Naming Rank-2 Scales using Mode Numbers

MOS Scales

Mode Numbers provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like 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.

MOS scales are formed from a segment of the 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 in "Rationale"). Unlike modal UDP notation, the generator isn't always chroma-positive. There are several disadvantages of only using chroma-positive generators, see the critique of UDP at the bottom of this page.

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 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.

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. A good alternative, especially for non-heptatonic and non-fifth-based scales, is to use + and - for forwards and backwards, as in 5th Meantone [7] +6 +7.

MODMOS names are ambiguous. The ascending melodic minor 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	"
Hungarian gypsy 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 D dorian to D 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"). The Bbb is "b7" not "bb7" because the 7th mode is Locrian, and Bbb is only one semitone flat of the Locrian mode's minor 7th Bb.

Ascending melodic minor modes:

1st Meantone [7] #5: C D E F# G# A B C

7th Meantone [7] b4: C Db Eb Fb Gb Ab Bb C (avoid "2nd Meantone [7] #1")

3rd Meantone [7] #4: C D E F# G A Bb C

4th Meantone [7] #7: C D Eb F G A B C

5th Meantone [7] #3: C D E F G Ab Bb C

6th Meantone [7] #6: C Db Eb F G A Bb C

7th Meantone [7] #2: C D Eb F Gb Ab Bb C

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, Srutal [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.

Srutal is only compatible with even-numbered frameworks. In order to preserve the primary meaning of ups and downs, which is up or down one key or fret, this notation is limited to frameworks in which the 4th is one key less than half an octave. These are 10, 12, 14, 16 and 18b, but not 18, 20, 22, 24, etc. For those frameworks, use double-ups.

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. Srutal'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 color notation. The srutal comma is 2048/2025 = sgg2, and the temperament's color 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 Srutal [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 Srutal [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 Srutal [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 Srutal [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 Srutal [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 Srutal [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 g42. The period is ~6/5 = Tg3. The color 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

In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y.

wF ------ yA

wC ------ yE

wG ------ yB

wD ------ yF#

wA ------ yC#

Both Blackwood modes, 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

Other rank-2 scales

These are scales that are neither MOS nor MODMOS. Some scales have too many or too few notes. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc.

However if there are chromatic alterations, and the genchain has gaps, there's no clear way to number the notes, and no clear way to name the scale. Such a scale must be named as a MOS scale 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 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:			(X = extra large)
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 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
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

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 possibility is a scale 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...

Non-heptatonic Scales

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 something like "#5" 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. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthwards side is sharp (assuming a fifth < 720¢). Use "+" for fifthwards and "-" for fourthwards.

Using J K L M N for note names, and arbitrarily centering the genchain on L, we get this genchain:

...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5...

...-K -N -L -J -M K N L J M +K +N +L +J +M ++K...

and these standard modes:

L 1st Meantone [5] = L M +N J +K L

L 2nd Meantone [5] = L M N J +K L

L 3rd Meantone [5] = L M N J K L

L 4th Meantone [5] = L -M N J K L

L 5th Meantone [5] = L -M N -J K L

The A C B D F E G A scale becomes L M -M N J +K K L, which has 3 possible names:

L 3rd Meantone [5] add -2, +5

L 2nd Meantone [5] add -2, -5

L 4th Meantone [5] add +2, +5

Sensi is a good example because it's nether heptatonic nor fifth-generated. Here's a Sensi [8] MOS and MODMOS in both heptatonic and octotonic notation. The generator, a heptatonic 3rd or octotonic 4th, is chroma-negative. In 19edo, generator = 7\19, L = 3\19, and s = 2\19.

notation	scale name	sL pattern	example in C	genchain
heptatonic	5th Sensi [8]	sL sL ssLs	C Db Eb E# Gb G# A B C	Gb B Eb G# C E# A Db
octotonic	5th Sensi [8]	"	C D E# F G# H A B# C	G# B# E# H C F A D
heptatonic	5th Sensi [8] +7	sL sL sssL	C Db Eb E# Gb G# A Bb C	Gb * Eb G# C E# A Db * Bb
octotonic	5th Sensi [8] +8	"	C D E# F G# H A B C	G# * E# H C F A D * B

Heptatonic fifth-based notation:

C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C

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

The heptatonic-notated MODMOS has "+7" because B is the 7th letter from C. Likewise octotonic has "+8" because with H, B is the 8th letter.

Rationale

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 added):

"Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio 3:2 (i.e., the untempered perfect fifth)." -- 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." -- en.wikipedia.org/wiki/Syntonic_temperament

"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths." --

en.wikipedia.org/wiki/Meantone_temperament

"In this system the perfect fifth is flattened by one quarter of a syntonic comma." -- en.wikipedia.org/wiki/Quarter-comma_meantone

"The term "well temperament" or "good temperament" usually means some sort of irregular temperament in which the tempered fifths are of different sizes." -- 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 it's well known that the generator is historically 3/2.

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

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 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] or Mohajira [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. Srutal [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 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.

Related links:

Jake Freivald has his own method of naming modes here:

http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method