Kite's Genchain mode numbering: Difference between revisions

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~~__FORCETOC__~~

'''Genchain mode numbering''' ('''GMN''' for short) provides 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. It also applies to abstract MOS patterns like 5L 3s.

=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|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~~.

This mode notation system was designed by [[Kite Giedraitis]].

[[~~MOSScales|~~MOS ~~scales~~]] are formed from a segment of the [[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.

== MOS scales ==

[[MOS scale]]s are formed from a segment of the [[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~~|'''Meantone'''~~]][7], using ~3/2 as the generator. ~~The~~ Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.

For example, here are all the modes of [[Meantone]][7], using ~3/2 as the generator. On this page, the Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.

{| class="wikitable"

|+ Meantone[7] modes on white keys

|-

! | old scale name

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| | 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".

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 term GMN can also be read as genchain mode number, and can refer to the numbers 1st, 2nd, 3rd etc., as in "Dorian's GMN is 4".

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).

{| class="wikitable"

|+ Meantone[7] modes in C

|-

! | old scale name

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|}

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|chroma-positive]]~~~~.~~'''~~ There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the "Rationale" section below.

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|§ Rationale]]). Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]. There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the [[#Rationale|§ Rationale]] section below.

Pentatonic meantone scales:

{| class="wikitable"

|+ Meantone[5] modes

|-

! | old scale name

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{| class="wikitable"

|+ Meantone[12] modes

|-

! | scale name

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| |

|}

~~'''~~[[Porcupine]] aka Triyo~~'''~~ has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2. The enharmonic ~~interval~~ is v3A1. Because the generator is a 2nd, the genchain resembles the scale. ~~Porcupine[7] modes, using [[Ups and Downs Notation|ups and downs notation]]:~~

[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is v3A1. Because the generator is a 2nd, the genchain resembles the scale.

{| class="wikitable"

|+ Porcupine[7]/Triyo[7] modes

|-

! | scale name

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| | D vE ^F G vA ^Bb '''C'''

|}

~~'''~~[[Sensi]] aka Sepgu~~'''~~ has pergen (P8, ~~WWP5~~/7). The ~9/7 generator is both a ^3d4 and a v4A3, and the enharmonic ~~interval~~ is ^7dd2. ~~Sensi[8] modes:~~

[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^3d4 and a v4A3, and the [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is ^7dd2.

{| class="wikitable"

|+ Sensi[8]/Sepgu[8] modes

|-

! | scale name

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| | F v4A# vD ^^Gb vvB ^Eb v3G# '''C'''

|}

~~=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 when the generator is chroma-positive, and N steps backwards when it isn't. This ensures a higher pitch. (Note that Meantone[5] is chroma-negative, more on this below.) However, an exception is made for superflat edos like 16edo when the generator is a 3/2 fifth, because in those edos, G# is actually flat of G. Another exception is when the generator is close to the "tipping point" between chroma-positive and chroma-negative. A good alternative in these and other situations, including non-heptatonic and non-fifth-generated scales, is to use + for forwards in the genchain and - for backwards, as in 5th Meantone[7] +6 +7.

== MODMOS scales ==

[[MODMOS scale]]s 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 when the generator is chroma-positive, and N steps backwards when it isn't. This ensures a higher pitch. (Note that Meantone[5] is chroma-negative, more on this below.) However, an exception is made for superflat edos like 16edo when the generator is a 3/2 fifth, because in those edos, G# is actually flat of G. Another exception is when the generator is close to the "tipping point" between chroma-positive and chroma-negative. A good alternative in these and other situations, including non-heptatonic and non-fifth-generated scales, is to use + for forwards in the genchain and - for backwards, as in 5th Meantone[7] +6 +7.

A MODMOS scale can have alternate names. The ascending melodic minor scale could also be called 2nd Meantone[7] b3 (major scale with a minor 3rd), or as 4th Meantone[7] #7 (dorian with a major 7th).

~~'''~~Meantone~~'''~~ MODMOS scales, with alternative names in italics and parentheses. Alternatives that have more alterations than the original aren't listed:

Meantone MODMOS scales, with alternative names in italics and parentheses. Alternatives that have more alterations than the original aren't listed:

{| class="wikitable"

|+ Meantone[7] MODMOS scale examples

|-

! | old scale name

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| | Bb F * G D '''A''' E * * C#

|}

As can be seen from the genchains, or from the LMs 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.

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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 ~~ ~~

* 1st Meantone[7] #2: C D# E F# G A B C

2nd Meantone[7] #:5 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)~~ ~~

* 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~~ ~~

* 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)~~ ~~

* 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)~~ ~~

* 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

* 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.

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Ascending melodic minor modes:

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

* 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")~~ ~~

* 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~~ ~~

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

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

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

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

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

~~'''~~Porcupine[7] MODMOS scales~~'''~~, not including alternative names because they all modify the 3rd or the 5th.

Porcupine[7] aka Triyo[7] MODMOS scales, not including alternative names because they all modify the 3rd or the 5th.

{| class="wikitable"

|+ Porcupine[7]/Triyo[7] MODMOS scale examples

|-

! | scale name

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| | D vE * G vA ^Bb '''C''' * * F

|}

~~=Temperaments with split octaves=~~

If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel~~, using~~ ups and downs. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice must be complete. The number in the brackets becomes two numbers, and the Ls pattern is ~~broken up into periods with~~ hyphens.

== Temperaments with split octaves ==

If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. Using ups and downs notation, each genchain has its own height. There is a plain one, an up one, perhaps a down one, etc. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice generated by the 5th and the up must be complete. The number in the brackets becomes two numbers, and the Ls pattern as written here is grouped by period, using hyphens.

~~'''~~[[Srutal]] aka Diaschismatic aka Sagugu~~'''~~ has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.

[[Srutal]] aka Diaschismatic aka Sagugu has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.

{| class="wikitable"

|+ Srutal[2x5]/Sagugu[2x5] modes

|-

! | scale name

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Srutal's period is written as a vA4, but could instead be written as an ^d5. The generator is written as a P5. 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. The generator could instead be written as ~16/15 (3/2 minus a period), because that would still create the same mode numbers and thus the same scale names. The first genchain of 1st Srutal[2x5] would be C vC# D vD# E, just like the first half of the scale.

~~'''~~[[Augmented family|Augmented]] aka Trigu~~'''~~ has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).

[[Augmented family|Augmented]] aka Trigu has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).

~~Augmented[3x3] modes:~~

{| class="wikitable"

|+ Augmented[3x3]/Trigu[3x3] modes

|-

! | scale name

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|}

'''[[Octatonic_scale|Diminished]] aka Quadgu''' has pergen (P8/4, P5) and a period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. The generator can't be ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, thus the 5th is flattened, and the 32/27 minor 3rd is sharpened. Therefore the 300¢ period is narrower than a m3, and must be a vm3. ~~Both Diminished[4x2] modes:~~

'''[[Octatonic_scale|Diminished]] aka Quadgu''' has pergen (P8/4, P5) and a period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. The generator can't be ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, thus the 5th is flattened, and the 32/27 minor 3rd is sharpened. Therefore the 300¢ period is narrower than a m3, and must be a vm3.

{| class="wikitable"

|+ Diminished[4x2]/Quadgu[4x2] modes

|-

! | scale name

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Using ~25/24 as the generator yields the same scales and mode numbers. 1st Diminished[4x2] would have genchains C – ^^C#, vEb – ^E, ^^F# – G and ^A – vBb, just like the scale.

[[Blackwood|'''Blackwood''']] '''aka ~~5-edo~~+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period. ~~Both Blackwood modes, using ups and downs:~~

[[Blackwood|'''Blackwood''']] '''aka Sawa+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.

{| class="wikitable"

|+ Blackwood[5x2]/5edo+ya[5x2]

|-

! | scale name

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|-

| | 2nd Blackwood[5x2]

|2nd ~~5-edo~~+ya[5x2]

|2nd 5edo+ya[5x2]

| | sL-sL-sL-sL-sL

| | C ^C D ^Eb F ^F G ^Ab A ^Bb C

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|}

=Other rank-2 scales=

== 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. But 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.

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. But 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. ~~Meantone examples:~~

{| class="wikitable"

|+ Non-MOS/MODMOS Meantone examples

|-

! | scale

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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=

== 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.

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and these standard modes:

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

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

* L 2nd 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 3rd Meantone[5] = L M N J K L~~ ~~

* L 4th 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

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

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

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

* L 4th 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. Below is a Sensi[8] MOS and a Sensi[8] MODMOS, each 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.

{| class="wikitable"

|+ Sensi[8]/Sepgu[8] MOS and MODMOS examples

|-

! | notation

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| | G# * E# H '''C''' F A D * B

|}

Heptatonic fifth-based notation:

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{| class="wikitable"

|+ Examples of MODMOS scales of split-octave temperaments

|-

! | scale name

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|}

=~~Rationale~~=

== Generalization to temperament-agnostic MOS scales ==

[[:Category:Abstract MOS patterns|Abstract MOS patterns]] like 5L 3s are not specific temperaments in which specific commas vanish. Thus there are no ratios other than the octave 2/1 (or more generally the equave 3/1 or whatever). Genchain mode numbers can be applied to these patterns. For example, 5L 3s has a generator in the 450-480¢ range. The "[8]" is redundant, so we drop it to get

* 1st 5L 3s = LLsLLsLs

* 2nd 5L 3s = LLsLsLLs

* 3rd 5L 3s = LsLLsLLs

* etc.

The modes of the sister MOS 3L 5s are the same, just exchange L and s:

* 1st 3L 5s = ssLssLsL

* 2nd 3L 5s = ssLsLssL

* 3rd 3L 5s = sLssLssL

* etc.

For a MOS pattern with a fifth-sized generator, the fifth is still prioritized over the fourth. Otherwise the generator is the mingen.

== Rationale ==

'''Why not number the modes in the order they occur in the scale?'''

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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''')." -- [https://en.wikipedia.org/wiki/Pythagorean_tuning]

"Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio ''3:2'' (i.e., the untempered perfect ''fifth'')." — [https://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]

"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]

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

"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect ''fifths''." — [https://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]

"In this system the perfect ''fifth'' is flattened by one quarter of a syntonic comma." — [https://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." -- [https://en.wikipedia.org/wiki/Well_temperament]

"The term "well temperament" or "good temperament" usually means some sort of irregular temperament in which the tempered ''fifths'' are of different sizes." — [https://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.

"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?'''

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See below.

'''Why not just use ~~Modal~~ UDP notation?'''

'''Why not just use modal UDP notation?'''

One problem with [[~~Modal_UDP_Notation|~~modal 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.

One problem with [[modal UDP notation]] 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.

{| class="wikitable"

|+ Comparison of meantone MOS scales in UDP and Mode Numbers

|-

! | scale

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{| class="wikitable"

|+ Comparison of various meantone scales in UDP and Mode Numbers

|-

! | scale

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| | C G D A E B F# C# G# D# A#

|-

| | Meantone[12] if generator ~~<~~ 700¢

| | 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¢

| | Meantone[12] if generator > 700¢

| | C G D A E B F# C# G# D# A# E#

| style="text-align:center;" | C G D A E B F# C# G# D# A# E#

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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 [[Wikipedia:Zero-based numbering|zero-based numbering]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.

~~=Other Naming Methods=~~

~~Modal UDP notation is explained here: [[Modal UDP Notation]]~~

~~Jake Freivald has his own method of naming modes, explained here:~~

[[~~Naming_Rank-2_Scales#~~Jake Freivald ~~method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method~~]]

== See also ==

* [[Comparison of mode notation systems]]

* [[Modal UDP notation]]

* [[Jake Freivald's mode numbering system]]

[[Category:Mode]]

[[Category:Rank 2]]

~~[[category:naming]]~~

@@ Line 1: / Line 1: @@
-__FORCETOC__
+'''Genchain mode numbering''' ('''GMN''' for short) provides 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. It also applies to abstract MOS patterns like 5L 3s.
-=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|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.
+This mode notation system was designed by [[Kite Giedraitis]].
-[[MOSScales|MOS scales]] are formed from a segment of the [[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.
+== MOS scales ==
+[[MOS scale]]s are formed from a segment of the [[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|'''Meantone''']][7], using ~3/2 as the generator. The Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.
+For example, here are all the modes of [[Meantone]][7], using ~3/2 as the generator. On this page, the Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.
 {| class="wikitable"
+|+ Meantone[7] modes on white keys
 |-
 ! | old scale name
@@ Line 58: / Line 59: @@
 | | F C G D A E <u>'''B'''</u>
 |}
-th 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".
+th 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 term GMN can also be read as genchain mode <u>number</u>, and can refer to the numbers 1st, 2nd, 3rd etc., as in "Dorian's GMN is 4".
 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).
 {| class="wikitable"
+|+ Meantone[7] modes in C
 |-
 ! | old scale name
@@ Line 113: / Line 116: @@
 |}
-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"). '''<u>Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]</u>.''' There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the "Rationale" section below.
+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|§&nbsp;Rationale]]). Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]. There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the [[#Rationale|§&nbsp;Rationale]] section below.
 Pentatonic meantone scales:
 {| class="wikitable"
+|+ Meantone[5] modes
 |-
 ! | old scale name
@@ Line 159: / Line 163: @@
 {| class="wikitable"
+|+ Meantone[12] modes
 |-
 ! | scale name
@@ Line 205: / Line 210: @@
 | |
 |}
-'''[[Porcupine]] aka Triyo''' has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2. The enharmonic interval is v<sup>3</sup>A1. Because the generator is a 2nd, the genchain resembles the scale. Porcupine[7] modes, using [[Ups and Downs Notation|ups and downs notation]]:
+[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is v<sup>3</sup>A1. Because the generator is a 2nd, the genchain resembles the scale.
 {| class="wikitable"
+|+ Porcupine[7]/Triyo[7] modes
 |-
 ! | scale name
@@ Line 257: / Line 264: @@
 | | D vE ^F G vA ^Bb <u>'''C'''</u>
 |}
-'''[[Sensi]] aka Sepgu''' has pergen (P8, WWP5/7). The ~9/7 generator is both a ^<sup>3</sup>d4 and a v<sup>4</sup>A3, and the enharmonic interval is ^<sup>7</sup>dd2. Sensi[8] modes:
+[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^<sup>3</sup>d4 and a v<sup>4</sup>A3, and the [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is ^<sup>7</sup>dd2.
 {| class="wikitable"
+|+ Sensi[8]/Sepgu[8] modes
 |-
 ! | scale name
@@ Line 314: / Line 323: @@
 | | F v<sup>4</sup>A# vD ^^Gb vvB ^Eb v<sup>3</sup>G# <u>'''C'''</u>
 |}
-=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 when the generator is chroma-positive, and N steps backwards when it isn't. This ensures a higher pitch. (Note that Meantone[5] is chroma-negative, more on this below.) However, an exception is made for superflat edos like 16edo when the generator is a 3/2 fifth, because in those edos, G# is actually flat of G. Another exception is when the generator is close to the "tipping point" between chroma-positive and chroma-negative. A good alternative in these and other situations, including non-heptatonic and non-fifth-generated scales, is to use + for forwards in the genchain and - for backwards, as in 5th Meantone[7] +6 +7.
+== MODMOS scales ==
+[[MODMOS scale]]s 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 when the generator is chroma-positive, and N steps backwards when it isn't. This ensures a higher pitch. (Note that Meantone[5] is chroma-negative, more on this below.) However, an exception is made for superflat edos like 16edo when the generator is a 3/2 fifth, because in those edos, G# is actually flat of G. Another exception is when the generator is close to the "tipping point" between chroma-positive and chroma-negative. A good alternative in these and other situations, including non-heptatonic and non-fifth-generated scales, is to use + for forwards in the genchain and - for backwards, as in 5th Meantone[7] +6 +7.
 A MODMOS scale can have alternate names. The ascending melodic minor scale could also be called 2nd Meantone[7] b3 (major scale with a minor 3rd), or as 4th Meantone[7] #7 (dorian with a major 7th).
-'''Meantone''' MODMOS scales, with alternative names in italics and parentheses. Alternatives that have more alterations than the original aren't listed:
+Meantone MODMOS scales, with alternative names in italics and parentheses. Alternatives that have more alterations than the original aren't listed:
 {| class="wikitable"
+|+ Meantone[7] MODMOS scale examples
 |-
 ! | old scale name
@@ Line 390: / Line 400: @@
 | | Bb F * G D <u>'''A'''</u> E * * C#
 |}
 As can be seen from the genchains, or from the LMs 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.
@@ Line 396: / Line 407: @@
 Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:
-st Meantone[7] #2: C D# E F# G A B C <br>
+* 1st Meantone[7] #2: C D# E F# G A B C
-nd Meantone[7] #:5 C D E F G# A B C<br>
+* 2nd Meantone[7] #5: C D E F G# A B C
-th Meantone[7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern, 7th mode not 3rd mode)<br>
+* 7th Meantone[7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern, 7th mode not 3rd mode)
-th Meantone[7] #4: C D Eb F# G A Bb C<br>
+* 4th Meantone[7] #4: C D Eb F# G A Bb C
-th Meantone[7] #7: C D Eb F G Ab B C (harmonic minor)<br>
+* 5th Meantone[7] #7: C D Eb F G Ab B C (harmonic minor)
-th Meantone[7] #3: C Db E F G Ab Bb C (phrygian dominant)<br>
+* 6th Meantone[7] #3: C Db E F G Ab Bb C (phrygian dominant)
-th Meantone[7] #6: C Db Eb F Gb A Bb C
+* 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.
@@ Line 408: / Line 419: @@
 Ascending melodic minor modes:
-st Meantone[7] #5: C D E F# G# A B C<br>
+* 1st Meantone[7] #5: C D E F# G# A B C
-th Meantone[7] b4: C Db Eb Fb Gb Ab Bb C (avoid "2nd Meantone[7] #1")<br>
+* 7th Meantone[7] b4: C Db Eb Fb Gb Ab Bb C (avoid "2nd Meantone[7] #1")
-rd Meantone[7] #4: C D E F# G A Bb C<br>
+* 3rd Meantone[7] #4: C D E F# G A Bb C
-th Meantone[7] #7: C D Eb F G A B C<br>
+* 4th Meantone[7] #7: C D Eb F G A B C
-th Meantone[7] #3: C D E F G Ab Bb C<br>
+* 5th Meantone[7] #3: C D E F G Ab Bb C
-th Meantone[7] #6: C Db Eb F G A Bb C<br>
+* 6th Meantone[7] #6: C Db Eb F G A Bb C
-th Meantone[7] #2: C D Eb F Gb Ab Bb C
+* 7th Meantone[7] #2: C D Eb F Gb Ab Bb C
-'''Porcupine[7] MODMOS scales''', not including alternative names because they all modify the 3rd or the 5th.
+Porcupine[7] aka Triyo[7] MODMOS scales, not including alternative names because they all modify the 3rd or the 5th.
 {| class="wikitable"
+|+ Porcupine[7]/Triyo[7] MODMOS scale examples
 |-
 ! | scale name
@@ Line 486: / Line 498: @@
 | | D vE * G vA ^Bb <u>'''C'''</u> * * F
 |}
-=Temperaments with split octaves=
-If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel, using ups and downs. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice must be complete. The number in the brackets becomes two numbers, and the Ls pattern is broken up into periods with hyphens.
+== Temperaments with split octaves ==
+If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. Using ups and downs notation, each genchain has its own height. There is a plain one, an up one, perhaps a down one, etc. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice generated by the 5th and the up must be complete. The number in the brackets becomes two numbers, and the Ls pattern as written here is grouped by period, using hyphens.
-'''[[Srutal]] aka Diaschismatic aka Sagugu''' has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.
+[[Srutal]] aka Diaschismatic aka Sagugu has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.
 {| class="wikitable"
+|+ Srutal[2x5]/Sagugu[2x5] modes
 |-
 ! | scale name
@@ Line 539: / Line 552: @@
 Srutal's period is written as a vA4, but could instead be written as an ^d5. The generator is written as a P5. 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. The generator could instead be written as ~16/15 (3/2 minus a period), because that would still create the same mode numbers and thus the same scale names. The first genchain of 1st Srutal[2x5] would be C vC# D vD# E, just like the first half of the scale.
-'''[[Augmented family|Augmented]] aka Trigu''' has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).
+[[Augmented family|Augmented]] aka Trigu has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).
-Augmented[3x3] modes:
 {| class="wikitable"
+|+ Augmented[3x3]/Trigu[3x3] modes
 |-
 ! | scale name
@@ Line 578: / Line 590: @@
 |}
-'''[[Octatonic_scale|Diminished]] aka Quadgu''' has pergen (P8/4, P5) and a period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. The generator can't be ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, thus the 5th is flattened, and the 32/27 minor 3rd is sharpened. Therefore the 300¢ period is narrower than a m3, and must be a vm3. Both Diminished[4x2] modes:
+'''[[Octatonic_scale|Diminished]] aka Quadgu''' has pergen (P8/4, P5) and a period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. The generator can't be ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, thus the 5th is flattened, and the 32/27 minor 3rd is sharpened. Therefore the 300¢ period is narrower than a m3, and must be a vm3.
 {| class="wikitable"
+|+ Diminished[4x2]/Quadgu[4x2] modes
 |-
 ! | scale name
@@ Line 612: / Line 625: @@
 Using ~25/24 as the generator yields the same scales and mode numbers. 1st Diminished[4x2] would have genchains  C – ^^C#, vEb – ^E, ^^F# – G and ^A – vBb, just like the scale.
-[[Blackwood|'''Blackwood''']] '''aka 5-edo+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period. Both Blackwood modes, using ups and downs:
+[[Blackwood|'''Blackwood''']] '''aka Sawa+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.
 {| class="wikitable"
+|+ Blackwood[5x2]/5edo+ya[5x2]
 |-
 ! | scale name
@@ Line 629: / Line 643: @@
 |-
 | | 2nd Blackwood[5x2]
-|2nd 5-edo+ya[5x2]
+|2nd 5edo+ya[5x2]
 | | sL-sL-sL-sL-sL
 | | C ^C D ^Eb F ^F G ^Ab A ^Bb C
@@ Line 635: / Line 649: @@
 |}
-=Other rank-2 scales=
+== 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. But 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.
-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. But 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. Meantone examples:
 {| class="wikitable"
+|+ Non-MOS/MODMOS Meantone examples
 |-
 ! | scale
@@ Line 721: / Line 735: @@
 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=
+== 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.
@@ Line 736: / Line 749: @@
 and these standard modes:
+* L 1st Meantone[5] = L M +N J +K L
-L 1st Meantone[5] = L M +N J +K L<br>
+* L 2nd Meantone[5] = L M N J +K L
-L 2nd Meantone[5] = L M N J +K L<br>
+* L 3rd Meantone[5] = L M N J K L
-L 3rd Meantone[5] = L M N J K L<br>
+* L 4th Meantone[5] = L -M N J K L
-L 4th Meantone[5] = L -M N J K L<br>
+* L 5th 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 3rd Meantone[5] add -2, +5<br>
+* L 2nd Meantone[5] add -2, -5
-L 2nd Meantone[5] add -2, -5<br>
+* L 4th 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. Below is a Sensi[8] MOS and a Sensi[8] MODMOS, each 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.
 {| class="wikitable"
+|+ Sensi[8]/Sepgu[8] MOS and MODMOS examples
 |-
 ! | notation
@@ Line 788: / Line 800: @@
 | | G# * E# H <u>'''C'''</u> F A D * B
 |}
 Heptatonic fifth-based notation:
@@ Line 801: / Line 814: @@
 {| class="wikitable"
+|+ Examples of MODMOS scales of split-octave temperaments
 |-
 ! | scale name
@@ Line 831: / Line 845: @@
 |}
-=Rationale=
+== Generalization to temperament-agnostic MOS scales ==
+[[:Category:Abstract MOS patterns|Abstract MOS patterns]] like 5L 3s are not specific temperaments in which specific commas vanish. Thus there are no ratios other than the octave 2/1 (or more generally the equave 3/1 or whatever). Genchain mode numbers can be applied to these patterns. For example, 5L 3s has a generator in the 450-480¢ range. The "[8]" is redundant, so we drop it to get
+* 1st 5L 3s = LLsLLsLs
+* 2nd 5L 3s = LLsLsLLs
+* 3rd 5L 3s = LsLLsLLs
+* etc.
+The modes of the sister MOS 3L 5s are the same, just exchange L and s:
+* 1st 3L 5s = ssLssLsL
+* 2nd 3L 5s = ssLsLssL
+* 3rd 3L 5s = sLssLssL
+* etc.
+For a MOS pattern with a fifth-sized generator, the fifth is still prioritized over the fourth. Otherwise the generator is the mingen.
+== Rationale ==
 '''Why not number the modes in the order they occur in the scale?'''
@@ Line 845: / Line 874: @@
 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''')." -- [https://en.wikipedia.org/wiki/Pythagorean_tuning]
+"Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio ''3:2'' (i.e., the untempered perfect ''fifth'')." — [https://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]
+"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]
-"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect '''fifths'''." -- [https://en.wikipedia.org/wiki/Meantone_temperament]
+"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect ''fifths''." — [https://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]
+"In this system the perfect ''fifth'' is flattened by one quarter of a syntonic comma." — [https://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." -- [https://en.wikipedia.org/wiki/Well_temperament]
+"The term "well temperament" or "good temperament" usually means some sort of irregular temperament in which the tempered ''fifths'' are of different sizes." — [https://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 <u>wise</u> consistency, it wouldn't reduce memorization, because it's well known that the generator is historically 3/2.
+"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?'''
@@ Line 865: / Line 894: @@
 See below.
-'''Why not just use Modal UDP notation?'''
+'''Why not just use modal UDP notation?'''
-One problem with [[Modal_UDP_Notation|modal 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.
+One problem with [[modal UDP notation]] 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.
 {| class="wikitable"
+|+ Comparison of meantone MOS scales in UDP and Mode Numbers
 |-
 ! | scale
@@ Line 917: / Line 947: @@
 {| class="wikitable"
+|+ Comparison of various meantone scales in UDP and Mode Numbers
 |-
 ! | scale
@@ Line 962: / Line 993: @@
 | | C G D A E B F# C# G# D# A#
 |-
-| | Meantone[12] if generator &lt; 700¢
+| | 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 &gt; 700¢
+| | Meantone[12] if generator > 700¢
 | | C G D A E B F# C# G# D# A# E#
 | style="text-align:center;" | C G D A E B F# C# G# D# A# E#
@@ Line 977: / Line 1,008: @@
 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 [[Wikipedia:Zero-based numbering|zero-based numbering]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.
-=Other Naming Methods=
-Modal UDP notation is explained here: [[Modal UDP Notation]]
-Jake Freivald has his own method of naming modes, explained here:
-[[Naming_Rank-2_Scales#Jake Freivald method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method]]
+== See also ==
+* [[Comparison of mode notation systems]]
+* [[Modal UDP notation]]
+* [[Jake Freivald's mode numbering system]]
+[[Category:Mode]]
 [[Category:Rank 2]]
-[[category:naming]]