Highly composite equal division: Difference between revisions
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A '''highly composite equal division''' is an [[equal tuning]] that divides a given [[equave]] into a [[ | A '''highly composite equal division''' is an [[equal tuning]] that divides a given [[equave]] into a [[highly composite number]] of pitches. The opposite of a highly composite equal division is a [[prime equal division]]. | ||
A '''highly composite edo''' therefore contains a highly composite number of pitches per octave, such as {{EDOs|6edo, 12edo and 60edo}}. | A '''highly composite edo''' therefore contains a highly composite number of pitches per octave, such as {{EDOs| 6edo, 12edo and 60edo }}. | ||
Generalizations of the concept of "anti-prime" edos include | Generalizations of the concept of "anti-prime" edos include {{w|superabundant number|superabundant}} edos, largely composite numbers, etc. | ||
== Properties == | == Properties == | ||
The defining characteristics of highly composite equal divisions are the following: | The defining characteristics of highly composite equal divisions are the following: | ||
* The tuning contains the largest count of notes in symmetrical chords, and correspondingly, in uniform equave-repeating scales, relative to its size. | * The tuning contains the largest count of notes in symmetrical chords, and correspondingly, in uniform equave-repeating scales, relative to its size. | ||
* The tuning has the largest amount of | * The tuning has the largest amount of {{w|mode of limited transposition|modes of limited transposition}} relative to its size. | ||
* The tuning has the largest amount of rank-2 temperaments whose period is a fraction of the equave, relative to its size. | * The tuning has the largest amount of rank-2 temperaments whose period is a fraction of the equave, relative to its size. | ||
* By the virtue of point 1, the tuning has the largest amount of familiar scales relative to its size | * By the virtue of point 1, the tuning has the largest amount of familiar scales relative to its size | ||
The factor of being relative to the number's size is important. For example, [[36edo|36]] has 9 divisors, while 132 has 12 divisors. However, 9/36 = 0.25, while 12/132 = 0. | The factor of being relative to the number's size is important. For example, [[36edo|36]] has 9 divisors, while 132 has 12 divisors. However, {{nowrap| 9/36 {{=}} 0.25 }}, while {{nowrap| 12/132 {{=}} 0.0909… }}, meaning that 36 carries a more impressive task in replicating sub-edos and symmetrical chords relative to its size. | ||
== Highly composite edo == | == Highly composite edo == | ||
[[12edo]], the predominantly used tuning in the world today, is the only known | [[12edo]], the predominantly used tuning in the world today, is currently the only known non-trivial highly composite edo that holds any zeta records and the only one with a step size above the [[just-noticeable difference]]. Others have not been found yet, and given the lack of such edos until hundreds of thousands it is likely if another one is found, it would not be of any harmonic use since its amount of steps would be astronomical. | ||
=== The first 41 highly composite edos === | === The first 41 highly composite edos === | ||
The first 19 highly composite edos are also the first 19 superabundant edos | The first 19 highly composite edos are also the first 19 superabundant edos: | ||
{{EDOs| 1, 2, 4, 6, 12, 24, 36, 48, 60, 120 | {{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, 11088 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160 }}… | ||
10080, 15120, 20160, 25200, | |||
166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160 | |||
Superabundant edos that are also highly composite, excluding the first 19: | Superabundant edos that are also highly composite, excluding the first 19: | ||
{{EDOs| 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 }}… The sequence is finite and has 430 terms starting with 10080. | |||
Highly composite | == Highly composite edf == | ||
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. | |||
[[Schismagonal]] temperament is useable in many highly composite edfs. | |||
{| class="wikitable" | === Highly composite edf–edo correspondence === | ||
|+Table of first highly melodic | The following is a table of first 19 highly composite/superabundant edfs and their corresponding edos. | ||
! | |||
! | {| class="wikitable right-1 right-2 center-3" | ||
!log2/log1. | |+Table of first highly melodic edf–edo correspondences | ||
(exact | ! Edf | ||
!Comments | ! Edo | ||
! log2/log1.5⋅edf<br>(exact edo) | |||
! Comments | |||
|- | |- | ||
|1 | | 1 | ||
|2 | | 2 | ||
|1.7095112 | | 1.7095112 | ||
|Trivial | | Trivial. | ||
|- | |- | ||
|2 | | 2 | ||
|3 | | 3 | ||
|3.4190226 | | 3.4190226 | ||
|Completely misses the octave. | | Completely misses the octave. | ||
|- | |- | ||
|4 | | 4 | ||
|[[7edo|7]] | | [[7edo|7]] | ||
|6.8380452 | | 6.8380452 | ||
| | | | ||
|- | |- | ||
|6 | | 6 | ||
|10 | | 10 | ||
|10.257068 | | 10.257068 | ||
|10edo, but with a heavy stretch | | 10edo, but with a heavy stretch. | ||
|- | |- | ||
|12 | | 12 | ||
| - | | - | ||
|20.514135 | | 20.514135 | ||
|Completely misses the octave | | Completely misses the octave. | ||
|- | |- | ||
|24 | | 24 | ||
|[[41edo|41]] | | [[41edo|41]] | ||
|41.028271 | | 41.028271 | ||
|24edf is equivalent to 41edo. Patent vals match through the 19-limit. | | 24edf is equivalent to 41edo. Patent vals match through the 19-limit. | ||
|- | |- | ||
|36 | | 36 | ||
| - | | - | ||
|61.542406 | | 61.542406 | ||
| | | | ||
|- | |- | ||
|48 | | 48 | ||
|[[82edo|82]] | | [[82edo|82]] | ||
|82.056542 | | 82.056542 | ||
|48edf is equivalent to 82edo. | | 48edf is equivalent to 82edo. | ||
|- | |- | ||
|60 | | 60 | ||
|[[103edo|103]] | | [[103edo|103]] | ||
|102.57067 | | 102.57067 | ||
|Surprisingly, it | | Surprisingly, it is a match to 103edo despite 60edf falling halfway between 102 and 103. | ||
|- | |- | ||
|120 | | 120 | ||
|[[205edo|205]] | | [[205edo|205]] | ||
|205.14135 | | 205.14135 | ||
| | | | ||
|- | |- | ||
|180 | | 180 | ||
|[[308edo|308]] | | [[308edo|308]] | ||
|307.71203 | | 307.71203 | ||
|Corresponds to 308edo, but with quite a stretch. | | Corresponds to 308edo, but with quite a stretch. | ||
|- | |- | ||
|240 | | 240 | ||
|[[410edo|410]] | | [[410edo|410]] | ||
|410.28271 | | 410.28271 | ||
| | | | ||
|- | |- | ||
|360 | | 360 | ||
| - | | - | ||
|615.42406 | | 615.42406 | ||
|Falls halfway between 615 and 616edo. Also, one step is quite close to the [[schisma]]. | | Falls halfway between 615 and 616edo. Also, one step is quite close to the [[schisma]]. | ||
|- | |- | ||
|720 | | 720 | ||
|[[1231edo|1231]] | | [[1231edo|1231]] | ||
|1230.8481 | | 1230.8481 | ||
| | | | ||
|- | |- | ||
|840 | | 840 | ||
|[[1436edo|1436]] | | [[1436edo|1436]] | ||
|1435.9895 | | 1435.9895 | ||
| | | | ||
|- | |- | ||
|1260 | | 1260 | ||
|2154 | | 2154 | ||
|2153.9842 | | 2153.9842 | ||
| | | | ||
|- | |- | ||
|1680 | | 1680 | ||
|2872 | | 2872 | ||
|2871.9789 | | 2871.9789 | ||
| | | | ||
|- | |- | ||
|2520 | | 2520 | ||
|4308 | | 4308 | ||
|4397.9685 | | 4397.9685 | ||
| | | | ||
|- | |- | ||
|5040 | | 5040 | ||
|8616 | | 8616 | ||
|8615.9369 | | 8615.9369 | ||
| | | | ||
|} | |} | ||
== Generalization == | == Generalization == | ||
{{Todo|inline=1|split page|comment=Move the definitions to [[Highly composite number]] and discuss the properties without regards to equal divisions. }} | |||
=== Extensions === | === 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. | 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- | 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 [[ | This also means that an ''N''-generalized highly composite number has the largest amount of [[regular temperament|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 ( | 2-generalized highly composite numbers are the following ({{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}}. | {{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 === | ||
{{w|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. | |||
Although neither sequence of these numbers is a subset of the other, the first 19 superabundant and highly composite numbers are the same, and there are more common terms further in the sequences. Therefore, many highly composite equal divisions are also superabundant and vice versa, but some equal divisions are only in one of the two categories (or neither at all). | Although neither sequence of these numbers is a subset of the other, the first 19 superabundant and highly composite numbers are the same, and there are more common terms further in the sequences. Therefore, many highly composite equal divisions are also superabundant and vice versa, but some equal divisions are only in one of the two categories (or neither at all). | ||
| Line 158: | Line 157: | ||
Highly composite edos have a record amount of divisors, where divisors correspond to the number of sub-edos, while superabundant edos have a record number of notes in those divisors if they were stretched end to end. | Highly composite edos have a record amount of divisors, where divisors correspond to the number of sub-edos, while superabundant edos have a record number of notes in those divisors if they were stretched end to end. | ||
The factor of being relative to the number's size is important with superabundant edos too. 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 with superabundant edos too. 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, {{nowrap| 91/36 = 2.527 }}, while {{nowrap| 252/130 = 1.938 }}, meaning that 36 carries a more impressive task in replicating sub-edos and symmetrical chords relative to it's size. | ||
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 | 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 are not 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 === | ||
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 | 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]]. 72edo has been used in | 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 {{w|Byzantine music|Byzantine chanting}}, has been theoreticized by {{w|Alois Hába}} and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. | ||
=== Highly factorable numbers === | === Highly factorable numbers === | ||
Highly factorable numbers are similar to highly composite numbers, but the difference is that they have the largest number of factorizations compared to the numbers before them, which means different ways of dividing edo into subsets as opposed to counting subsets themselves. | Highly factorable numbers are similar to highly composite numbers, but the difference is that they have the largest number of factorizations compared to the numbers before them, which means different ways of dividing edo into subsets as opposed to counting subsets themselves. | ||
For example, [[24edo]] has ways of division as 3 | For example, [[24edo]] has ways of division as {{nowrap| 3 × 8 }}, {{nowrap| 4 × 6 }}, {{nowrap| 2 × 12 }}, {{nowrap| 2 × 2 × 6 }}, {{nowrap| 2 × 3 × 4 }}, and {{nowrap| 2 × 2 × 2 × 3 }}, which is a total of 6 ways of factorizing, and is greater than any number before it. | ||
First few are (OEIS | 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 == | ||
| Line 182: | Line 180: | ||
Example: [[1848edo]] is more prominent than [[3456edo]] despite both having 32 divisors, due to 1848 having 2, 3, 7, and 11 as prime factors, while 3456 having only 2 and 3. | Example: [[1848edo]] is more prominent than [[3456edo]] despite both having 32 divisors, due to 1848 having 2, 3, 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. | 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 == | ||
* | * {{OEIS|A002182}} – highly composite numbers | ||
* | * {{OEIS|A004394}} – superabundant numbers | ||
* | * {{OEIS|A067128}} – largely composite numbers | ||
[[Category:Highly composite]] | |||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||