Highly composite equal division: Difference between revisions

Line 154:

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. 1-generalized, superabundant equal divisions, have the largest amount of notes in those scales, 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]]):

{| class="wikitable"

|<code>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</code>

|}

=== Largely composite numbers ===

@@ Line 154: / Line 154: @@
 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. 1-generalized, superabundant equal divisions, have the largest amount of notes in those scales, 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.
+-generalized highly composite numbers are the following ([[oeis:A208767|OEIS: A208767]]):
+{| class="wikitable"
+|<code>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</code>
+|}
 === Largely composite numbers ===