreasing, as opposed to strictly increasing. | ||
First few are (OEIS: A067128): {{EDOs|1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240}}. | First few are (OEIS: A067128): {{EDOs|1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240}}. | ||
While it contains the well known highly composite numbers, it also contains some historically notable temperaments such as [[72edo]] and [[96edo]]. | While it contains the well known highly composite numbers, it also contains some historically notable temperaments such as [[72edo]] and [[96edo]]. 72edo has been used in [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|Alois Haba]] and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. | ||
=== 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 EDO can provide, since a more prominent number is capable of approximating more distinct temperaments efficiently for its size. | |||
Example: [[1848edo]] is more prominent than [[3456edo]] despite both having 32 divisors, due to 1848 having 2, 3, 5, 7, and 11 as prime factors, while 3456 having only 2 and 3. | |||
Just as there are generalized N-highly composite numbers, one can generalize N,M-highly prominent numbers, since this time there is another dimension. | |||
== External links == | == External links == | ||