View source for User:Overthink/Neutral scale theory

Neutral scales are generated by a neutral third of around 350 cents, which exists in 17edo, 24edo, and many other tunings. The generator can be taken to represent [[11/9]], and two generators reach [[3/2]], tempering out the comma [[243/242]]. This generator leads to [[MOS]]es of size 4, 7, 10, 17, etc. Here we will mainly be considering the 7-note [[3L 4s]] and 10-note [[7L 3s]] MOSes.

''Note: Similar theory can be used in the 2.3.13 temperament tempering out 512/507.''

== 3L 4s (mosh) ==

=== Intervals ===

The 3L 4s MOS has a step pattern of LssLsLs, where L is a major second and s is a neutral second. We borrow note names from [[5L 2s]] (diatonic) as C, D, E, F, G, A, B. We use sharp (typed as #) and flat (typed as b) to denote an increment and decrement by {{nowrap|L-s}}, or a quarter tone. A table of intervals of this scale in 24edo is below. Note that other tunings can be used, but overall structures are the same.

{| class="wikitable mw-collapsible mw-collapsed"
|+ 24edo&nbsp;mosh&nbsp;scale
|-
! Note !! Cents !! Ratio
|-
| C || 0 || 1/1
|-
| D || 200 || 9/8
|-
| E || 350 || 11/9
|-
| F || 500 || 4/3
|-
| G || 700 || 3/2
|-
| A || 850 || 18/11
|-
| B || 1050 || 11/6
|-
| C || 1200 || 2/1
|}

Just like diatonic, this scale has major and minor intervals, which are found on different notes of the scale. A table of such intervals is below.

{| class="wikitable mw-collapsible mw-collapsed"
|+ Intervals&nbsp;of&nbsp;mosh
|-
! Interval !! Cents !! Ratio !! note on C
|-
| P1 || 0 || 1/1 || C
|-
| A1 || 50 || 33/32 || C#
|-
| m2 || 150 || 12/11 || Db
|-
| M2 || 200 || 9/8 || D
|-
| d3 || 300 || 32/27 || Eb
|-
| P3 || 350 || 11/9 || E
|-
| m4 || 500 || 4/3 || F
|-
| M4 || 550 || 11/8 || F#
|-
| m5 || 650 || 16/11 || Gb 
|-
| M5 || 700 || 3/2 || G
|-
| P6 || 850 || 18/11 || A
|-
| A6 || 900 || 27/16 || A#
|-
| m7 || 1000 || 16/9 || Bb
|-
| M7 || 1050 || 11/6 || B
|-
| d8 || 1150 || 64/33 || Cb
|-
| P8 || 1200 || 2/1 || C
|}

=== Chords ===

In diatonic, chords are built by Root-3rd-5th. We can try to use the same logic here. This gives us neutral (P1-P3-M5) chords on C, E, F, G, and A, a P1-d3-m5 chord on D, and a P1-P3-m5 chord on B. This construction, however, is flawed, as none of these chords can be written with particularly simple ratios, and therefore these chords aren't very consonant.

This scale is in the [[2.3.11 subgroup]], with the fundamental otonal consonance being [[8:9:11:12]] (or one of its different voicings), which in our scale is P1-M2-M4-M5. A different construction, Root-2nd-4th-5th, is now needed. The utonal inverse of 8:9:11:12 is 1/(8:9:11:12) = [[66:72:88:99]], or P1-m2-m4-M5. The otonal 8:9:11:12 chord occurs on F and A, and the utonal 1/(8:9:11:12) chord occurs on E and G. Neither chord occurs on C, B, or D, with a P1-M2-m4-M5 chord on C, and a P1-m2-m4-m5 chord on B and D. Of course, this is only one possible construction. Note that chords can also be constructed as a chain of three fourths (4/3 or 11/8) or fifths (3/2 or 16/11), with the chain being 4th-Root-5th-2nd for fifths and the reverse for fourths.

We need names for these chords, and once again we can borrow some from diatonic. The major and minor chords are otonal and utonal respectively. The suspended chord is [[3-limit]], like the suspended chord in diatonic, with a [[32/27]] dissonance here compared to a [[9/8]] dissonance (which we consider a consonance here) in diatonic. The '''Ambi''' chord is named for being ambitonal, and is notably an [[anomalous saturated suspension]].  

{| class="wikitable"
|+ Mosh chord names
! colspan=2 |Chord
! rowspan=2 |Name
|-
! Root up
! Chain of 5ths
|-
| P1-M2-M4-M5
| M4-P1-M5-M2
| '''Major / M''' (or no symbol)
|-
| P1-m2-m4-M5
| m4-P1-M5-m2
| '''Minor / m'''
|-
| P1-M2-m4-M5
| m4-P1-M5-M2
| '''Suspended / sus'''
|-
| P1-m2-m4-m5
| m4-P1-m5-m2
| '''Ambi / am'''
|}

On the white keys (C, D, E, ...), we have Csus, Dam, Em, F, Gm, A, and Bam. While this mode has a 3/2 interval above the I, IV, and V, there is a lack of tonality with a sus chord on the root.

=== Modes ===

This scale has 7 modes. They are listed below, with the intervals of the mode and chords on each note of the scale.

{| class="wikitable"
|+ Modes&nbsp;of&nbsp;mosh (brightest to darkest)
|-
! Name<ref>[https://en.xen.wiki/w/3L_4s#Proposed_names 3L 4s] on Xenharmonic Wiki</ref>
! Intervals
! Chords (from I to VII)
|-
| Dalmatian
| P1, M2, P3, M4, M5, A6, M7
| I, ii, III, IVam, Vsus, VIam, vii
|-
| Galatian
| P1, M2, P3, M4, M5, P6, M7
| I, IIam, IIIsus, IVam, v, VI, vii
|-
| Cilician
| P1, M2, P3, m4, M5, P6, M7
| Isus, IIam, iii, IV, v, VI, VIIam
|-
| Bithynian
| P1, m2, P3, m4, M5, P6, M7
| i, II, iii, IV, Vam, VIsus, VIIam
|-
| Pisidian
| P1, m2, P3, m4, M5, P6, m7
| i, II, IIIam, IVsus, Vam, vi, VII
|-
| Illyrian
| P1, m2, P3, m4, m5, P6, m7
| Iam, iisus, IIIam, iv, V, vi, VII
|-
| Lycian
| P1, m2, d3, m4, m5, P6, m7
| Iam, ii, III, iv, V, VIam, VIIsus
|}

The Dalmation and Galatian modes on I, Bithynian and Pisidian have minor on I, Illyrian and Lycian have ambi on I, and Cicilian has sus on I.

== References ==
<references/>