View source for User:Zhenlige/Chromatic notation

This page introduces a proposed notation system centering the [[5L 7s]] chromatic scale. This system is suitable for temperaments with consonant intervals far away in the [[chain of fifths]], such as [[schismatic]]. Adding more naturals is usually unnecessary, since the [[12-comma]] is much smaller than the [[256/243|limma]] in JI. For the meantone chromatic scale [[7L 5s]], there are two approches to use this system, in anology to using the standard notation system for the [[2L 5s|antidiatonic]] scale.

== Intervals ==
The degrees are named as 0-indexed numbers of “sem”, which stands for “'''sem'''itone”, referring to the chromatic step. [[12edo]] diatonic interval names can also be used for representing degrees, with A4 and d5 merged into “tritone”. The [[5L 7s]] major, minor, augmented and diminished are renamed to wide (W), narrow (n), super (S) and sub (s) respectively, to avoid ambiguity. Note that diatonic major is always wide and diatonic minor is always narrow, which also applies to fourths and fifths as if they were named the same as other intervals. Mid or neutral degrees should be abbreviated as “m” or “~” to avoid confusion with “n” which stands for narrow.
{| class="wikitable"
|+ Chromatic interval degrees
! colspan="3" | Interval
! rowspan="2" | Abbrev.
! colspan="2" | Steps
! rowspan="2" | 29edo<br>tuning<br>(steps)
! rowspan="2" | Equivalent<br>diatonic<br>degree
! colspan="2" | Approximate ratios
|-
! Degree !! Degree name<br>from [[12edo]] diatonic !! Quality
! L !! s
! [[schismatic]] !! additional<br>in [[garibaldi]]
|-
| Unison<br>0-tone<br>0-sem
| Unison
|Perfect|| P0s / P1 || 0 || 0 || 0 || '''[[1/1|P1]]''' || [[1/1]] ||
|-
| rowspan="2" | Semitone<br>Sem / 1-sem
| rowspan="2" | Minor second
|Narrow|| n1s / nm2 || 0 || 1 || 2 || '''m2''' || [[135/128]], [[256/243]] || [[21/20]]
|-
|Wide|| W1s / Wm2 || 1 || 0 || 3 || A1 || [[16/15]] || [[15/14]]
|-
| rowspan="2" | Tone / 1-tone<br>2-sem
| rowspan="2" | Major second
|Narrow|| n2s / nM2 || 0 || 2 || 4 || d3 || [[10/9]] ||
|-
|Wide|| W2s / WM2 || 1 || 1 || 5 || '''M2''' || [[9/8]] ||
|-
| rowspan="2" | 3-sem
| rowspan="2" | Minor third
|Narrow|| n3s / nm3 || 1 || 2 || 7 || '''m3''' || [[32/27]] ||
|-
|Wide|| W3s / Wm3 || 2 || 1 || 8 || A2 || [[6/5]] ||
|-
| rowspan="2" | Ditone / 2-tone<br>4-sem
| rowspan="2" | Major third
|Narrow|| n4s / nM3 || 1 || 3 || 9 || d4 || [[5/4]] || [[56/45]]
|-
|Wide|| W4s / WM3 || 2 || 2 || 10 || '''M3''' || [[81/64]] || [[80/63]], [[63/50]]
|-
| rowspan="2" | 5-sem
| rowspan="2" | Fourth
|Perfect|| P5s / P4 || 2 || 3 || 12 || '''P4''' || [[4/3]] ||
|-
|Super|| S5s / S4 || 3 || 2 || 13 || A3 || [[27/20]] ||
|-
| rowspan="2" | [[Tritone]] / 3-tone<br>6-sem
| rowspan="2" | Tritone
|Narrow|| n6s / nT || 2 || 4 || 14 || '''d5''' || [[45/32]] || [[7/5]]
|-
|Wide|| W6s / WT || 3 || 3 || 15 || '''A4''' || [[64/45]] || [[10/7]]
|-
| rowspan="2" | 7-sem
| rowspan="2" | Fifth
|Sub|| s7s / s5 || 2 || 5 || 16 || d6 || [[40/27]] ||
|-
|Perfect|| P7s / P5 || 3 || 4 || 17 || '''[[Perfect fifth (diatonic interval category)|P5]]''' || [[3/2]] ||
|-
| rowspan="2" | Tetratone / 4-tone<br>8-sem
| rowspan="2" | Minor sixth
|Narrow|| n8s / nm6 || 3 || 5 || 19 || '''m6''' || [[128/81]] || [[63/40]], [[100/63]]
|-
|Wide|| W8s / Wm6 || 4 || 4 || 20 || A5 || [[8/5]] || [[45/28]]
|-
| rowspan="2" | 9-sem
| rowspan="2" | Major sixth
|Narrow|| n9s / nM6 || 3 || 6 || 21 || d7 || [[5/3]] ||
|-
|Wide|| W9s / WM6 || 4 || 5 || 22 || '''M6''' || [[27/16]] ||
|-
| rowspan="2" | Pentatone / 5-tone<br>10-sem
| rowspan="2" | Minor seventh
|Narrow|| n10s / nm7 || 4 || 6 || 24 || '''m7''' || [[16/9]] ||
|-
|Wide|| W10s / Wm7 || 5 || 5 || 25 || A6 || [[9/5]] ||
|-
| rowspan="2" | 11-sem
| rowspan="2" | Major seventh
|Narrow|| n11s / nM7 || 4 || 7 || 26 || d8 || [[15/8]] || [[28/15]]
|-
|Wide|| W11s / WM7 || 5 || 6 || 27 || '''M7''' || [[243/128]], [[256/135]] || [[40/21]]
|-
| Hexatone / 6-tone<br>12-sem
| Octave
|Perfect|| P12s / P8 || 5 || 7 || 29 || '''[[2/1|P8]]''' || [[2/1]] ||
|}

The 12edo diatonic based interval names along the chain of fifths is P1-P5-WM2-WM6-WM3-WM7-WT-Wm2-Wm6-Wm3-Wm7-S4-S1-S5-SM2-… with the quality goes as perfect - wide major - wide minor - super - super major - super minor - double super … .

== Note names ==
Latin letters ABCDEFG and Greek letters αβγδε are used. Latin letters represent the same notes as diatonic. Each Greek letter represents the note a [[2187/2048|diatonic chroma]] below the note marked with the corresponding Latin letter. Ups and downs are used for altering [[12-comma]]s.
{| class="wikitable"
|-
! Chromatic !! Diatonic
|-
| '''D''' || '''D'''
|-
| '''ε''' || E♭
|-
| ^ε || D♯
|-
| '''E''' || '''E'''
|-
| '''F''' || '''F'''
|-
| '''γ''' || G♭
|-
| ^γ || F♯
|-
| '''G''' || '''G'''
|-
| '''α''' || A♭
|-
| ^α || G♯
|-
| '''A''' || '''A'''
|-
| '''β''' || B♭
|-
| ^β || A♯
|-
| '''B''' || '''B'''
|-
| '''C''' || '''C'''
|-
| '''δ''' || D♭
|-
| ^δ || C♯
|-
| '''D''' || '''D'''
|}

== Staves ==
A 9-lined staff is used. To improve readability, the lines alter between thin and thick. The first and last lines are thick. Only C-clef is used.  C is always on a thick line. For the standard clef, the notes on the lines are '''E'''γ'''α'''β'''C'''D'''E'''γ'''α'''.

== Applications ==
The chromatic notation is useful for notating temperaments that require many fifths to get some primes, making diatonic notation very unintuitive, or that have interdiatonic degrees such as the half octave.

{| class="wikitable"
|+ Interval mappings in some temperaments<br>(12edo diatonic based names)
! Interval !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || nM3 || nM3 || Sm3 || ~M3
|-
| [[7/4]] || 3SM6 || sm7 || SM6 || 1/2-sm7
|-
| [[11/8]] || 7s5 || 2S4 || S4 || 1/2-sT
|-
| [[13/8]] || || Sm6 || Wm6 || 1/2-sM6
|-
| [[17/16]] || || || 2S1 || ~m2
|}

{| class="wikitable"
|+ Interval mappings in some temperaments<br>(n-sem names)
! Interval !! [[Ponta]] !! [[Cassandra]] !! [[Leapday]] !! [[Diaschismic]]
|-
| [[5/4]] || n4s || n4s || S3s || ~4s
|-
| [[7/4]] || 3S9s || s10s || S9s || 1/2-s10s
|-
| [[11/8]] || 7s7s || 2S5s || S5s || 1/2-s6s
|-
| [[13/8]] || || S8s || W8s || 1/2-s9s
|-
| [[17/16]] || || || 2S0s || ~1s
|}