Kite's Genchain mode numbering: Difference between revisions

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==MOS Scales== 

[[toc]]
**Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] to create a generator-chain, and adds a way to number each mode uniquely.

[[xenharmonic/MOSScales|MOS scales]] are formed from a segment of the [[xenharmonic/periods and generators|generator-chain]], or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions.

For example, here are all the modes of Meantone[7], using ~3/2 as the generator:
|| old scale name || new scale name || sL pattern || example on white keys || genchain ||
|| Lydian || 1st Meantone [7] || LLLs LLs || F G A B C D E F || __**F**__ C G D A E B ||
|| Ionian (major) || 2nd Meantone [7] || LLsL LLs || C D E F G A B C || F __**C**__ G D A E B ||
|| Mixolydian || 3rd Meantone [7] || LLsL LsL || G A B C D E F G || F C __**G**__ D A E B ||
|| Dorian || 4th Meantone [7] || LsLL LsL || D E F G A B C D || F C G __**D**__ A E B ||
|| Aeolian (minor) || 5th Meantone [7] || LsLL sLL || A B C D E F G A || F C G D __**A**__ E B ||
|| Phrygian || 6th Meantone [7] || sLLL sLL || E F G A B C D E || F C G D A __**E**__ B ||
|| Locrian || 7th Meantone [7] || sLLs LLL || B C D E F G A B || F C G D A E __**B**__ ||
4th Meantone [7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".

The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian).
|| old scale name || new scale name || sL pattern || example in C || ------------------- genchain --------------- ||
|| Lydian || 1st Meantone [7] || LLLs LLs || C D E F# G A B C ||> __**C**__ G D A E B F# ||
|| Ionian (major) || 2nd Meantone [7] || LLsL LLs || C D E F G A B C ||> F __**C**__ G D A E B ---- ||
|| Mixolydian || 3rd Meantone [7] || LLsL LsL || C D E F G A Bb C ||> Bb F __**C**__ G D A E ------- ||
|| Dorian || 4th Meantone [7] || LsLL LsL || C D Eb F G A Bb C || ------------- Eb Bb F __**C**__ G D A ||
|| Aeolian (minor) || 5th Meantone [7] || LsLL sLL || C D Eb F G Ab Bb C || --------- Ab Eb Bb F __**C**__ G D ||
|| Phrygian || 6th Meantone [7] || sLLL sLL || C Db Eb F G Ab Bb C || ---- Db Ab Eb Bb F __**C**__ G ||
|| Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ ||

The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are several disadvantages of only using chroma-positive generators, see [[Naming Rank-2 Scales using Mode Numbers#Why%20not%20just%20use%20UDP%20notation?|Why not just use UDP notation?]] below.

Pentatonic meantone scales:
|| old scale name || new scale name || sL pattern || example in C || --------- genchain ------- ||
|| major pentatonic || 1st Meantone [5] || ssL sL || C D E G A C ||> __**C**__ G D A E ||
||=   || 2nd Meantone [5] || sLs sL || C D F G A C ||> F __**C**__ G D A -- ||
||=   || 3rd Meantone [5] || sLs Ls || C D F G Bb C || -------- Bb F __**C**__ G D ||
|| minor pentatonic || 4th Meantone [5] || Lss Ls || C Eb F G Bb C || ---- Eb Bb F __**C**__ G ||
||=   || 5th Meantone [5] || LsL ss || C Eb F Ab Bb C || Ab Eb Bb F __**C**__ ||

Chromatic meantone scales.
|| scale name || sL pattern
(assumes 3/2 < 700¢) || example in C || genchain ||
|| 1st Meantone [12] || sLsL sLL sLsLL || C C# D D# E E# F# G G# A A# B C || __**C**__ G D A E B F# C# G# D# A# E# ||
|| 2nd Meantone [12] || sLsL LsL sLsLL || C C# D D# E F F# G G# A A# B C || F __**C**__ G D A E B F# C# G# D# A# ||
|| 3rd Meantone [12] || sLsL LsL sLLsL || C C# D D# E F F# G G# A Bb B C || Bb F __**C**__ G D A E B F# C# G# D# ||
|| 4th Meantone [12] || sLLs LsL sLLsL || C C# D Eb E F F# G G# A Bb B C || Eb Bb F __**C**__ G D A E B F# C# G# ||
|| 5th Meantone [12] || sLLs LsL LsLsL || C C# D Eb E F F# G Ab A Bb B C || Ab Eb Bb F __**C**__ G D A E B F# C# ||
|| 6th Meantone [12] || LsLs LsL LsLsL || C Db D Eb E F F# G Ab A Bb B C || Db Ab Eb Bb F __**C**__ G D A E B F# ||
|| 7th Meantone [12] || LsLs LLs LsLsL || C Db D Eb E F Gb G Ab A Bb B C || Gb Db Ab Eb Bb F __**C**__ G D A E B ||
||= etc. ||   ||   ||   ||
If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.


[[Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)
|| scale name || sL pattern || example in C || genchain ||
|| 1st Sensi [8] || ssL ssL sL || C Db D# E# F# G A Bb C || __**C**__ E# A Db F# Bb D# G ||
|| 2nd Sensi [8] || ssL sL ssL || C Db D# E# F# G# A Bb C || G# __**C**__ E# A Db F# Bb D# ||
|| 3rd Sensi [8] || sL ssL ssL || C Db Eb E# F# G# A Bb C || Eb G# __**C**__ E# A Db F# Bb ||
|| 4th Sensi [8] || sL ssL sL s || C Db Eb E# F# G# A B C || B Eb G# __**C**__ E# A Db F# ||
|| 5th Sensi [8] || sL sL ssL s || C Db Eb E# Gb G# A B C || Gb B Eb G# __**C**__ E# A Db ||
|| 6th Sensi [8] || Lss Lss Ls || C D Eb E# Gb G# A B C || D Gb B Eb G# __**C**__ E# A ||
|| 7th Sensi [8] || Lss Ls Lss || C D Eb E# Gb G# A# B C || A# D Gb B Eb G# __**C**__ E# ||
|| 8th Sensi [8] || Ls Lss Lss || C D Eb F Gb G# A# B C || F A# D Gb B Eb G# __**C**__ ||
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
The modes would follow a more regular pattern if using octotonic fourth-based notation:
C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A - A#/Bb - B - B# - Cb -C
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.


Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[xenharmonic/ups and downs notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.
|| scale name || sL pattern || example in C || genchain ||
|| 1st Porcupine [7] || ssss ssL || C Dv Eb^ F Gv Ab^ Bb C || __**C**__ Dv Eb^ F Gv Ab^ Bb ||
|| 2nd Porcupine [7] || ssss sLs || C Dv Eb^ F Gv Ab^ Bb^ C || Bb^ __**C**__ Dv Eb^ F Gv Ab^ ||
|| 3rd Porcupine [7] || ssss Lss || C Dv Eb^ F Gv Av Bb^ C || Av Bb^ __**C**__ Dv Eb^ F Gv ||
|| 4th Porcupine [7] || sssL sss || C Dv Eb^ F G Av Bb^ C || G Av Bb^ __**C**__ Dv Eb^ F ||
|| 5th Porcupine [7] || ssLs sss || C Dv Eb^ F^ G Av Bb^ C ||= F^ G Av Bb^ __**C**__ Dv Eb^ ||
|| 6th Porcupine [7] || sLss sss || C Dv Ev F^ G Av Bb^ C || Ev F^ G Av Bb^ __**C**__ Dv ||
|| 7th Porcupine [7] || Lsss sss || C D Ev F^ G Av Bb^ C || D Ev F^ G Av Bb^ __**C**__ ||
Again, the modes would follow a more regular pattern if using the appropriate notation, in this case 2nd-based:
C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.


==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**== 

[[MODMOS Scales|MODMOS]] scales are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The "#" symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G.

MODMOS names are ambiguous. This scale could also be written as 2nd Meantone [7] b3 (major scale with a minor 3rd), or as 4th Meantone [7] #7 (dorian with a major 7th).

|| old scale name || example in A || genchain || new scale name || sML pattern ||
|| Harmonic minor || A B C D E F G# A || F C * D __**A**__ E B * * G# || 5th Meantone [7] #7 || MsMM sLs ||
|| Ascending melodic minor || A B C D E F# G# A || C * D __**A**__ E B F# * G# || 5th Meantone [7] #6 #7 || LsLL LLs ||
||= " ||= " ||= " || 2nd Meantone [7] b3 || " ||
||= " ||= " ||= " || 4th Meantone [7] #7 || " ||
|| Double harmonic minor || A B C D# E F G# A || F C * * __**A**__ E B * * G# D# || 5th Meantone [7] #4 #7 || MsLs sLs ||
||= " ||= " || " || 1st Meantone [7] b3 b6 || " ||
|| Double harmonic major || A Bb C# D E F G# A || Bb F * * D __**A**__ E * * C# G# || 2nd Meantone [7] b2 b6 || sLsM sLs ||
||= " ||= " || " || 6th Meantone [7] #3 #7 || " ||
|| <span class="mw-redirect">Hungarian gypsy </span>minor || A B C D# E F G A || F C G * __**A**__ E B * * * D# || 5th Meantone [7] #4 || MsLs sMM ||
|| Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || 6th Meantone [7] #3 || sLsM sMM ||
As can be seen from the genchains or from the sML patterns, the harmonic minor and the phrygian dominant are modes of each other, as are the double harmonic minor and the double harmonic major. Unfortunately the scale names do not indicate this.

The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from A dorian to A melodic minor. In this context, melodic minor might better be described as an altered dorian scale.

Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:
1st Meantone [7] #2: C D# E F# G A B C
2nd Meantone [7] #:5 C D E F G# A B C
7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern, 7th mode not 3rd mode)
4th Meantone [7] #4: C D Eb F# G A Bb C
5th Meantone [7] #7: C D Eb F G Ab B C (harmonic minor)
6th Meantone [7] #3: C Db E F G Ab Bb C (phrygian dominant)
7th Meantone [7] #6: C Db Eb F Gb A Bb C
The 3rd scale breaks the pattern to avoid an altered tonic ("3rd Meantone [7] #1").

Ascending melodic minor modes:
1st Meantone [7] #6 #7: C D E F# G# A B C
2nd Meantone [7] #6 #7: C Db Eb Fb Gb Ab Bb C
3rd Meantone [7] #6 #7: C D E F# G A Bb C
4th Meantone [7] #6 #7: C D Eb F G A B C
5th Meantone [7] #6 #7: C D E F G Ab Bb C
6th Meantone [7] #6 #7: C Db Eb F G A Bb C
7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C


==[[#Fractional-octave periods]]**__Fractional-octave periods__**== 

Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3

When the period is an octave, the genweb octave-reduces to a single horizontal genchain:
F --- C --- G --- D --- A --- E --- B

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, shrutal [10] might look like this:
F^3 --- C^4 --- G^4 --- D^5 --- A^5
C3 ---- G3 ----- D4 ---- A4 ---- E5
F^2 --- C^3 --- G^3 --- D^4 --- A^4
C2 ---- G2 ----- D3 ---- A3 ---- E3
F^1 --- C^2 --- G^2 --- D^3 --- A^3
C1 ---- G1 ----- D2 ---- A2 ---- E2

which octave-reduces to two genchains:
F^ --- C^ --- G^ --- D^ --- A^
C ---- G ----- D ---- A ---- E

Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of "up" here is "a half-octave minus a fourth", with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.

It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.
Gv --- Dv --- Av --- Ev --- Bv
C ----- G ----- D ---- A ---- E

It would also be valid to exchange the two rows:
C ----- G ----- D ---- A ---- E
Gv --- Dv --- Av --- Ev --- Bv

Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.

In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.

If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:
F^ -- G --- G^ -- A --- A^
C --- C^ -- D --- D^ -- E

Another alternative is to use [[Kite's color notation|color notation]]. The shrutal comma is 2048/2025 = sgg2, and the temperament name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:

yF# --- yC# --- yG# --- yD# --- yA#
wC ---- wG ---- wD ---- wA ---- wE

As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:
wC ---- wG ---- wD ----- wA ---- wE
gGb --- gDb --- gAb --- gEb --- gBb

All five Shrutal [10] modes, using ups and downs. Every other scale note has an up.
|| scale name || sL pattern || example in C || 1st genchain || 2nd genchain ||
|| 1st Shrutal [10] || ssssL-ssssL || C C^ D D^ E F^ G G^ A A^ C || __**C**__ G D A E || F^ C^ G^ D^ A^ ||
|| 2nd Shrutal [10] || sssLs-sssLs || C C^ D D^ F F^ G G^ A Bb^ C || F __**C**__ G D A || Bb^ F^ C^ G^ D^ ||
|| 3rd Shrutal [10] || ssLss-ssLss || C C^ D Eb^ F F^ G G^ Bb Bb^ C || Bb F __**C**__ G D || Eb^ Bb^ F^ C^ G^ ||
|| 4th Shrutal [10] || sLsss-sLsss || C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C || Eb Bb F __**C**__ G || Ab^ Eb^ Bb^ F^ C^ ||
|| 5th Shrutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ ||


The Diminished [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.
Gb^^ ----- Db^^
Eb^ ------- Bb^
C ---------- G
Av --------- Ev
The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. "Up" means "a quarter-octave minus a ~32/27".

Using ~25/24 as the generator yields the same scales and mode numbers:
Gb^^ ----- G
Eb^ ------- Ev
C ---------- Db^^
Av --------- Bb^
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The name is 4-EDO+y [8].
ggGb ----- ggDb
gEb ------- gBb
wC -------- wG
yA --------- yE

Both Diminished [8] modes, using ups and downs:
|| scale name || sL pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain ||
|| 1st Diminished[ 8] || sLsL sLsL || C Db^^ Eb^ Ev Gb^^ G Av Bb^ C ||= __**C**__ G || Eb^ Bb^ || Gb^^ Db^^ || Av Ev ||
|| 2nd Diminished [8] || LsLs LsLs || C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C ||= F __**C**__ || Ab^ Eb^ || Cb^^ Gb^^ || Dv Av ||


There are only two Blackwood [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":
E^^ ------- G#^^
D^ -------- F#^
C ---------- E
Bbv ------- Fv
Gvv ------- Dvv

Ups and downs could indicate the generator instead of the period:
F ------ Av
D ------ F#v
C ------ Ev
A ------ C#v
G ------ Bv

Assuming octave equivalence, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.
F ------ Av
C ------ Ev
G ------ Bv
D ------ F#v
A ------ C#v

Using color notation. The name is 5-EDO+y.
wF ------ yA
wC ------ yE
wG ------ yB
wD ------ yF#
wA ------ yC#

Using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4":
|| scale name || sL pattern || example in C || genchains ||
|| 1st Blackwood [10] || Ls-Ls-Ls-Ls-Ls || C C#v D Ev F F#v G Av A Bv C ||= __**C**__-Ev, D-F#v, F-Av, G-Bv, A-C#v ||
|| 2nd Blackwood [10] || sL-sL-sL-sL-sL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^-__**C**__, Bb^-D, C^-E, Eb^-G, F^-A ||


==[[#Rank-2 scales that are neither MOS nor MODMOS]]**__Other rank-2 scales__**== 

Some scales have too many or too few notes to be MOS or MODMOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets can be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}.

However chromatic alterations create genchains with gaps that are very difficult to name. These scales must be named as MOS scales with notes added or removed, using "add" and "no", analogous to chord names. As with MODMOS scales, there is often more than one name for a scale.

|| scale || genchain || name || sMLX pattern ||
|| octotonic: ||   ||   || (assumes 3/2 < 700¢) ||
|| C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone {8} || LLMs MLLM ||
|| " || " || C 2nd Meantone [7] add #4 ||   ||
||= " ||= " || C 1st Meantone [7] add b4 * ||   ||
|| C D E F F# G A Bb C || Bb F __**C**__ G D A E * F# || C 3rd Meantone [7] add #4 || LLMs MLML ||
|| A B C D D# E F G# A || F C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 || LMLs MMXM ||
|| A B C D D# E G# A || C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 no6 || LMLs MXM ||
|| nonotonic: ||   ||   ||   ||
|| A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone {9} || LLMsM LMsM ||
|| " || " || A 3rd Meantone [7] add #4, #7 ||   ||
||= " ||= " || A 2nd Meantone [7] add #4, b7 ||   ||
||= " ||= " || A 1st Meantone [7] add b4, b7 ||   ||
|| A B C D D# E F G G# A || F C G D __**A**__ E B * * G# D# || A 5th Meantone [7] add #4, #7 || LMLsM MLsM ||
|| hexatonic: ||   ||   ||   ||
|| F G A C D E F || __**F**__ C G D A E || F 1st Meantone {6} || MML MMs ||
|| " || " || F 2nd Meantone [7] no4 ||   ||
||= " ||= " || F 1st Meantone [7] no4 ||   ||
|| G A C D E F# G || C __**G**__ D A E * F# || G 2nd Meantone [7] no3 || MLM MMs ||
|| pentatonic: ||   ||   ||   ||
|| F G A C E F || __**F**__ C G * A E || F 2nd Meantone [7] no4 no6 || MML Xs ||
||= " ||= " || F 1st Meantone [7] no4 no6 ||   ||
|| A B C E F A || F C * * __**A**__ E B || A 5th Meantone [7] no4 no7 || MsL sL ||
* In the 3rd row, "add b4" means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th.

The sML notation requires X = extra-large for various intervals.

The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be "F 1st Meantone [5] no5 add b6" and "A 3rd Meantone [5] no4 no7 add #5, #2".

Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4.

Another category is scales that would be MOS, but the generator is too sharp or flat. For example, a genchain F C G D A E B of 8\13 fifths makes an out-of-order scale A C B D F E G A. This scale is best named as Meantone [5] with added notes: Which brings us to...


==[[#Numbering considerations]]__Numbering considerations__== 

As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what "#4" means in a pentatonic or hexatonic context.

If the scale is written using heptatonically using 7 /note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone [5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is bering used.

The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Using the numbers 1-5 both as note names and as scale degrees, we get this genchain:
...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b...
and these standard modes:
1 1st Meantone [5] = 1 2 b3 4 b5 1
1 2nd Meantone [5] = 1 2 3 4 b5 1
1 3rd Meantone [5] = 1 2 3 4 5 1
1 4th Meantone [5] = 1 #2 3 4 5 1
1 5th Meantone [5] = 1 #2 3 #4 5 1
The initial "1" is the tonic of the scale.

The A C B D F E G A scale becomes 1 2 2# 3 4 b5 5 1, which has 3 possible names:
1 3rd Meantone [5] add #2, b5
1 2nd Meantone [5] add #2, #5
1 4th Meantone [5] add b2, b5




==[[#Explanation / Rationale]]__Explanation__== 

**__Why not number the modes in the order they occur in the scale?__**

Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.

__Genchain-based__: if the Meantone[7] genchain were notated 1 2 3 4 5 6 7, the Lydian scale would be 1 3 5 7 2 4 6 1, and the major scale would be 2 4 6 1 3 5 7 2.

__Scale-based__: if the Meantone[7] major scale were notated 1 2 3 4 5 6 7 1, the genchain would be 4 1 5 2 6 3 7.

The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The modes are ordered in a spectrum, and the 1st and last modes are always the two most extreme. For MOS scales, adjacent modes differ by only one note.

The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.

__**Why make an exception for 3/2 vs 4/3 as the generator?**__

There are centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):

"Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio __**<span class="mw-redirect">3:2</span>**__ (i.e., the untempered perfect __**fifth**__)." -- [[https://en.wikipedia.org/wiki/Pythagorean_tuning|en.wikipedia.org/wiki/Pythagorean_tuning]]

"The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect __**fifth**__." -- [[https://en.wikipedia.org/wiki/Syntonic_temperament|en.wikipedia.org/wiki/Syntonic_temperament]]

"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect __**fifths**__." --
[[https://en.wikipedia.org/wiki/Meantone_temperament|en.wikipedia.org/wiki/Meantone_temperament]]

"In this system the perfect __**fifth**__ is flattened by one quarter of a syntonic comma." -- [[https://en.wikipedia.org/wiki/Quarter-comma_meantone|en.wikipedia.org/wiki/Quarter-comma_meantone]]

"The term "well temperament" or "good temperament" usually means some sort of <span class="new">irregular temperament</span> in which the tempered __**fifths**__ are of different sizes." -- [[https://en.wikipedia.org/wiki/Well_temperament|en.wikipedia.org/wiki/Well_temperament]]

"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a __wise__ consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.

__**Then why not always choose the larger of the two generators?**__

Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine [7] above.)

**__Why not always choose the chroma-positive generator?__**

See below.

__**Why not just use UDP notation?**__

One problem with [[Modal UDP Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.
|| scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain ||
|| Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E ||
|| Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# ||
|| Meantone[12] in 31edo ||= 4/3 || E# A# D# G# C# F#
B E A D G C ||= 3/2 || C G D A E B F# C# G#
D# A# E# ||
|| Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C#
G# D# A# E# B#
FxCx Gx Dx Ax Ex ||= 3/2 || C G D A E B F# C# G#
D# A# E# B# Fx Cx Gx
Dx Ax Ex ||

A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. Mode Numbers notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful.
|| scale || UDP genchain || Mode Numbers genchain ||
|| Meantone [2] || C G || C G ||
|| Meantone [3] || D G C || C G D ||
|| Meantone [4] || ??? || C G D A ||
|| Meantone [5] || E A D G C || C G D A E ||
|| Meantone [6] || ??? || G C D A E B ||
|| Meantone [7] || C G D A E B F# || C G D A E B F# ||
|| Meantone [8] || ??? || C G D A E B F# C# ||
|| Meantone [9] || ??? || C G D A E B F# C# G# ||
|| Meantone [10] || ??? || C G D A E B F# C# G# D# ||
|| Meantone [11] || ??? || C G D A E B F# C# G# D# A# ||
|| Meantone [12] if generator < 700¢ || E# A# D# G# C# F# B E A D G C || C G D A E B F# C# G# D# A# E# ||
||< Meantone [12] if generator > 700¢ || C G D A E B F# C# G# D# A# E# ||= C G D A E B F# C# G# D# A# E# ||

An even larger problem is that the notation is overly tuning-dependent. Meantone [12] generated by 701¢ has a different genchain than Meantone [12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant [12]" or "9th Dominant [12]". Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.

Three other problems with UDP are more issues of taste. The most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Shrutal [10] 6|2(2), use four numbers where only two are needed.

Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.

Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.

Original HTML content:

<html><head><title>Naming Rank-2 Scales using Mode Numbers</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-MOS Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->MOS Scales</h2>
 <br />
<!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 2em;"><a href="#x-MOS Scales">MOS Scales</a></div>
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 2em;"><a href="#x-MODMOS scales">MODMOS scales</a></div>
<!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 2em;"><a href="#x-Fractional-octave periods">Fractional-octave periods</a></div>
<!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 2em;"><a href="#x-Other rank-2 scales">Other rank-2 scales</a></div>
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 2em;"><a href="#x-Numbering considerations">Numbering considerations</a></div>
<!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#x-Explanation">Explanation</a></div>
<!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --></div>
<!-- ws:end:WikiTextTocRule:19 --><strong>Mode numbers</strong> provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely.<br />
<br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/periods%20and%20generators">generator-chain</a>, or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions.<br />
<br />
For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br />


<table class="wiki_table">
    <tr>
        <td>old scale name<br />
</td>
        <td>new scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example on white keys<br />
</td>
        <td>genchain<br />
</td>
    </tr>
    <tr>
        <td>Lydian<br />
</td>
        <td>1st Meantone [7]<br />
</td>
        <td>LLLs LLs<br />
</td>
        <td>F G A B C D E F<br />
</td>
        <td><u><strong>F</strong></u> C G D A E B<br />
</td>
    </tr>
    <tr>
        <td>Ionian (major)<br />
</td>
        <td>2nd Meantone [7]<br />
</td>
        <td>LLsL LLs<br />
</td>
        <td>C D E F G A B C<br />
</td>
        <td>F <u><strong>C</strong></u> G D A E B<br />
</td>
    </tr>
    <tr>
        <td>Mixolydian<br />
</td>
        <td>3rd Meantone [7]<br />
</td>
        <td>LLsL LsL<br />
</td>
        <td>G A B C D E F G<br />
</td>
        <td>F C <u><strong>G</strong></u> D A E B<br />
</td>
    </tr>
    <tr>
        <td>Dorian<br />
</td>
        <td>4th Meantone [7]<br />
</td>
        <td>LsLL LsL<br />
</td>
        <td>D E F G A B C D<br />
</td>
        <td>F C G <u><strong>D</strong></u> A E B<br />
</td>
    </tr>
    <tr>
        <td>Aeolian (minor)<br />
</td>
        <td>5th Meantone [7]<br />
</td>
        <td>LsLL sLL<br />
</td>
        <td>A B C D E F G A<br />
</td>
        <td>F C G D <u><strong>A</strong></u> E B<br />
</td>
    </tr>
    <tr>
        <td>Phrygian<br />
</td>
        <td>6th Meantone [7]<br />
</td>
        <td>sLLL sLL<br />
</td>
        <td>E F G A B C D E<br />
</td>
        <td>F C G D A <u><strong>E</strong></u> B<br />
</td>
    </tr>
    <tr>
        <td>Locrian<br />
</td>
        <td>7th Meantone [7]<br />
</td>
        <td>sLLs LLL<br />
</td>
        <td>B C D E F G A B<br />
</td>
        <td>F C G D A E <u><strong>B</strong></u><br />
</td>
    </tr>
</table>

4th Meantone [7] is spoken as &quot;fourth meantone heptatonic&quot;, or possibly &quot;fourth meantone seven&quot;. If in D, as above, it would be &quot;D fourth meantone heptatonic&quot;.<br />
<br />
The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian).<br />


<table class="wiki_table">
    <tr>
        <td>old scale name<br />
</td>
        <td>new scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>------------------- genchain ---------------<br />
</td>
    </tr>
    <tr>
        <td>Lydian<br />
</td>
        <td>1st Meantone [7]<br />
</td>
        <td>LLLs LLs<br />
</td>
        <td>C D E F# G A B C<br />
</td>
        <td style="text-align: right;"><u><strong>C</strong></u> G D A E B F#<br />
</td>
    </tr>
    <tr>
        <td>Ionian (major)<br />
</td>
        <td>2nd Meantone [7]<br />
</td>
        <td>LLsL LLs<br />
</td>
        <td>C D E F G A B C<br />
</td>
        <td style="text-align: right;">F <u><strong>C</strong></u> G D A E B ----<br />
</td>
    </tr>
    <tr>
        <td>Mixolydian<br />
</td>
        <td>3rd Meantone [7]<br />
</td>
        <td>LLsL LsL<br />
</td>
        <td>C D E F G A Bb C<br />
</td>
        <td style="text-align: right;">Bb F <u><strong>C</strong></u> G D A E -------<br />
</td>
    </tr>
    <tr>
        <td>Dorian<br />
</td>
        <td>4th Meantone [7]<br />
</td>
        <td>LsLL LsL<br />
</td>
        <td>C D Eb F G A Bb C<br />
</td>
        <td>------------- Eb Bb F <u><strong>C</strong></u> G D A<br />
</td>
    </tr>
    <tr>
        <td>Aeolian (minor)<br />
</td>
        <td>5th Meantone [7]<br />
</td>
        <td>LsLL sLL<br />
</td>
        <td>C D Eb F G Ab Bb C<br />
</td>
        <td>--------- Ab Eb Bb F <u><strong>C</strong></u> G D<br />
</td>
    </tr>
    <tr>
        <td>Phrygian<br />
</td>
        <td>6th Meantone [7]<br />
</td>
        <td>sLLL sLL<br />
</td>
        <td>C Db Eb F G Ab Bb C<br />
</td>
        <td>---- Db Ab Eb Bb F <u><strong>C</strong></u> G<br />
</td>
    </tr>
    <tr>
        <td>Locrian<br />
</td>
        <td>7th Meantone [7]<br />
</td>
        <td>sLLs LLL<br />
</td>
        <td>C Db Eb F Gb Ab Bb C<br />
</td>
        <td>Gb Db Ab Eb Bb F <u><strong>C</strong></u><br />
</td>
    </tr>
</table>

<br />
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below). <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> There are several disadvantages of only using chroma-positive generators, see <a class="wiki_link" href="/Naming%20Rank-2%20Scales%20using%20Mode%20Numbers#Why%20not%20just%20use%20UDP%20notation?">Why not just use UDP notation?</a> below.<br />
<br />
Pentatonic meantone scales:<br />


<table class="wiki_table">
    <tr>
        <td>old scale name<br />
</td>
        <td>new scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>--------- genchain -------<br />
</td>
    </tr>
    <tr>
        <td>major pentatonic<br />
</td>
        <td>1st Meantone [5]<br />
</td>
        <td>ssL sL<br />
</td>
        <td>C D E G A C<br />
</td>
        <td style="text-align: right;"><u><strong>C</strong></u> G D A E<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td>2nd Meantone [5]<br />
</td>
        <td>sLs sL<br />
</td>
        <td>C D F G A C<br />
</td>
        <td style="text-align: right;">F <u><strong>C</strong></u> G D A --<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td>3rd Meantone [5]<br />
</td>
        <td>sLs Ls<br />
</td>
        <td>C D F G Bb C<br />
</td>
        <td>-------- Bb F <u><strong>C</strong></u> G D<br />
</td>
    </tr>
    <tr>
        <td>minor pentatonic<br />
</td>
        <td>4th Meantone [5]<br />
</td>
        <td>Lss Ls<br />
</td>
        <td>C Eb F G Bb C<br />
</td>
        <td>---- Eb Bb F <u><strong>C</strong></u> G<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td>5th Meantone [5]<br />
</td>
        <td>LsL ss<br />
</td>
        <td>C Eb F Ab Bb C<br />
</td>
        <td>Ab Eb Bb F <u><strong>C</strong></u><br />
</td>
    </tr>
</table>

<br />
Chromatic meantone scales.<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>sL pattern<br />
(assumes 3/2 &lt; 700¢)<br />
</td>
        <td>example in C<br />
</td>
        <td>genchain<br />
</td>
    </tr>
    <tr>
        <td>1st Meantone [12]<br />
</td>
        <td>sLsL sLL sLsLL<br />
</td>
        <td>C C# D D# E E# F# G G# A A# B C<br />
</td>
        <td><u><strong>C</strong></u> G D A E B F# C# G# D# A# E#<br />
</td>
    </tr>
    <tr>
        <td>2nd Meantone [12]<br />
</td>
        <td>sLsL LsL sLsLL<br />
</td>
        <td>C C# D D# E F F# G G# A A# B C<br />
</td>
        <td>F <u><strong>C</strong></u> G D A E B F# C# G# D# A#<br />
</td>
    </tr>
    <tr>
        <td>3rd Meantone [12]<br />
</td>
        <td>sLsL LsL sLLsL<br />
</td>
        <td>C C# D D# E F F# G G# A Bb B C<br />
</td>
        <td>Bb F <u><strong>C</strong></u> G D A E B F# C# G# D#<br />
</td>
    </tr>
    <tr>
        <td>4th Meantone [12]<br />
</td>
        <td>sLLs LsL sLLsL<br />
</td>
        <td>C C# D Eb E F F# G G# A Bb B C<br />
</td>
        <td>Eb Bb F <u><strong>C</strong></u> G D A E B F# C# G#<br />
</td>
    </tr>
    <tr>
        <td>5th Meantone [12]<br />
</td>
        <td>sLLs LsL LsLsL<br />
</td>
        <td>C C# D Eb E F F# G Ab A Bb B C<br />
</td>
        <td>Ab Eb Bb F <u><strong>C</strong></u> G D A E B F# C#<br />
</td>
    </tr>
    <tr>
        <td>6th Meantone [12]<br />
</td>
        <td>LsLs LsL LsLsL<br />
</td>
        <td>C Db D Eb E F F# G Ab A Bb B C<br />
</td>
        <td>Db Ab Eb Bb F <u><strong>C</strong></u> G D A E B F#<br />
</td>
    </tr>
    <tr>
        <td>7th Meantone [12]<br />
</td>
        <td>LsLs LLs LsLsL<br />
</td>
        <td>C Db D Eb E F Gb G Ab A Bb B C<br />
</td>
        <td>Gb Db Ab Eb Bb F <u><strong>C</strong></u> G D A E B<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">etc.<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.<br />
<br />
<br />
<a class="wiki_link" href="/Sensi">Sensi</a> [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>genchain<br />
</td>
    </tr>
    <tr>
        <td>1st Sensi [8]<br />
</td>
        <td>ssL ssL sL<br />
</td>
        <td>C Db D# E# F# G A Bb C<br />
</td>
        <td><u><strong>C</strong></u> E# A Db F# Bb D# G<br />
</td>
    </tr>
    <tr>
        <td>2nd Sensi [8]<br />
</td>
        <td>ssL sL ssL<br />
</td>
        <td>C Db D# E# F# G# A Bb C<br />
</td>
        <td>G# <u><strong>C</strong></u> E# A Db F# Bb D#<br />
</td>
    </tr>
    <tr>
        <td>3rd Sensi [8]<br />
</td>
        <td>sL ssL ssL<br />
</td>
        <td>C Db Eb E# F# G# A Bb C<br />
</td>
        <td>Eb G# <u><strong>C</strong></u> E# A Db F# Bb<br />
</td>
    </tr>
    <tr>
        <td>4th Sensi [8]<br />
</td>
        <td>sL ssL sL s<br />
</td>
        <td>C Db Eb E# F# G# A B C<br />
</td>
        <td>B Eb G# <u><strong>C</strong></u> E# A Db F#<br />
</td>
    </tr>
    <tr>
        <td>5th Sensi [8]<br />
</td>
        <td>sL sL ssL s<br />
</td>
        <td>C Db Eb E# Gb G# A B C<br />
</td>
        <td>Gb B Eb G# <u><strong>C</strong></u> E# A Db<br />
</td>
    </tr>
    <tr>
        <td>6th Sensi [8]<br />
</td>
        <td>Lss Lss Ls<br />
</td>
        <td>C D Eb E# Gb G# A B C<br />
</td>
        <td>D Gb B Eb G# <u><strong>C</strong></u> E# A<br />
</td>
    </tr>
    <tr>
        <td>7th Sensi [8]<br />
</td>
        <td>Lss Ls Lss<br />
</td>
        <td>C D Eb E# Gb G# A# B C<br />
</td>
        <td>A# D Gb B Eb G# <u><strong>C</strong></u> E#<br />
</td>
    </tr>
    <tr>
        <td>8th Sensi [8]<br />
</td>
        <td>Ls Lss Lss<br />
</td>
        <td>C D Eb F Gb G# A# B C<br />
</td>
        <td>F A# D Gb B Eb G# <u><strong>C</strong></u><br />
</td>
    </tr>
</table>

The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:<br />
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C<br />
The modes would follow a more regular pattern if using octotonic fourth-based notation:<br />
C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A - A#/Bb - B - B# - Cb -C<br />
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.<br />
<br />
<br />
Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using <a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation">ups and downs notation</a>. Because the generator is a 2nd, the genchain resembles the scale.<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>genchain<br />
</td>
    </tr>
    <tr>
        <td>1st Porcupine [7]<br />
</td>
        <td>ssss ssL<br />
</td>
        <td>C Dv Eb^ F Gv Ab^ Bb C<br />
</td>
        <td><u><strong>C</strong></u> Dv Eb^ F Gv Ab^ Bb<br />
</td>
    </tr>
    <tr>
        <td>2nd Porcupine [7]<br />
</td>
        <td>ssss sLs<br />
</td>
        <td>C Dv Eb^ F Gv Ab^ Bb^ C<br />
</td>
        <td>Bb^ <u><strong>C</strong></u> Dv Eb^ F Gv Ab^<br />
</td>
    </tr>
    <tr>
        <td>3rd Porcupine [7]<br />
</td>
        <td>ssss Lss<br />
</td>
        <td>C Dv Eb^ F Gv Av Bb^ C<br />
</td>
        <td>Av Bb^ <u><strong>C</strong></u> Dv Eb^ F Gv<br />
</td>
    </tr>
    <tr>
        <td>4th Porcupine [7]<br />
</td>
        <td>sssL sss<br />
</td>
        <td>C Dv Eb^ F G Av Bb^ C<br />
</td>
        <td>G Av Bb^ <u><strong>C</strong></u> Dv Eb^ F<br />
</td>
    </tr>
    <tr>
        <td>5th Porcupine [7]<br />
</td>
        <td>ssLs sss<br />
</td>
        <td>C Dv Eb^ F^ G Av Bb^ C<br />
</td>
        <td style="text-align: center;">F^ G Av Bb^ <u><strong>C</strong></u> Dv Eb^<br />
</td>
    </tr>
    <tr>
        <td>6th Porcupine [7]<br />
</td>
        <td>sLss sss<br />
</td>
        <td>C Dv Ev F^ G Av Bb^ C<br />
</td>
        <td>Ev F^ G Av Bb^ <u><strong>C</strong></u> Dv<br />
</td>
    </tr>
    <tr>
        <td>7th Porcupine [7]<br />
</td>
        <td>Lsss sss<br />
</td>
        <td>C D Ev F^ G Av Bb^ C<br />
</td>
        <td>D Ev F^ G Av Bb^ <u><strong>C</strong></u><br />
</td>
    </tr>
</table>

Again, the modes would follow a more regular pattern if using the appropriate notation, in this case 2nd-based:<br />
C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C<br />
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:20:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-MODMOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-MODMOS scales&quot;/&gt; --><a name="How to name rank-2 scales-MODMOS scales"></a><!-- ws:end:WikiTextAnchorRule:20 --><strong><u>MODMOS scales</u></strong></h2>
 <br />
<a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The &quot;#&quot; symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G.<br />
<br />
MODMOS names are ambiguous. This scale could also be written as 2nd Meantone [7] b3 (major scale with a minor 3rd), or as 4th Meantone [7] #7 (dorian with a major 7th).<br />
<br />


<table class="wiki_table">
    <tr>
        <td>old scale name<br />
</td>
        <td>example in A<br />
</td>
        <td>genchain<br />
</td>
        <td>new scale name<br />
</td>
        <td>sML pattern<br />
</td>
    </tr>
    <tr>
        <td>Harmonic minor<br />
</td>
        <td>A B C D E F G# A<br />
</td>
        <td>F C * D <u><strong>A</strong></u> E B * * G#<br />
</td>
        <td>5th Meantone [7] #7<br />
</td>
        <td>MsMM sLs<br />
</td>
    </tr>
    <tr>
        <td>Ascending melodic minor<br />
</td>
        <td>A B C D E F# G# A<br />
</td>
        <td>C * D <u><strong>A</strong></u> E B F# * G#<br />
</td>
        <td>5th Meantone [7] #6 #7<br />
</td>
        <td>LsLL LLs<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>2nd Meantone [7] b3<br />
</td>
        <td>&quot;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>4th Meantone [7] #7<br />
</td>
        <td>&quot;<br />
</td>
    </tr>
    <tr>
        <td>Double harmonic minor<br />
</td>
        <td>A B C D# E F G# A<br />
</td>
        <td>F C * * <u><strong>A</strong></u> E B * * G# D#<br />
</td>
        <td>5th Meantone [7] #4 #7<br />
</td>
        <td>MsLs sLs<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>&quot;<br />
</td>
        <td>1st Meantone [7] b3 b6<br />
</td>
        <td>&quot;<br />
</td>
    </tr>
    <tr>
        <td>Double harmonic major<br />
</td>
        <td>A Bb C# D E F G# A<br />
</td>
        <td>Bb F * * D <u><strong>A</strong></u> E * * C# G#<br />
</td>
        <td>2nd Meantone [7] b2 b6<br />
</td>
        <td>sLsM sLs<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>&quot;<br />
</td>
        <td>6th Meantone [7] #3 #7<br />
</td>
        <td>&quot;<br />
</td>
    </tr>
    <tr>
        <td><span class="mw-redirect">Hungarian gypsy </span>minor<br />
</td>
        <td>A B C D# E F G A<br />
</td>
        <td>F C G * <u><strong>A</strong></u> E B * * * D#<br />
</td>
        <td>5th Meantone [7] #4<br />
</td>
        <td>MsLs sMM<br />
</td>
    </tr>
    <tr>
        <td>Phrygian dominant<br />
</td>
        <td>A Bb C# D E F G A<br />
</td>
        <td>Bb F * G D <u><strong>A</strong></u> E * * C#<br />
</td>
        <td>6th Meantone [7] #3<br />
</td>
        <td>sLsM sMM<br />
</td>
    </tr>
</table>

As can be seen from the genchains or from the sML patterns, the harmonic minor and the phrygian dominant are modes of each other, as are the double harmonic minor and the double harmonic major. Unfortunately the scale names do not indicate this.<br />
<br />
The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from A dorian to A melodic minor. In this context, melodic minor might better be described as an altered dorian scale.<br />
<br />
Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:<br />
1st Meantone [7] #2: C D# E F# G A B C<br />
2nd Meantone [7] #:5 C D E F G# A B C<br />
7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern, 7th mode not 3rd mode)<br />
4th Meantone [7] #4: C D Eb F# G A Bb C<br />
5th Meantone [7] #7: C D Eb F G Ab B C (harmonic minor)<br />
6th Meantone [7] #3: C Db E F G Ab Bb C (phrygian dominant)<br />
7th Meantone [7] #6: C Db Eb F Gb A Bb C<br />
The 3rd scale breaks the pattern to avoid an altered tonic (&quot;3rd Meantone [7] #1&quot;).<br />
<br />
Ascending melodic minor modes:<br />
1st Meantone [7] #6 #7: C D E F# G# A B C<br />
2nd Meantone [7] #6 #7: C Db Eb Fb Gb Ab Bb C<br />
3rd Meantone [7] #6 #7: C D E F# G A Bb C<br />
4th Meantone [7] #6 #7: C D Eb F G A B C<br />
5th Meantone [7] #6 #7: C D E F G Ab Bb C<br />
6th Meantone [7] #6 #7: C Db Eb F G A Bb C<br />
7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:21:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Fractional-octave periods&quot; title=&quot;Anchor: Fractional-octave periods&quot;/&gt; --><a name="Fractional-octave periods"></a><!-- ws:end:WikiTextAnchorRule:21 --><strong><u>Fractional-octave periods</u></strong></h2>
 <br />
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.<br />
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5<br />
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4<br />
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3<br />
<br />
When the period is an octave, the genweb octave-reduces to a single horizontal genchain:<br />
F --- C --- G --- D --- A --- E --- B<br />
<br />
But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, shrutal [10] might look like this:<br />
F^3 --- C^4 --- G^4 --- D^5 --- A^5<br />
C3 ---- G3 ----- D4 ---- A4 ---- E5<br />
F^2 --- C^3 --- G^3 --- D^4 --- A^4<br />
C2 ---- G2 ----- D3 ---- A3 ---- E3<br />
F^1 --- C^2 --- G^2 --- D^3 --- A^3<br />
C1 ---- G1 ----- D2 ---- A2 ---- E2<br />
<br />
which octave-reduces to two genchains:<br />
F^ --- C^ --- G^ --- D^ --- A^<br />
C ---- G ----- D ---- A ---- E<br />
<br />
Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of &quot;up&quot; here is &quot;a half-octave minus a fourth&quot;, with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.<br />
<br />
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.<br />
Gv --- Dv --- Av --- Ev --- Bv<br />
C ----- G ----- D ---- A ---- E<br />
<br />
It would also be valid to exchange the two rows:<br />
C ----- G ----- D ---- A ---- E<br />
Gv --- Dv --- Av --- Ev --- Bv<br />
<br />
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.<br />
<br />
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.<br />
<br />
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:<br />
F^ -- G --- G^ -- A --- A^<br />
C --- C^ -- D --- D^ -- E<br />
<br />
Another alternative is to use <a class="wiki_link" href="/Kite%27s%20color%20notation">color notation</a>. The shrutal comma is 2048/2025 = sgg2, and the temperament name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:<br />
<br />
yF# --- yC# --- yG# --- yD# --- yA#<br />
wC ---- wG ---- wD ---- wA ---- wE<br />
<br />
As always, y means &quot;81/80 below w&quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:<br />
wC ---- wG ---- wD ----- wA ---- wE<br />
gGb --- gDb --- gAb --- gEb --- gBb<br />
<br />
All five Shrutal [10] modes, using ups and downs. Every other scale note has an up.<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>1st genchain<br />
</td>
        <td>2nd genchain<br />
</td>
    </tr>
    <tr>
        <td>1st Shrutal [10]<br />
</td>
        <td>ssssL-ssssL<br />
</td>
        <td>C C^ D D^ E F^ G G^ A A^ C<br />
</td>
        <td><u><strong>C</strong></u> G D A E<br />
</td>
        <td>F^ C^ G^ D^ A^<br />
</td>
    </tr>
    <tr>
        <td>2nd Shrutal [10]<br />
</td>
        <td>sssLs-sssLs<br />
</td>
        <td>C C^ D D^ F F^ G G^ A Bb^ C<br />
</td>
        <td>F <u><strong>C</strong></u> G D A<br />
</td>
        <td>Bb^ F^ C^ G^ D^<br />
</td>
    </tr>
    <tr>
        <td>3rd Shrutal [10]<br />
</td>
        <td>ssLss-ssLss<br />
</td>
        <td>C C^ D Eb^ F F^ G G^ Bb Bb^ C<br />
</td>
        <td>Bb F <u><strong>C</strong></u> G D<br />
</td>
        <td>Eb^ Bb^ F^ C^ G^<br />
</td>
    </tr>
    <tr>
        <td>4th Shrutal [10]<br />
</td>
        <td>sLsss-sLsss<br />
</td>
        <td>C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C<br />
</td>
        <td>Eb Bb F <u><strong>C</strong></u> G<br />
</td>
        <td>Ab^ Eb^ Bb^ F^ C^<br />
</td>
    </tr>
    <tr>
        <td>5th Shrutal [10]<br />
</td>
        <td>Lssss-Lssss<br />
</td>
        <td>C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C<br />
</td>
        <td>Ab Eb Bb F <u><strong>C</strong></u><br />
</td>
        <td>Db^ Ab^ Eb^ Bb^ F^<br />
</td>
    </tr>
</table>

<br />
<br />
The Diminished [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.<br />
Gb^^ ----- Db^^<br />
Eb^ ------- Bb^<br />
C ---------- G<br />
Av --------- Ev<br />
The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. &quot;Up&quot; means &quot;a quarter-octave minus a ~32/27&quot;.<br />
<br />
Using ~25/24 as the generator yields the same scales and mode numbers:<br />
Gb^^ ----- G<br />
Eb^ ------- Ev<br />
C ---------- Db^^<br />
Av --------- Bb^<br />
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The name is 4-EDO+y [8].<br />
ggGb ----- ggDb<br />
gEb ------- gBb<br />
wC -------- wG<br />
yA --------- yE<br />
<br />
Both Diminished [8] modes, using ups and downs:<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>1st chain<br />
</td>
        <td>2nd chain<br />
</td>
        <td>3rd chain<br />
</td>
        <td>4th chain<br />
</td>
    </tr>
    <tr>
        <td>1st Diminished[ 8]<br />
</td>
        <td>sLsL sLsL<br />
</td>
        <td>C Db^^ Eb^ Ev Gb^^ G Av Bb^ C<br />
</td>
        <td style="text-align: center;"><u><strong>C</strong></u> G<br />
</td>
        <td>Eb^ Bb^<br />
</td>
        <td>Gb^^ Db^^<br />
</td>
        <td>Av Ev<br />
</td>
    </tr>
    <tr>
        <td>2nd Diminished [8]<br />
</td>
        <td>LsLs LsLs<br />
</td>
        <td>C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C<br />
</td>
        <td style="text-align: center;">F <u><strong>C</strong></u><br />
</td>
        <td>Ab^ Eb^<br />
</td>
        <td>Cb^^ Gb^^<br />
</td>
        <td>Dv Av<br />
</td>
    </tr>
</table>

<br />
<br />
There are only two Blackwood [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different &quot;height&quot;:<br />
E^^ ------- G#^^<br />
D^ -------- F#^<br />
C ---------- E<br />
Bbv ------- Fv<br />
Gvv ------- Dvv<br />
<br />
Ups and downs could indicate the generator instead of the period:<br />
F ------ Av<br />
D ------ F#v<br />
C ------ Ev<br />
A ------ C#v<br />
G ------ Bv<br />
<br />
Assuming octave equivalence, the lattice rows can be reordered to make a &quot;pseudo-period&quot; of 3\5 = ~3/2.<br />
F ------ Av<br />
C ------ Ev<br />
G ------ Bv<br />
D ------ F#v<br />
A ------ C#v<br />
<br />
Using color notation. The name is 5-EDO+y.<br />
wF ------ yA<br />
wC ------ yE<br />
wG ------ yB<br />
wD ------ yF#<br />
wA ------ yC#<br />
<br />
Using ups and downs to mean &quot;raised/lowered by 2/5 of an octave minus ~5/4&quot;:<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>sL pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>genchains<br />
</td>
    </tr>
    <tr>
        <td>1st Blackwood [10]<br />
</td>
        <td>Ls-Ls-Ls-Ls-Ls<br />
</td>
        <td>C C#v D Ev F F#v G Av A Bv C<br />
</td>
        <td style="text-align: center;"><u><strong>C</strong></u>-Ev, D-F#v, F-Av, G-Bv, A-C#v<br />
</td>
    </tr>
    <tr>
        <td>2nd Blackwood [10]<br />
</td>
        <td>sL-sL-sL-sL-sL<br />
</td>
        <td>C C^ D Eb^ E F^ G Ab^ A Bb^ C<br />
</td>
        <td style="text-align: center;">Ab^-<u><strong>C</strong></u>, Bb^-D, C^-E, Eb^-G, F^-A<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Other rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:22:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank-2 scales that are neither MOS nor MODMOS&quot; title=&quot;Anchor: Rank-2 scales that are neither MOS nor MODMOS&quot;/&gt; --><a name="Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextAnchorRule:22 --><strong><u>Other rank-2 scales</u></strong></h2>
 <br />
Some scales have too many or too few notes to be MOS or MODMOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets can be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}.<br />
<br />
However chromatic alterations create genchains with gaps that are very difficult to name. These scales must be named as MOS scales with notes added or removed, using &quot;add&quot; and &quot;no&quot;, analogous to chord names. As with MODMOS scales, there is often more than one name for a scale.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>scale<br />
</td>
        <td>genchain<br />
</td>
        <td>name<br />
</td>
        <td>sMLX pattern<br />
</td>
    </tr>
    <tr>
        <td>octotonic:<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>(assumes 3/2 &lt; 700¢)<br />
</td>
    </tr>
    <tr>
        <td>C D E F F# G A B C<br />
</td>
        <td>F <u><strong>C</strong></u> G D A E B F#<br />
</td>
        <td>C 2nd Meantone {8}<br />
</td>
        <td>LLMs MLLM<br />
</td>
    </tr>
    <tr>
        <td>&quot;<br />
</td>
        <td>&quot;<br />
</td>
        <td>C 2nd Meantone [7] add #4<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>C 1st Meantone [7] add b4 *<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>C D E F F# G A Bb C<br />
</td>
        <td>Bb F <u><strong>C</strong></u> G D A E * F#<br />
</td>
        <td>C 3rd Meantone [7] add #4<br />
</td>
        <td>LLMs MLML<br />
</td>
    </tr>
    <tr>
        <td>A B C D D# E F G# A<br />
</td>
        <td>F C * D <u><strong>A</strong></u> E B * * G# D#<br />
</td>
        <td>A 5th Meantone [7] #7 add #4<br />
</td>
        <td>LMLs MMXM<br />
</td>
    </tr>
    <tr>
        <td>A B C D D# E G# A<br />
</td>
        <td>C * D <u><strong>A</strong></u> E B * * G# D#<br />
</td>
        <td>A 5th Meantone [7] #7 add #4 no6<br />
</td>
        <td>LMLs MXM<br />
</td>
    </tr>
    <tr>
        <td>nonotonic:<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>A B C# D D# E F# G G# A<br />
</td>
        <td>G D <u><strong>A</strong></u> E B F# C# G# D#<br />
</td>
        <td>A 3rd Meantone {9}<br />
</td>
        <td>LLMsM LMsM<br />
</td>
    </tr>
    <tr>
        <td>&quot;<br />
</td>
        <td>&quot;<br />
</td>
        <td>A 3rd Meantone [7] add #4, #7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>A 2nd Meantone [7] add #4, b7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>A 1st Meantone [7] add b4, b7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>A B C D D# E F G G# A<br />
</td>
        <td>F C G D <u><strong>A</strong></u> E B * * G# D#<br />
</td>
        <td>A 5th Meantone [7] add #4, #7<br />
</td>
        <td>LMLsM MLsM<br />
</td>
    </tr>
    <tr>
        <td>hexatonic:<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>F G A C D E F<br />
</td>
        <td><u><strong>F</strong></u> C G D A E<br />
</td>
        <td>F 1st Meantone {6}<br />
</td>
        <td>MML MMs<br />
</td>
    </tr>
    <tr>
        <td>&quot;<br />
</td>
        <td>&quot;<br />
</td>
        <td>F 2nd Meantone [7] no4<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>F 1st Meantone [7] no4<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>G A C D E F# G<br />
</td>
        <td>C <u><strong>G</strong></u> D A E * F#<br />
</td>
        <td>G 2nd Meantone [7] no3<br />
</td>
        <td>MLM MMs<br />
</td>
    </tr>
    <tr>
        <td>pentatonic:<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>F G A C E F<br />
</td>
        <td><u><strong>F</strong></u> C G * A E<br />
</td>
        <td>F 2nd Meantone [7] no4 no6<br />
</td>
        <td>MML Xs<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>F 1st Meantone [7] no4 no6<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>A B C E F A<br />
</td>
        <td>F C * * <u><strong>A</strong></u> E B<br />
</td>
        <td>A 5th Meantone [7] no4 no7<br />
</td>
        <td>MsL sL<br />
</td>
    </tr>
</table>

<ul><li>In the 3rd row, &quot;add b4&quot; means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th.</li></ul><br />
The sML notation requires X = extra-large for various intervals.<br />
<br />
The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be &quot;F 1st Meantone [5] no5 add b6&quot; and &quot;A 3rd Meantone [5] no4 no7 add #5, #2&quot;.<br />
<br />
Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4.<br />
<br />
Another category is scales that would be MOS, but the generator is too sharp or flat. For example, a genchain F C G D A E B of 8\13 fifths makes an out-of-order scale A C B D F E G A. This scale is best named as Meantone [5] with added notes: Which brings us to...<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Numbering considerations"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:23:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Numbering considerations&quot; title=&quot;Anchor: Numbering considerations&quot;/&gt; --><a name="Numbering considerations"></a><!-- ws:end:WikiTextAnchorRule:23 --><u>Numbering considerations</u></h2>
 <br />
As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what &quot;#4&quot; means in a pentatonic or hexatonic context.<br />
<br />
If the scale is written using heptatonically using 7 /note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone [5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is bering used.<br />
<br />
The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Using the numbers 1-5 both as note names and as scale degrees, we get this genchain:<br />
...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b...<br />
and these standard modes:<br />
1 1st Meantone [5] = 1 2 b3 4 b5 1<br />
1 2nd Meantone [5] = 1 2 3 4 b5 1<br />
1 3rd Meantone [5] = 1 2 3 4 5 1<br />
1 4th Meantone [5] = 1 #2 3 4 5 1<br />
1 5th Meantone [5] = 1 #2 3 #4 5 1<br />
The initial &quot;1&quot; is the tonic of the scale.<br />
<br />
The A C B D F E G A scale becomes 1 2 2# 3 4 b5 5 1, which has 3 possible names:<br />
1 3rd Meantone [5] add #2, b5<br />
1 2nd Meantone [5] add #2, #5<br />
1 4th Meantone [5] add b2, b5<br />
<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Explanation"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:24:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Explanation / Rationale&quot; title=&quot;Anchor: Explanation / Rationale&quot;/&gt; --><a name="Explanation / Rationale"></a><!-- ws:end:WikiTextAnchorRule:24 --><u>Explanation</u></h2>
 <br />
<strong><u>Why not number the modes in the order they occur in the scale?</u></strong><br />
<br />
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br />
<br />
<u>Genchain-based</u>: if the Meantone[7] genchain were notated 1 2 3 4 5 6 7, the Lydian scale would be 1 3 5 7 2 4 6 1, and the major scale would be 2 4 6 1 3 5 7 2.<br />
<br />
<u>Scale-based</u>: if the Meantone[7] major scale were notated 1 2 3 4 5 6 7 1, the genchain would be 4 1 5 2 6 3 7.<br />
<br />
The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The modes are ordered in a spectrum, and the 1st and last modes are always the two most extreme. For MOS scales, adjacent modes differ by only one note.<br />
<br />
The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.<br />
<br />
<u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u><br />
<br />
There are centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):<br />
<br />
&quot;Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <u><strong><span class="mw-redirect">3:2</span></strong></u> (i.e., the untempered perfect <u><strong>fifth</strong></u>).&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</a><br />
<br />
&quot;The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect <u><strong>fifth</strong></u>.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Syntonic_temperament" rel="nofollow">en.wikipedia.org/wiki/Syntonic_temperament</a><br />
<br />
&quot;Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect <u><strong>fifths</strong></u>.&quot; --<br />
<a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Meantone_temperament" rel="nofollow">en.wikipedia.org/wiki/Meantone_temperament</a><br />
<br />
&quot;In this system the perfect <u><strong>fifth</strong></u> is flattened by one quarter of a syntonic comma.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Quarter-comma_meantone" rel="nofollow">en.wikipedia.org/wiki/Quarter-comma_meantone</a><br />
<br />
&quot;The term &quot;well temperament&quot; or &quot;good temperament&quot; usually means some sort of <span class="new">irregular temperament</span> in which the tempered <u><strong>fifths</strong></u> are of different sizes.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Well_temperament" rel="nofollow">en.wikipedia.org/wiki/Well_temperament</a><br />
<br />
&quot;A foolish consistency is the hobgoblin of little minds&quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a <u>wise</u> consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br />
<br />
<u><strong>Then why not always choose the larger of the two generators?</strong></u><br />
<br />
Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine [7] above.)<br />
<br />
<strong><u>Why not always choose the chroma-positive generator?</u></strong><br />
<br />
See below.<br />
<br />
<u><strong>Why not just use UDP notation?</strong></u><br />
<br />
One problem with <a class="wiki_link" href="/Modal%20UDP%20Notation">UDP</a> is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.<br />


<table class="wiki_table">
    <tr>
        <td>scale<br />
</td>
        <td>UDP generator<br />
</td>
        <td>UDP genchain<br />
</td>
        <td>Mode Numbers generator<br />
</td>
        <td>Mode Numbers genchain<br />
</td>
    </tr>
    <tr>
        <td>Meantone[5] in 31edo<br />
</td>
        <td style="text-align: center;">4/3<br />
</td>
        <td>E A D G C<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>C G D A E<br />
</td>
    </tr>
    <tr>
        <td>Meantone[7] in 31edo<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>C G D A E B F#<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>C G D A E B F#<br />
</td>
    </tr>
    <tr>
        <td>Meantone[12] in 31edo<br />
</td>
        <td style="text-align: center;">4/3<br />
</td>
        <td>E# A# D# G# C# F#<br />
B E A D G C<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>C G D A E B F# C# G#<br />
D# A# E#<br />
</td>
    </tr>
    <tr>
        <td>Meantone[19] in 31edo<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>C G D A E B F# C#<br />
G# D# A# E# B#<br />
FxCx Gx Dx Ax Ex<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>C G D A E B F# C# G#<br />
D# A# E# B# Fx Cx Gx<br />
Dx Ax Ex<br />
</td>
    </tr>
</table>

<br />
A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. Mode Numbers notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful.<br />


<table class="wiki_table">
    <tr>
        <td>scale<br />
</td>
        <td>UDP genchain<br />
</td>
        <td>Mode Numbers genchain<br />
</td>
    </tr>
    <tr>
        <td>Meantone [2]<br />
</td>
        <td>C G<br />
</td>
        <td>C G<br />
</td>
    </tr>
    <tr>
        <td>Meantone [3]<br />
</td>
        <td>D G C<br />
</td>
        <td>C G D<br />
</td>
    </tr>
    <tr>
        <td>Meantone [4]<br />
</td>
        <td>???<br />
</td>
        <td>C G D A<br />
</td>
    </tr>
    <tr>
        <td>Meantone [5]<br />
</td>
        <td>E A D G C<br />
</td>
        <td>C G D A E<br />
</td>
    </tr>
    <tr>
        <td>Meantone [6]<br />
</td>
        <td>???<br />
</td>
        <td>G C D A E B<br />
</td>
    </tr>
    <tr>
        <td>Meantone [7]<br />
</td>
        <td>C G D A E B F#<br />
</td>
        <td>C G D A E B F#<br />
</td>
    </tr>
    <tr>
        <td>Meantone [8]<br />
</td>
        <td>???<br />
</td>
        <td>C G D A E B F# C#<br />
</td>
    </tr>
    <tr>
        <td>Meantone [9]<br />
</td>
        <td>???<br />
</td>
        <td>C G D A E B F# C# G#<br />
</td>
    </tr>
    <tr>
        <td>Meantone [10]<br />
</td>
        <td>???<br />
</td>
        <td>C G D A E B F# C# G# D#<br />
</td>
    </tr>
    <tr>
        <td>Meantone [11]<br />
</td>
        <td>???<br />
</td>
        <td>C G D A E B F# C# G# D# A#<br />
</td>
    </tr>
    <tr>
        <td>Meantone [12] if generator &lt; 700¢<br />
</td>
        <td>E# A# D# G# C# F# B E A D G C<br />
</td>
        <td>C G D A E B F# C# G# D# A# E#<br />
</td>
    </tr>
    <tr>
        <td style="text-align: left;">Meantone [12] if generator &gt; 700¢<br />
</td>
        <td>C G D A E B F# C# G# D# A# E#<br />
</td>
        <td style="text-align: center;">C G D A E B F# C# G# D# A# E#<br />
</td>
    </tr>
</table>

<br />
An even larger problem is that the notation is overly tuning-dependent. Meantone [12] generated by 701¢ has a different genchain than Meantone [12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example &quot;Dominant 8|3&quot; could mean either &quot;4th Dominant [12]&quot; or &quot;9th Dominant [12]&quot;. Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.<br />
<br />
Three other problems with UDP are more issues of taste. The most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Shrutal [10] 6|2(2), use four numbers where only two are needed.<br />
<br />
Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.<br />
<br />
Furthermore, UDP uses the more mathematical <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.</body></html>

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+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-24 04:54:30 UTC</tt>.<br>
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 Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.
-Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.
+Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.</pre></div>
-__**Other links:**__
-[[xenharmonic/Naming Rank-2 Scales#Jake%20Freivald%20method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method]]</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Naming Rank-2 Scales using Mode Numbers&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-MOS Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;MOS Scales&lt;/h2&gt;
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 Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.&lt;br /&gt;
 &lt;br /&gt;
-Furthermore, UDP uses the more mathematical &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow"&gt;zero-based counting&lt;/a&gt; and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.&lt;br /&gt;
+Furthermore, UDP uses the more mathematical &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow"&gt;zero-based counting&lt;/a&gt; and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.&lt;/body&gt;&lt;/html&gt;</pre></div>
-&lt;br /&gt;
-&lt;u&gt;&lt;strong&gt;Other links:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Jake%20Freivald%20method"&gt;http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Kite's Genchain mode numbering: Difference between revisions

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