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Proposed terms: mega-edo explicitly refers to divisions in the millions; deka-, hecto-, and kilo-edo for divisions in the tens, hundreds, and thousands
 
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=== Nk note name notation ===
=== N(k) note name notation (work-in-progress) ===
Rather than using alphabetical names, notes of the form Nk are used. These are used to indicate position on a staff, where N0 is middle C, unless specified otherwise. These names serve as an alternative to using different notations for different scales, but may be interpreted as blanks for one to fill in with different, more specific notation. If k is unbounded, then this notation denotes position on a staff. However, k may be bounded within the range [0, n), where n is the note count, to indicate pitch classes.
Rather than using alphabetical names, notes of the form N(k) are used. These are used to indicate position on a staff, where N(0) is the root. These names serve as an alternative to using different notations for different scales, but may be interpreted as blanks for one to fill in with different, more specific notation. If k is unbounded, then this notation denotes position on a staff. However, k may be bounded within the range [0, n), where n is the note count, to indicate pitch classes.


For a given mos xL ys, note names are based on a mode u|p; the choice of mode is up to the user. Starting at N0, or 0-mosstep, successive pitch classes are named N1, N2, and so on. If two pitches, reached by going up or down some quantity of mossteps, have the same remainder when divided by xL+ys (which is the same as octave-reducing), then they are in the same pitch class.
For a given mos xL ys, note names are based on a mode u|p; the choice of mode is up to the user. Starting at the root of N(0), successive pitch classes are named N(1), N(2), and so on. If note names are given and assuming N(0) is the root, then N(k) can be thought of as a function that returns an unaltered note name corresponding to the k-mosdegree of a mos xL ys in the mode u|p. In standard notation, N(0) is C, N(1), is D, and so on. Since this is cyclical, N(7) and N(0) are both the same value of C.
 
If two pitches, reached by going up or down some quantity of mossteps, have the same remainder when divided by xL+ys (which is the same as octave-reducing), then they are in the same pitch class.
{| class="wikitable"
{| class="wikitable"
|+Example for 5L 2s (LLsLLLs, mode 5|Example with standard notation (5L 2s, mode 5|1)
|+ Example for 5L 2s (LLsLLLs, mode 5 |Example with standard notation (5L 2s, mode 5|1)
!Mossteps from root
!Mossteps from root
!Substring
!Substring
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|0
|0
|C
|C
|N0
|N(0)
|-
|-
|1
|1
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|L
|L
|D
|D
|N1
|N(1)
|-
|-
|2
|2
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|2L
|2L
|E
|E
|N2
|N(2)
|-
|-
|3
|3
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|2L+s
|2L+s
|F
|F
|N3
|N(3)
|-
|-
|4
|4
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|3L+s
|3L+s
|G
|G
|N4
|N(4)
|-
|-
|5
|5
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|4L+s
|4L+s
|A
|A
|N5
|N(5)
|-
|-
|6
|6
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|5L+s
|5L+s
|B
|B
|N6
|N(6)
|-
|-
|7
|7
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|5L+2s
|5L+2s
|C
|C
|N7 (same as N0)
|N(7) (same as N(0))
|}
|}
Chromas are denoted using the letter c, and are expressed as a multiple of c being added (or subtracted) from a note Nk. Half-accidentals are denoted as fractions (such as c/2) or decimals (such as 0.5c). Dieses, if present, are expressed similarly using the letter d. If this notation denotes position on a staff, then chromas and dieses don't change position on a staff, but modify the pitch at that position. If this notation is treated as placeholders for more specific notation, then adding or subtracting c represents the use of sharp or flat (or equivalent) accidentals.
Chromas are denoted using the letter c, and are expressed as a multiple of c being added (or subtracted) from a note N(k). Half-accidentals are denoted as fractions (such as c/2) or decimals (such as 0.5c). Dieses, if present, are expressed similarly using the letter d. If this notation denotes position on a staff, then chromas and dieses don't change position on a staff, but modify the pitch at that position. If this notation is treated as placeholders for more specific notation, then adding or subtracting c represents the use of sharp or flat (or equivalent) accidentals.


Since chromas and dieses can be expressed in terms of L and s – where a chroma is L - s and a diesis is the absolute value of L - 2s – modifying a note by a chroma or diesis can equivalently expressed as going up (or down) some interval iL+js. If, for a given step ratio L:s, two pitch classes Np and Nq are modified by different amounts of chromas uc and vc to produce pitch classes Np+uc and Nq+vc, if dividing both by xL+ys produces the same remainder, then the two pitches are enharmonic equivalents.
Since chromas and dieses can be expressed in terms of L and s – where a chroma is L - s and a diesis is the absolute value of L - 2s – modifying a note by a chroma or diesis can equivalently expressed as going up (or down) some interval iL+js. If, for a given step ratio L:s, two pitch classes Np and Nq are modified by different amounts of chromas uc and vc to produce pitch classes N(p)+uc and N(q)+vc, if dividing both by xL+ys produces the same remainder, then the two pitches are enharmonic equivalents.


As an example, the table below denotes diatonic (5L 2s) pitch classes as sums of L's and s's, and shows how different step ratios produce different enharmonic equivalences; namely, in 12edo, C# and Db are equivalent, but in 19edo, C# and Db are not equivalent but B# and Cb are equivalent.
As an example, the table below denotes diatonic (5L 2s) pitch classes as sums of L's and s's, and shows how different step ratios produce different enharmonic equivalences; namely, in 12edo, C# and Db are equivalent, but in 19edo, C# and Db are not equivalent but B# and Cb are equivalent.
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|+Examples with standard diatonic notation
|+Examples with standard diatonic notation
!Note name
!Note name
!Nk note name with chroma
!N(k) note name with chroma
!Mosstep sum
!Mosstep sum
!Like terms combined
!Like terms combined
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|-
|-
|C
|C
|N0
|N(0)
|0
|0
|0
|0
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|-
|-
|C#
|C#
|N0+c
|N(0)+c
|L-s
|L-s
|L-s
|L-s
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|-
|-
|Db
|Db
|N1-c
|N(1)-c
|L-(L-s)
|L-(L-s)
|s
|s
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|-
|-
|D
|D
|N1
|N(1)
|L
|L
|L
|L
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|-
|-
|B
|B
|N6
|N(6)
|5L+s
|5L+s
|5L+s
|5L+s
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|-
|-
|B#
|B#
|N6+c
|N(6)+c
|5L+s+(L-s)
|5L+s+(L-s)
|6L
|6L
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|-
|-
|Cb
|Cb
|N7-c
|N(7)-c
|5L+2s-(L-s)
|5L+2s-(L-s)
|4L+3s
|4L+3s
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|-
|-
|C (one octave up)
|C (one octave up)
|N7 (same as N0, as a pitch class)
|N(7) (same as N(0), as a pitch class)
|5L+2s (reduced to 0 due to modular arithmetic)
|5L+2s (reduced to 0 due to modular arithmetic)
|5L+2s (reduced to 0)
|5L+2s (reduced to 0)
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|19 (reduced to 0)
|19 (reduced to 0)
|}
|}
N(k) notation can also be used to build a genchain that is agnostic of the size (in cents) of the generator and equave. For example, the genchain for standard notation can be written as N(0), N(4), N(8), N(12), N(16), N(20), N(24)+c, N(28)+c for the ascending chain. The descending chain can be written as N(0), N(3), N(6)-c, N(9)-c, N(12)-c, N(15)-c, N(18)-c, N(21)-c, or as N(0), N(-4), N(-8)-c, N(-12)-c, N(-16)-c, N(-20)-c, N(-24)-c, N(-28)-c. The value k isn't entered into the function, but rather its remainder when divided by the number of steps in the mos (modulo 7, for the case of standard notation), so N(8) is equivalent to N(1) for example.
Since the gamut on C is based on the ionian mode, or produced using 5 generators going up and 1 going down, the first note after N(20) has a chroma added, producing N(24)+c. Simply put, the first 5 notes after the root have zero chromas added, the next 6 after that have 1 chroma added, the next 6 have 2 chromas added, and so on. For the descending chain, accidentals are subtracted after the first note, and every 6 notes thereafter has one more chroma subtracted.
Ups and downs may also be represented, using the variable u. Up-C-sharp, or ^C#, is written as N(0)+c+u, where u is an edostep.


=== Chord notation using mossteps ===
=== Chord notation using mossteps ===
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|}
|}
If the quantities of mossteps s1 and s2 are different, then the symmetric chrods are quasisymmetric instead. The interval sizes don't need to be major or minor, either; they can also be augmented, perfect, or diminished if it's a generator.
If the quantities of mossteps s1 and s2 are different, then the symmetric chrods are quasisymmetric instead. The interval sizes don't need to be major or minor, either; they can also be augmented, perfect, or diminished if it's a generator.
== Proposal (wip): strict and weak definitions for a chromatic pair ==
=== Strict definition ===
A '''chromatic pair''' is a pair of mosses zL ws and xL ys within some temperament, such that x = z + w and y = z, where zL ws is a '''haplotonic''' '''scale''' and xL ys is an '''albitonic''' '''scale'''. The large steps of the albitonic scale are such that haplotonic scale can be found within the large steps, forming a '''chromatic scale''' of either xL (x+y)s or (x+y)L xs, or more generally, xA (x+y)B.
=== Weak definition ===
A chromatic pair, under the weak definition, is a pair of mosses zL ws and xL ys, such that x = nz + w and y = z. The strict definition is such that n = 1. However, rather than the mosses zL ys and xL ys that form the chromatic scale of xA (x+y)B, it's the mosses zL ((n-1)z+w)s and xL ys that form the chromatic scale.
=== Things to consider ===
* A haplotonic scale's note count should be 4 or 5 notes, corresponding to the note counts of the grandchild mosses of 1L 1s: 2L 3s, 3L 2s, 1L 3s, and 3L 1s.
* An albitonic scale's note count should be around 7 notes.
== Warped scales ==
A somewhat generalized notion of warping, described by the addition, removal, or substitution of a single step. The most common scales of 12edo are used as examples: 5L 2s, the whole-tone scale (effectively 6edo), the chromatic scale (effectively 12edo), and the diminished scale (4L 4s, hardness of 2).
The simplest ways to warp a scale are through the addition of a step and the removal of a step. Substitution of a step, where one step is changed for a step of a different size, can be thought of removing a step of one size and adding a step of a different size.
{| class="wikitable"
|+Warped 5L 2s
! rowspan="2" |Small step changes
! colspan="3" |Large step changes
|-
!-1L
!+0L
!+1L
|-
!-1s
|
|5L 1s
|6L 1s
|-
!+0s
|5L 1s
|'''5L 2s'''
|6L 2s
|-
!+1s
|4L 3s
|5L 3s
|
|}
{| class="wikitable"
|+Warped 6edo
(equal-tempered whole-tone scale)
! rowspan="2" |Small step changes
! colspan="3" |Large step changes
|-
!-1L
!+0L
!+1L
|-
!-1s
|
|
|1L 5s
|-
!+0s
|
|'''6edo'''
|1L 6s
|-
!+1s
|5L 1s
|6L 1s
|
|}
{| class="wikitable"
|+Warped 12edo
(equal-tempered chromatic scale)
! rowspan="2" |Small step changes
! colspan="3" |Large step changes
|-
!-1L
!+0L
!+1L
|-
!-1s
|
|
|1L 11s
|-
!+0s
|
|'''12edo'''
|1L 12s
|-
!+1s
|1L 11s
|12L 1s
|
|}
{| class="wikitable"
|+Warped 4L 4s
! rowspan="2" |Small step changes
! colspan="3" |Large step changes
|-
!-1L
!+0L
!+1L
|-
!-1s
|
|4L 3s
|5L 3s
|-
!+0s
|3L 4s
|'''4L 4s'''
|5L 4s
|-
!+1s
|3L 5s
|4L 5s
|
|}
== EDO/ED classifications ==
* Deka-edo (deka-division): an equal division of the octave (or equave) where the number of divisions is in the tens.
* Hecto-edo (hecto-division): an equal division of the octave (or equave) where the number of divisions is in the hundreds.
* Kilo-edo (kilo-division): an equal division of the octave (or equave) where the number of divisions is in the thousands.
* Mega-edo (mega-division): an equal division of the octave (or equave) where the number of divisions is in the millions.
** This term already exists to refer to a large edo, but how large is subjective. Since the terms deka-, hecto-, and kilo-edo (and deka-, hecto-, and kilo-division) explicitly refer to specific powers of 10 (specifically, tens, hundreds, and thousands), so should mega-edo and mega-division to refer to divisions in the millions.
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