Comparison of mode notation systems: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
Several mode numbering or notation systems have been designed in an attempt to generalize the [[mode]]s of the [[5L 2s|diatonic]] scale. These systems aim to work for given scale, or at least any scale with certain properties such as [[MOS]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-23 02:51:05 UTC</tt>.<br>
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">Mode numbers are 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. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:
|| old scale name || new scale name || Ls 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**__ ||


These [[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 mode #1, the 2nd note is the tonic of mode #2, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as "fourth meantone heptatonic". If in D, as above, it would be "D fourth meantone heptatonic".
Here are the most common systems used proposed by the xenharmonic community:
* [[Modal UDP notation]];
* [[Genchain mode numbering]] (GMN);
* [[Jake Freivald's mode numbering system]].


The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The modes proceed from sharper (Lydian) to flatter (Locrian).
== Comparison tables ==
|| old scale name || new scale name || Ls pattern || example in C || ------------------- genchain --------------- ||
This section compares the different notations enumerated above. As an example, the seven diatonic modes (Ionian, Dorian, Phrygian, etc) will be used, first using the tempered meantone[7] scale and then using the abstract 5L 2s scale.
|| Lydian || 1st Meantone[7] || LLLs LLs || C D E F# G A B C ||&gt; __**C**__ G D A E B F# ||
{| class="wikitable sortable center-all"
|| Ionian (major) || 2nd Meantone[7] || LLsL LLs || C D E F G A B C ||&gt; F __**C**__ G D A E B ---- ||
|+ Comparison for meantone[7] modes
|| Mixolydian || 3rd Meantone[7] || LLsL LsL || C D E F G A Bb C ||&gt; Bb F __**C**__ G D A E ------- ||
!Mode name
|| Dorian || 4th Meantone[7] || LsLL LsL || C D Eb F G A Bb C || ------------- Eb Bb F __**C**__ G D A ||
!Step pattern
|| Aeolian (minor) || 5th Meantone[7] || LsLL sLL || C D Eb F G Ab Bb C || --------- Ab Eb Bb F __**C**__ G D ||
!Modal UDP notation
|| Phrygian || 6th Meantone[7] || sLLL sLL || C Db Eb F G Ab Bb C || ---- Db Ab Eb Bb F __**C**__ G ||
!Genchain mode numbering
|| Locrian || 7th Meantone[7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ ||
!Freivald's method
|-
|Lydian
|LLLsLLs
|meantone[7] <nowiki>6|0</nowiki>
|1st meantone[7]
|meantone[7] mode 4
|-
|Ionian
|LLsLLLs
|meantone[7] <nowiki>5|1</nowiki>
|2nd meantone[7]
|meantone[7] mode 1
|-
|Mixolydian
|LLsLLsL
|meantone[7] <nowiki>4|2</nowiki>
|3rd meantone[7]
|meantone[7] mode 5
|-
|Dorian
|LsLLLsL
|meantone[7] <nowiki>3|3</nowiki>
|4th meantone[7]
|meantone[7] mode 2
|-
|Aeolian
|LsLLsLL
|meantone[7] <nowiki>2|4</nowiki>
|5th meantone[7]
|meantone[7] mode 6
|-
|Phrygian
|sLLLsLL
|meantone[7] <nowiki>1|5</nowiki>
|6th meantone[7]
|meantone[7] mode 3
|-
|Locrian
|sLLsLLL
|meantone[7] <nowiki>0|6</nowiki>
|7th meantone[7]
|meantone[7] mode 7
|}


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. **Unlike modal UDP notation, the generator isn't always chroma-positive.** This is necessary to keep the same generator for different MOS's of the same [[Regular Temperaments|temperament]], which guarantees that the smaller MOS will always be a subset of the larger MOS.
{| class="wikitable sortable center-all"
|+ Comparison for 5L 2s modes
!Mode name
!Step pattern
!Modal UDP notation
!Genchain mode numbering
!Freivald's method
|-
|Lydian
|LLLsLLs
|5L 2s <nowiki>6|0</nowiki>
|1st 5L 2s
|5L 2s mode 4
|-
|Ionian
|LLsLLLs
|5L 2s <nowiki>5|1</nowiki>
|2nd 5L 2s
|5L 2s mode 1
|-
|Mixolydian
|LLsLLsL
|5L 2s <nowiki>4|2</nowiki>
|3rd 5L 2s
|5L 2s mode 5
|-
|Dorian
|LsLLLsL
|5L 2s <nowiki>3|3</nowiki>
|4th 5L 2s
|5L 2s mode 2
|-
|Aeolian
|LsLLsLL
|5L 2s <nowiki>2|4</nowiki>
|5th 5L 2s
|5L 2s mode 6
|-
|Phrygian
|sLLLsLL
|5L 2s <nowiki>1|5</nowiki>
|6th 5L 2s
|5L 2s mode 3
|-
|Locrian
|sLLsLLL
|5L 2s <nowiki>0|6</nowiki>
|7th 5L 2s
|5L 2s mode 7
|}


For example, Meantone[5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes proceed from flatter to sharper. Because Meantone[5] and Meantone[7]have the same generator, C 2nd Meantone[5] = CDFGAC is a subset of C 2nd Meantone[7] = CDEFGABC.
It should be noted that the MOS pattern 5L 2s is supported by multiple temperaments, and as such, the name "meantone" is not meant to exclusively describe 5L 2s.


Pentatonic meantone scales:
[[Category:Mode]]
|| old scale name || new scale name || Ls pattern || example in C || --------- genchain ------- ||
{{todo|expand}} <!-- add tables for other scales to show each system's strengths and weaknesses -->
|| major pentatonic || 1st Meantone[5] || ssL sL || C D E G A C ||&gt; __**C**__ G D A E ||
||= ??? || 2nd Meantone[5] || sLs sL || C D F G A C ||&gt; 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. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.
|| scale name || Ls pattern || example in C || genchain ||
|| 1st Meantone[12] || sLsLsLL 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] || sLsLLsL 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] || sLsLLsL 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] || sLLsLsL 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] || sLLsLsL 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] || LsLsLsL 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] || LsLsLLs 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. ||  ||  ||  ||
 
Sensi[8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)
|| scale name || Ls 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**__ ||
 
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 looks like the scale.
|| scale name || Ls 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**__ ||
 
 
==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**==
As in modal UDP notation, these are written as MOS scales with chromatic alterations. To find the scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the compacted genchain, then add the appropriate alterations.
|| old scale name || example in A || genchain (* marks a gap) || compacted genchain || new scale name ||
|| Harmonic minor || A B C D E F G# A || F C * D __**A**__ E B * * G# || F C G D __**A**__ E B || 5th Meantone[7] #7 ||
|| Melodic minor || A B C D E F# G# A || C * D __**A**__ E B F# * G# || F C G D __**A**__ E B || 5th Meantone[7] #6 #7 ||
||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7] b3 ||
|| Japanese pentatonic || A B C E F A || F C * * __**A**__ E B || __**A**__ E B F# C# || 1st Meantone[5] b3 b6 ||
|| (a mode of the above) || F A B C E F || __**F**__ C * * A E B || Ab Eb Bb __**F**__ C || 4th Meantone[5] #2 #3 #6 ||
 
The Japanese pentatonic has b6, not b5, because heptatonic scale degrees are used, even though the scale is pentatonic. The rationale for this is that the notation uses 7 letters, so the notation is still essentially heptatonic. In other words, F is the 5th note of the scale, but F is the 6th letter counting from the tonic A. If the notation used only 5 letters, perhaps H J K L M, the alteration would be written "b5".
 
The F mode of Japanese pentatonic alters three notes, not two, to avoid "b1 b5". Unfortunately, it's not apparent from the scale names that the last two examples are modes of each other.
 
 
==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**==
Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:
Eb -- Bb -- F --- C --- G
A --- E --- B --- F# -- C#
 
Or alternatively, using 16/15 not 3/2 as the generator:
Eb -- E --- F --- F# -- G
A --- Bb -- B --- C --- C#
 
Multiple genchains occur because rank-2 really is 2 dimensional, with a "genweb" running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. When the period is an octave, this octave-reduces to a single horizontal genchain. But shrutal has a genweb with vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth.
 
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 all other generators, even though that makes the generator larger than the period. Shrutal's generator is 3/2, not 16/15. However, 16/15 would still create the same mode numbers and thus the same scale names.
 
All five Shrutal[10] modes:
|| scale name || Ls 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 B C || F __**C**__ G D A || B F# C# G# D# ||
|| 3rd Shrutal[10] || ssLss-ssLss || C C# D E F F# G G# Bb B C || Bb F __**C**__ G D || E B F# C# G# ||
|| 4th Shrutal[10] || sLsss-sLsss || C C# Eb E F F# G A Bb B C || Eb Bb F __**C**__ G || A E B F# C# ||
|| 5th Shrutal[10] || Lssss-Lssss || C D Eb E F F# Ab A Bb B C || Ab Eb Bb F __**C**__ || D A E B F# ||
 
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4 = 386¢. L = 146¢ and s = 94¢. There are five short genchains. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.
|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || 5th chain ||
|| 1st Blackwood[10] || LsLsLs LsLs || 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] || sLsLsL sLsL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^ __**C**__ || Bb^ D || C^ E || Eb^ G || F^ A ||
 
 
==[[#How to name rank-2 scales-Non-MOS scales]]**__Non-MOS non-MODMOS scales__**==
Compact the genchain to remove any gaps via chromatic alterations. The mode number is derived from the compacted genchain. Examples:
 
C D E F F# G A B C, which has a genchain F __**C**__ G D A E B F#, and is named C 2nd Meantone[8].
 
C D E F F# G A Bb C, with genchain Bb F __**C**__ G D A E * F#. Alter Bb to get an unbroken genchain: F __**C**__ G D A E B F#. The scale is C 2nd Meantone[8] b7. Even though the scale is octotonic, heptatonic scale degrees are used for the alteration (b7 not b8), because only seven note names are used.
 
A B C D D# E F G G# A, with genchain F C G D __**A**__ E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D __**A**__ E B F# C# G# D#, giving the name A 3rd Meantone[9] b3 b7.
 
F G A C E F, with genchain __**F**__ C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6.
 
 
==__Explanation / Rationale__==
 
**__Why not number the modes in the order they occur in the scale?__**
This would order the modes Ionian, Dorian, Phrygian, etc. The advantage of scale-based numbering is that the modes are easier to find on one's instrument. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown.
 
__**Why make an exception for 3/2 vs 4/3 as the generator?**__
Because of 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 &lt;span class="mw-redirect"&gt;generator&lt;/span&gt; is the ratio __**&lt;span class="mw-redirect"&gt;3:2&lt;/span&gt;**__." [[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]]
 
__**Then why not choose the larger of the two generators?**__
Because the interval arithmetic is easier with smaller intervals. It's easier to compute the sum of stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain often is identical to the scale, simplifying mode numbering.
 
__**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. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction.
|| scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain ||
|| Meantone [2] || 3/2 || C G || 3/2 || C G ||
|| Meantone [3] || 4/3 || D G C || 3/2 || C G D ||
|| Meantone [5] || 4/3 || E A D G C || 3/2 || C G D A E ||
|| Meantone [7] || 3/2 || C G D A E B F# || 3/2 || C G D A E B F# ||
 
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 &lt; 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# ||
||&lt; Meantone [12] if generator &gt; 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 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. In other words the notation is overly tuning-dependent. 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.
 
A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. For example, 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). Also, as noted above, 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. 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.
 
 
 
 
Admins: Please delete the blank page at http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position</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&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Mode numbers are a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation"&gt;Modal UDP notation&lt;/a&gt;, it starts with the convention of using &lt;em&gt;some-temperament-name&lt;/em&gt;[&lt;em&gt;some-number&lt;/em&gt;] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;old scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;new scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example on white keys&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Lydian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1st Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LLLs LLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F G A B C D E F&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;/u&gt; C G D A E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Ionian (major)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2nd Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LLsL LLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D E F G A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Mixolydian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3rd Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LLsL LsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G A B C D E F G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C &lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Dorian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLL LsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D E F G A B C D&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C G &lt;u&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;/u&gt; A E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Aeolian (minor)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLL sLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A B C D E F G A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Phrygian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLLL sLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;E F G A B C D E&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C G D A &lt;u&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;/u&gt; B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Locrian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLLs LLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;B C D E F G A B&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C G D A E &lt;u&gt;&lt;strong&gt;B&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
These &lt;a class="wiki_link" href="/MOSScales"&gt;MOS scales&lt;/a&gt; are formed from a segment of the &lt;a class="wiki_link" href="/periods%20and%20generators"&gt;generator-chain&lt;/a&gt;, or genchain. The first note in the genchain is the tonic of mode #1, the 2nd note is the tonic of mode #2, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as &amp;quot;fourth meantone heptatonic&amp;quot;. If in D, as above, it would be &amp;quot;D fourth meantone heptatonic&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The modes proceed from sharper (Lydian) to flatter (Locrian).&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;old scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;new scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;------------------- genchain ---------------&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Lydian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1st Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LLLs LLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D E F# G A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Ionian (major)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2nd Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LLsL LLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D E F G A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B ----&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Mixolydian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3rd Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LLsL LsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D E F G A Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E -------&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Dorian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLL LsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Eb F G A Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;------------- Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Aeolian (minor)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLL sLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Eb F G Ab Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;--------- Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Phrygian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLLL sLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db Eb F G Ab Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;---- Db Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Locrian&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7th Meantone[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLLs LLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db Eb F Gb Ab Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gb Db Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
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. &lt;strong&gt;Unlike modal UDP notation, the generator isn't always chroma-positive.&lt;/strong&gt; This is necessary to keep the same generator for different MOS's of the same &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;temperament&lt;/a&gt;, which guarantees that the smaller MOS will always be a subset of the larger MOS.&lt;br /&gt;
&lt;br /&gt;
For example, Meantone[5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes proceed from flatter to sharper. Because Meantone[5] and Meantone[7]have the same generator, C 2nd Meantone[5] = CDFGAC is a subset of C 2nd Meantone[7] = CDEFGABC.&lt;br /&gt;
&lt;br /&gt;
Pentatonic meantone scales:&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;old scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;new scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;--------- genchain -------&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;major pentatonic&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1st Meantone[5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssL sL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D E G A C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2nd Meantone[5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLs sL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D F G A C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A --&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3rd Meantone[5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLs Ls&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D F G Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-------- Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;minor pentatonic&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4th Meantone[5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Lss Ls&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Eb F G Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;---- Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5th Meantone[5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsL ss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Eb F Ab Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
Chromatic meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1st Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLsLsLL sLsLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D D# E E# F# G G# A A# B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F# C# G# D# A# E#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2nd Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLsLLsL sLsLL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D D# E F F# G G# A A# B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F# C# G# D# A#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3rd Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLsLLsL sLLsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D D# E F F# G G# A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F# C# G# D#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4th Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLLsLsL sLLsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D Eb E F F# G G# A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F# C# G#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5th Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLLsLsL LsLsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D Eb E F F# G Ab A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F# C#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6th Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLsLsL LsLsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db D Eb E F F# G Ab A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Db Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7th Meantone[12]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLsLLs LsLsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db D Eb E F Gb G Ab A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gb Db Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;etc.&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
Sensi[8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1st Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssL ssL sL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db D# E# F# G A Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E# A Db F# Bb D# G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2nd Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssL sL ssL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db D# E# F# G# A Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E# A Db F# Bb D#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3rd Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sL ssL ssL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db Eb E# F# G# A Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Eb G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E# A Db F# Bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4th Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sL ssL sL s&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db Eb E# F# G# A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;B Eb G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E# A Db F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5th Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sL sL ssL s&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Db Eb E# Gb G# A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gb B Eb G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E# A Db&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6th Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Lss Lss Ls&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Eb E# Gb G# A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D Gb B Eb G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E# A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7th Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Lss Ls Lss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Eb E# Gb G# A# B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A# D Gb B Eb G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; E#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8th Sensi[8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls Lss Lss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Eb F Gb G# A# B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F A# D Gb B Eb G# &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
Porcupine[7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation"&gt;ups and downs notation&lt;/a&gt;.&lt;br /&gt;
Because the generator is a 2nd, the genchain looks like the scale.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1st Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssss ssL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Dv Eb^ F Gv Ab^ Bb C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Dv Eb^ F Gv Ab^ Bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2nd Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssss sLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Dv Eb^ F Gv Ab^ Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Bb^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Dv Eb^ F Gv Ab^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3rd Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssss Lss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Dv Eb^ F Gv Av Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Av Bb^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Dv Eb^ F Gv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4th Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sssL sss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Dv Eb^ F G Av Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G Av Bb^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Dv Eb^ F&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5th Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssLs sss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Dv Eb^ F^ G Av Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^ G Av Bb^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Dv Eb^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6th Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLss sss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C Dv Ev F^ G Av Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ev F^ G Av Bb^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Dv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7th Porcupine[7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Lsss sss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Ev F^ G Av Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D Ev F^ G Av Bb^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-MODMOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:8:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@How to name rank-2 scales-MODMOS scales&amp;quot; title=&amp;quot;Anchor: How to name rank-2 scales-MODMOS scales&amp;quot;/&amp;gt; --&gt;&lt;a name="How to name rank-2 scales-MODMOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:8 --&gt;&lt;strong&gt;&lt;u&gt;MODMOS scales&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
As in modal UDP notation, these are written as MOS scales with chromatic alterations. To find the scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the compacted genchain, then add the appropriate alterations.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;old scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;genchain (* marks a gap)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;compacted genchain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;new scale name&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Harmonic minor&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A B C D E F G# A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C * D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B * * G#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5th Meantone[7] #7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Melodic minor&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A B C D E F# G# A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C * D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B F# * G#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5th Meantone[7] #6 #7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B F# C# G#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2nd Meantone[7] b3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Japanese pentatonic&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A B C E F A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F C * * &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B F# C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1st Meantone[5] b3 b6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;(a mode of the above)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F A B C E F&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;/u&gt; C * * A E B&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ab Eb Bb &lt;u&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;/u&gt; C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4th Meantone[5] #2 #3 #6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
The Japanese pentatonic has b6, not b5, because heptatonic scale degrees are used, even though the scale is pentatonic. The rationale for this is that the notation uses 7 letters, so the notation is still essentially heptatonic. In other words, F is the 5th note of the scale, but F is the 6th letter counting from the tonic A. If the notation used only 5 letters, perhaps H J K L M, the alteration would be written &amp;quot;b5&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
The F mode of Japanese pentatonic alters three notes, not two, to avoid &amp;quot;b1 b5&amp;quot;. Unfortunately, it's not apparent from the scale names that the last two examples are modes of each other.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Fractional-octave periods"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:9:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@How to name rank-2 scales-Fractional-octave periods&amp;quot; title=&amp;quot;Anchor: How to name rank-2 scales-Fractional-octave periods&amp;quot;/&amp;gt; --&gt;&lt;a name="How to name rank-2 scales-Fractional-octave periods"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:9 --&gt;&lt;strong&gt;&lt;u&gt;Fractional-octave periods&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:&lt;br /&gt;
Eb -- Bb -- F --- C --- G&lt;br /&gt;
A --- E --- B --- F# -- C#&lt;br /&gt;
&lt;br /&gt;
Or alternatively, using 16/15 not 3/2 as the generator:&lt;br /&gt;
Eb -- E --- F --- F# -- G&lt;br /&gt;
A --- Bb -- B --- C --- C#&lt;br /&gt;
&lt;br /&gt;
Multiple genchains occur because rank-2 really is 2 dimensional, with a &amp;quot;genweb&amp;quot; running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. When the period is an octave, this octave-reduces to a single horizontal genchain. But shrutal has a genweb with vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
If the period is a fraction of an octave, 3/2 is still preferred over all other generators, even though that makes the generator larger than the period. Shrutal's generator is 3/2, not 16/15. However, 16/15 would still create the same mode numbers and thus the same scale names.&lt;br /&gt;
&lt;br /&gt;
All five Shrutal[10] modes:&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1st genchain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2nd genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1st Shrutal[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssssL-ssssL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D D# E F# G G# A A# C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F# C# G# D# A#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2nd Shrutal[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sssLs-sssLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D D# F F# G G# A B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;B F# C# G# D#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3rd Shrutal[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ssLss-ssLss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# D E F F# G G# Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;E B F# C# G#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4th Shrutal[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLsss-sLsss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C# Eb E F F# G A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A E B F# C#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5th Shrutal[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Lssss-Lssss&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C D Eb E F F# Ab A Bb B C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D A E B F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4 = 386¢. L = 146¢ and s = 94¢. There are five short genchains. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ls pattern&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1st chain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2nd chain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3rd chain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4th chain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5th chain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1st Blackwood[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;LsLsLs LsLs&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C#v D Ev F F#v G Av A Bv C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; Ev&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D F#v&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F Av&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G Bv&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A C#v&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2nd Blackwood[10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;sLsLsL sLsL&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C C^ D Eb^ E F^ G Ab^ A Bb^ C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ab^ &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Bb^ D&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C^ E&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Eb^ G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F^ A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Non-MOS non-MODMOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:10:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales&amp;quot; title=&amp;quot;Anchor: How to name rank-2 scales-Non-MOS scales&amp;quot;/&amp;gt; --&gt;&lt;a name="How to name rank-2 scales-Non-MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:10 --&gt;&lt;strong&gt;&lt;u&gt;Non-MOS non-MODMOS scales&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
Compact the genchain to remove any gaps via chromatic alterations. The mode number is derived from the compacted genchain. Examples:&lt;br /&gt;
&lt;br /&gt;
C D E F F# G A B C, which has a genchain F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F#, and is named C 2nd Meantone[8].&lt;br /&gt;
&lt;br /&gt;
C D E F F# G A Bb C, with genchain Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E * F#. Alter Bb to get an unbroken genchain: F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E B F#. The scale is C 2nd Meantone[8] b7. Even though the scale is octotonic, heptatonic scale degrees are used for the alteration (b7 not b8), because only seven note names are used.&lt;br /&gt;
&lt;br /&gt;
A B C D D# E F G G# A, with genchain F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B F# C# G# D#, giving the name A 3rd Meantone[9] b3 b7.&lt;br /&gt;
&lt;br /&gt;
F G A C E F, with genchain &lt;u&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;/u&gt; C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Explanation / Rationale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;u&gt;Explanation / Rationale&lt;/u&gt;&lt;/h2&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;Why not number the modes in the order they occur in the scale?&lt;/u&gt;&lt;/strong&gt;&lt;br /&gt;
This would order the modes Ionian, Dorian, Phrygian, etc. The advantage of scale-based numbering is that the modes are easier to find on one's instrument. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. &lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Why make an exception for 3/2 vs 4/3 as the generator?&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show [emphasis mine]:&lt;br /&gt;
&lt;br /&gt;
&amp;quot;Pythagorean tuning is a tuning of the syntonic temperament in which the &lt;span class="mw-redirect"&gt;generator&lt;/span&gt; is the ratio &lt;u&gt;&lt;strong&gt;&lt;span class="mw-redirect"&gt;3:2&lt;/span&gt;&lt;/strong&gt;&lt;/u&gt;.&amp;quot; &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow"&gt;en.wikipedia.org/wiki/Pythagorean_tuning&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&amp;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 &lt;u&gt;&lt;strong&gt;fifth&lt;/strong&gt;&lt;/u&gt;.&amp;quot; &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Syntonic_temperament" rel="nofollow"&gt;en.wikipedia.org/wiki/Syntonic_temperament&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&amp;quot;Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect &lt;u&gt;&lt;strong&gt;fifths&lt;/strong&gt;&lt;/u&gt;&amp;quot;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Meantone_temperament" rel="nofollow"&gt;en.wikipedia.org/wiki/Meantone_temperament&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&amp;quot;In this system the perfect &lt;u&gt;&lt;strong&gt;fifth&lt;/strong&gt;&lt;/u&gt; is flattened by one quarter of a syntonic comma&amp;quot; &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Quarter-comma_meantone" rel="nofollow"&gt;en.wikipedia.org/wiki/Quarter-comma_meantone&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Then why not choose the larger of the two generators?&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
Because the interval arithmetic is easier with smaller intervals. It's easier to compute the sum of stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain often is identical to the scale, simplifying mode numbering.&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Why not just use UDP notation?&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;UDP generator&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;UDP genchain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Mode Numbers generator&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Mode Numbers genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D G C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;E A D G C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
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.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;scale&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;UDP genchain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Mode Numbers genchain&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D G C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [4]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [5]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;E A D G C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [6]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G C D A E B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [7]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [8]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F# C#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [9]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F# C# G#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [10]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F# C# G# D#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [11]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;???&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F# C# G# D# A#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Meantone [12] if generator &amp;lt; 700¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;E# A# D# G# C# F# B E A D G C&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F# C# G# D# A# E#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: left;"&gt;Meantone [12] if generator &amp;gt; 700¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C G D A E B F# C# G# D# A# E#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C G D A E B F# C# G# D# A# E#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
An even larger problem is that 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. In other words the notation is overly tuning-dependent. 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 &amp;quot;Dominant 8|3&amp;quot; could mean either &amp;quot;4th Dominant[12]&amp;quot; or &amp;quot;9th Dominant[12]&amp;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.&lt;br /&gt;
&lt;br /&gt;
A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. For example, 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). Also, as noted above, 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. 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Admins: Please delete the blank page at &lt;!-- ws:start:WikiTextUrlRule:1383:http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position"&gt;http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:1383 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 08:44, 5 June 2025

Several mode numbering or notation systems have been designed in an attempt to generalize the modes of the diatonic scale. These systems aim to work for given scale, or at least any scale with certain properties such as MOS.

Here are the most common systems used proposed by the xenharmonic community:

Comparison tables

This section compares the different notations enumerated above. As an example, the seven diatonic modes (Ionian, Dorian, Phrygian, etc) will be used, first using the tempered meantone[7] scale and then using the abstract 5L 2s scale.

Comparison for meantone[7] modes
Mode name Step pattern Modal UDP notation Genchain mode numbering Freivald's method
Lydian LLLsLLs meantone[7] 6|0 1st meantone[7] meantone[7] mode 4
Ionian LLsLLLs meantone[7] 5|1 2nd meantone[7] meantone[7] mode 1
Mixolydian LLsLLsL meantone[7] 4|2 3rd meantone[7] meantone[7] mode 5
Dorian LsLLLsL meantone[7] 3|3 4th meantone[7] meantone[7] mode 2
Aeolian LsLLsLL meantone[7] 2|4 5th meantone[7] meantone[7] mode 6
Phrygian sLLLsLL meantone[7] 1|5 6th meantone[7] meantone[7] mode 3
Locrian sLLsLLL meantone[7] 0|6 7th meantone[7] meantone[7] mode 7
Comparison for 5L 2s modes
Mode name Step pattern Modal UDP notation Genchain mode numbering Freivald's method
Lydian LLLsLLs 5L 2s 6|0 1st 5L 2s 5L 2s mode 4
Ionian LLsLLLs 5L 2s 5|1 2nd 5L 2s 5L 2s mode 1
Mixolydian LLsLLsL 5L 2s 4|2 3rd 5L 2s 5L 2s mode 5
Dorian LsLLLsL 5L 2s 3|3 4th 5L 2s 5L 2s mode 2
Aeolian LsLLsLL 5L 2s 2|4 5th 5L 2s 5L 2s mode 6
Phrygian sLLLsLL 5L 2s 1|5 6th 5L 2s 5L 2s mode 3
Locrian sLLsLLL 5L 2s 0|6 7th 5L 2s 5L 2s mode 7

It should be noted that the MOS pattern 5L 2s is supported by multiple temperaments, and as such, the name "meantone" is not meant to exclusively describe 5L 2s.

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