Comparison of mode notation systems: Difference between revisions

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**Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] to create a generator-chain, and adds a way to number each mode uniquely. 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 the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as "fourth meantone heptatonic" or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".

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

The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. **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.

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.

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

Chromatic meantone scales. 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__**== 
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. For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. "+" and "-" are used to indicate the altered notes. For example, Meantone[7,+3,-6] means that the 3rd note in the compacted genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. "+" and "-" are preferred over "#" and "b" because in the case of a chroma-negative generator, "+" makes the note flatter.
|| old scale name || example in A || genchain || 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,+3] ||
|| Ascending 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,+1,+3] ||
||= " ||= " ||= " || C G D __**A**__ E B F# || 4th Meantone[7,+2] ||
||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7,-6] ||
|| Double harmonic minor || A B C D# E F G# A || F C * * __**A**__ E B * * G# D# || F C G D __**A**__ E B || 5th Meantone[7,+3,+4] ||
||= " ||= " ||= " || __**A**__ E B F# C# G# D# || 1st Meantone[7,-4,-5] ||
|| Double harmonic major || A Bb C# D E F G# A || Bb F * * D __**A**__ E * * C# G# || Bb F C G D __**A**__ E || 6th Meantone[7,+3,+4] ||
||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7,-4,-5] ||
|| <span class="mw-redirect">Hungarian gypsy </span>minor || A B C D# E F G A || F C G * __**A**__ E B * * * D# || F C G D __**A**__ E B || 5th Meantone[7,+4] ||
|| Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || Bb F C G D __**A**__ E || 6th Meantone[7,+3] ||
|| Japanese pentatonic || A B C E F A || F C * * __**A**__ E B || __**A**__ E B F# C# || 1st Meantone[5,-4,-5] ||
|| (a mode of the above) || F A B C E F || __**F**__ C * * A E B || Ab Eb Bb __**F**__ C || 4th Meantone[5,-4,-5] ||
Use the compacted genchain to find the mode number so that harmonic minor and melodic minor are both the 5th mode, just as standard minor is. Also so that two notes a (possibly aug or dim) fifth apart will still have adjacent mode numbers.


//A point to consider://

The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.


==[[#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 a just 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.


==[[#How to name rank-2 scales-Non-MOS scales]]__Explanation / Rationale__== 

===**__Why not number the modes in the order they occur in the scale?__**=== 
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The advantage of scale-based numbering is that the modes are easier to find on one's instrument.

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

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

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

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

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

===__**Then why not always choose the larger of the two generators?**__=== 
Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)

===__Why not always choose the chroma-positive generator?__=== 
See below.

===__**Why not just use UDP notation?**__=== 
One problem with 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 < 700¢ || E# A# D# G# C# F# B E A D G C || C G D A E B F# C# G# D# A# E# ||
||< Meantone [12] if generator > 700¢ || C G D A E B F# C# G# D# A# E# ||= C G D A E B F# C# G# D# A# E# ||

An even larger problem is that 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

Original HTML content:

<html><head><title>Naming Rank-2 Scales</title></head><body><strong>This page is a work in progress...</strong><br />
<br />
<strong>Mode numbers</strong> provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br />


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

<br />
These <a class="wiki_link" href="/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="/periods%20and%20generators">generator-chain</a>, or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as &quot;fourth meantone heptatonic&quot; or possibly &quot;fourth meantone seven&quot;. If in D, as above, it would be &quot;D fourth meantone heptatonic&quot;.<br />
<br />
The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The modes proceed from sharper (Lydian) to flatter (Locrian).<br />


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

<br />
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. <strong>Unlike modal UDP notation, the generator isn't always chroma-positive.</strong> This is necessary to keep the same generator for different MOS's of the same <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>, which guarantees that the smaller MOS will always be a subset of the larger MOS.<br />
<br />
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.<br />
<br />
Pentatonic meantone scales:<br />


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

<br />
Chromatic meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>Ls pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>genchain<br />
</td>
    </tr>
    <tr>
        <td>1st Meantone[12]<br />
</td>
        <td>sLsLsLL sLsLL<br />
</td>
        <td>C C# D D# E E# F# G G# A A# B C<br />
</td>
        <td><u><strong>C</strong></u> G D A E B F# C# G# D# A# E#<br />
</td>
    </tr>
    <tr>
        <td>2nd Meantone[12]<br />
</td>
        <td>sLsLLsL sLsLL<br />
</td>
        <td>C C# D D# E F F# G G# A A# B C<br />
</td>
        <td>F <u><strong>C</strong></u> G D A E B F# C# G# D# A#<br />
</td>
    </tr>
    <tr>
        <td>3rd Meantone[12]<br />
</td>
        <td>sLsLLsL sLLsL<br />
</td>
        <td>C C# D D# E F F# G G# A Bb B C<br />
</td>
        <td>Bb F <u><strong>C</strong></u> G D A E B F# C# G# D#<br />
</td>
    </tr>
    <tr>
        <td>4th Meantone[12]<br />
</td>
        <td>sLLsLsL sLLsL<br />
</td>
        <td>C C# D Eb E F F# G G# A Bb B C<br />
</td>
        <td>Eb Bb F <u><strong>C</strong></u> G D A E B F# C# G#<br />
</td>
    </tr>
    <tr>
        <td>5th Meantone[12]<br />
</td>
        <td>sLLsLsL LsLsL<br />
</td>
        <td>C C# D Eb E F F# G Ab A Bb B C<br />
</td>
        <td>Ab Eb Bb F <u><strong>C</strong></u> G D A E B F# C#<br />
</td>
    </tr>
    <tr>
        <td>6th Meantone[12]<br />
</td>
        <td>LsLsLsL LsLsL<br />
</td>
        <td>C Db D Eb E F F# G Ab A Bb B C<br />
</td>
        <td>Db Ab Eb Bb F <u><strong>C</strong></u> G D A E B F#<br />
</td>
    </tr>
    <tr>
        <td>7th Meantone[12]<br />
</td>
        <td>LsLsLLs LsLsL<br />
</td>
        <td>C Db D Eb E F Gb G Ab A Bb B C<br />
</td>
        <td>Gb Db Ab Eb Bb F <u><strong>C</strong></u> G D A E B<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">etc.<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
Sensi[8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)<br />


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

<br />
Porcupine[7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using <a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation">ups and downs notation</a>.<br />
Because the generator is a 2nd, the genchain looks like the scale.<br />


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

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule:18:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-MODMOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-MODMOS scales&quot;/&gt; --><a name="How to name rank-2 scales-MODMOS scales"></a><!-- ws:end:WikiTextAnchorRule:18 --><strong><u>MODMOS scales</u></strong></h2>
 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 <u>compacted</u> genchain. For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. &quot;+&quot; and &quot;-&quot; are used to indicate the altered notes. For example, Meantone[7,+3,-6] means that the 3rd note in the compacted genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. &quot;+&quot; and &quot;-&quot; are preferred over &quot;#&quot; and &quot;b&quot; because in the case of a chroma-negative generator, &quot;+&quot; makes the note flatter.<br />


<table class="wiki_table">
    <tr>
        <td>old scale name<br />
</td>
        <td>example in A<br />
</td>
        <td>genchain<br />
</td>
        <td>compacted genchain<br />
</td>
        <td>new scale name<br />
</td>
    </tr>
    <tr>
        <td>Harmonic minor<br />
</td>
        <td>A B C D E F G# A<br />
</td>
        <td>F C * D <u><strong>A</strong></u> E B * * G#<br />
</td>
        <td>F C G D <u><strong>A</strong></u> E B<br />
</td>
        <td>5th Meantone[7,+3]<br />
</td>
    </tr>
    <tr>
        <td>Ascending melodic minor<br />
</td>
        <td>A B C D E F# G# A<br />
</td>
        <td>C * D <u><strong>A</strong></u> E B F# * G#<br />
</td>
        <td>F C G D <u><strong>A</strong></u> E B<br />
</td>
        <td>5th Meantone[7,+1,+3]<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>C G D <u><strong>A</strong></u> E B F#<br />
</td>
        <td>4th Meantone[7,+2]<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>D <u><strong>A</strong></u> E B F# C# G#<br />
</td>
        <td>2nd Meantone[7,-6]<br />
</td>
    </tr>
    <tr>
        <td>Double harmonic minor<br />
</td>
        <td>A B C D# E F G# A<br />
</td>
        <td>F C * * <u><strong>A</strong></u> E B * * G# D#<br />
</td>
        <td>F C G D <u><strong>A</strong></u> E B<br />
</td>
        <td>5th Meantone[7,+3,+4]<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td><u><strong>A</strong></u> E B F# C# G# D#<br />
</td>
        <td>1st Meantone[7,-4,-5]<br />
</td>
    </tr>
    <tr>
        <td>Double harmonic major<br />
</td>
        <td>A Bb C# D E F G# A<br />
</td>
        <td>Bb F * * D <u><strong>A</strong></u> E * * C# G#<br />
</td>
        <td>Bb F C G D <u><strong>A</strong></u> E<br />
</td>
        <td>6th Meantone[7,+3,+4]<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td>D <u><strong>A</strong></u> E B F# C# G#<br />
</td>
        <td>2nd Meantone[7,-4,-5]<br />
</td>
    </tr>
    <tr>
        <td><span class="mw-redirect">Hungarian gypsy </span>minor<br />
</td>
        <td>A B C D# E F G A<br />
</td>
        <td>F C G * <u><strong>A</strong></u> E B * * * D#<br />
</td>
        <td>F C G D <u><strong>A</strong></u> E B<br />
</td>
        <td>5th Meantone[7,+4]<br />
</td>
    </tr>
    <tr>
        <td>Phrygian dominant<br />
</td>
        <td>A Bb C# D E F G A<br />
</td>
        <td>Bb F * G D <u><strong>A</strong></u> E * * C#<br />
</td>
        <td>Bb F C G D <u><strong>A</strong></u> E<br />
</td>
        <td>6th Meantone[7,+3]<br />
</td>
    </tr>
    <tr>
        <td>Japanese pentatonic<br />
</td>
        <td>A B C E F A<br />
</td>
        <td>F C * * <u><strong>A</strong></u> E B<br />
</td>
        <td><u><strong>A</strong></u> E B F# C#<br />
</td>
        <td>1st Meantone[5,-4,-5]<br />
</td>
    </tr>
    <tr>
        <td>(a mode of the above)<br />
</td>
        <td>F A B C E F<br />
</td>
        <td><u><strong>F</strong></u> C * * A E B<br />
</td>
        <td>Ab Eb Bb <u><strong>F</strong></u> C<br />
</td>
        <td>4th Meantone[5,-4,-5]<br />
</td>
    </tr>
</table>

Use the compacted genchain to find the mode number so that harmonic minor and melodic minor are both the 5th mode, just as standard minor is. Also so that two notes a (possibly aug or dim) fifth apart will still have adjacent mode numbers.<br />
<br />
<br />
<em>A point to consider:</em><br />
<br />
The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:19:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-Fractional-octave periods&quot; title=&quot;Anchor: How to name rank-2 scales-Fractional-octave periods&quot;/&gt; --><a name="How to name rank-2 scales-Fractional-octave periods"></a><!-- ws:end:WikiTextAnchorRule:19 --><strong><u>Fractional-octave periods</u></strong></h2>
 Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br />
Eb -- Bb -- F --- C --- G<br />
A --- E --- B --- F# -- C#<br />
<br />
Or alternatively, using 16/15 not 3/2 as the generator:<br />
Eb -- E --- F --- F# -- G<br />
A --- Bb -- B --- C --- C#<br />
<br />
Multiple genchains occur because rank-2 really is 2 dimensional, with a &quot;genweb&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.<br />
<br />
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.<br />
<br />
If the period is a fraction of an octave, 3/2 is still preferred over 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.<br />
<br />
All five Shrutal[10] modes:<br />


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

<br />
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. 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.<br />


<table class="wiki_table">
    <tr>
        <td>scale name<br />
</td>
        <td>Ls pattern<br />
</td>
        <td>example in C<br />
</td>
        <td>1st chain<br />
</td>
        <td>2nd chain<br />
</td>
        <td>3rd chain<br />
</td>
        <td>4th chain<br />
</td>
        <td>5th chain<br />
</td>
    </tr>
    <tr>
        <td>1st Blackwood[10]<br />
</td>
        <td>LsLsLs LsLs<br />
</td>
        <td>C C#v D Ev F F#v G Av A Bv C<br />
</td>
        <td style="text-align: center;"><u><strong>C</strong></u> Ev<br />
</td>
        <td>D F#v<br />
</td>
        <td>F Av<br />
</td>
        <td>G Bv<br />
</td>
        <td>A C#v<br />
</td>
    </tr>
    <tr>
        <td>2nd Blackwood[10]<br />
</td>
        <td>sLsLsL sLsL<br />
</td>
        <td>C C^ D Eb^ E F^ G Ab^ A Bb^ C<br />
</td>
        <td style="text-align: center;">Ab^ <u><strong>C</strong></u><br />
</td>
        <td>Bb^ D<br />
</td>
        <td>C^ E<br />
</td>
        <td>Eb^ G<br />
</td>
        <td>F^ A<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:20:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-Non-MOS scales&quot;/&gt; --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:20 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2>
 Compact the genchain to remove any gaps via chromatic alterations. The mode number is derived from the compacted genchain. Examples:<br />
<br />
C D E F F# G A B C, which has a genchain F <u><strong>C</strong></u> G D A E B F#, and is named C 2nd Meantone[8].<br />
<br />
C D E F F# G A Bb C, with genchain Bb F <u><strong>C</strong></u> G D A E * F#. Alter Bb to get an unbroken genchain: F <u><strong>C</strong></u> 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.<br />
<br />
A B C D D# E F G G# A, with genchain F C G D <u><strong>A</strong></u> E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D <u><strong>A</strong></u> E B F# C# G# D#, giving the name A 3rd Meantone[9] b3 b7.<br />
<br />
F G A C E F, with genchain <u><strong>F</strong></u> C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:21:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-Non-MOS scales&quot;/&gt; --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:21 --><u>Explanation / Rationale</u></h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3>
 Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The advantage of scale-based numbering is that the modes are easier to find on one's instrument.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:10 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3>
 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]:<br />
<br />
&quot;Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <u><strong><span class="mw-redirect">3:2</span></strong></u> (i.e., the untempered perfect <u><strong>fifth</strong></u>).&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</a><br />
<br />
&quot;The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect <u><strong>fifth</strong></u>.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Syntonic_temperament" rel="nofollow">en.wikipedia.org/wiki/Syntonic_temperament</a><br />
<br />
&quot;Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect <u><strong>fifths</strong></u>.&quot; --<br />
<a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Meantone_temperament" rel="nofollow">en.wikipedia.org/wiki/Meantone_temperament</a><br />
<br />
&quot;In this system the perfect <u><strong>fifth</strong></u> is flattened by one quarter of a syntonic comma.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Quarter-comma_meantone" rel="nofollow">en.wikipedia.org/wiki/Quarter-comma_meantone</a><br />
<br />
&quot;The term &quot;well temperament&quot; or &quot;good temperament&quot; usually means some sort of <span class="new">irregular temperament</span> in which the tempered <u><strong>fifths</strong></u> are of different sizes.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Well_temperament" rel="nofollow">en.wikipedia.org/wiki/Well_temperament</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:12 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3>
 Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Explanation / Rationale-Why not always choose the chroma-positive generator?"></a><!-- ws:end:WikiTextHeadingRule:14 --><u>Why not always choose the chroma-positive generator?</u></h3>
 See below.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Why not just use UDP notation?</strong></u></h3>
 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.<br />


<table class="wiki_table">
    <tr>
        <td>scale<br />
</td>
        <td>UDP generator<br />
</td>
        <td>UDP genchain<br />
</td>
        <td>Mode Numbers generator<br />
</td>
        <td>Mode Numbers genchain<br />
</td>
    </tr>
    <tr>
        <td>Meantone [2]<br />
</td>
        <td>3/2<br />
</td>
        <td>C G<br />
</td>
        <td>3/2<br />
</td>
        <td>C G<br />
</td>
    </tr>
    <tr>
        <td>Meantone [3]<br />
</td>
        <td>4/3<br />
</td>
        <td>D G C<br />
</td>
        <td>3/2<br />
</td>
        <td>C G D<br />
</td>
    </tr>
    <tr>
        <td>Meantone [5]<br />
</td>
        <td>4/3<br />
</td>
        <td>E A D G C<br />
</td>
        <td>3/2<br />
</td>
        <td>C G D A E<br />
</td>
    </tr>
    <tr>
        <td>Meantone [7]<br />
</td>
        <td>3/2<br />
</td>
        <td>C G D A E B F#<br />
</td>
        <td>3/2<br />
</td>
        <td>C G D A E B F#<br />
</td>
    </tr>
</table>

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


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

<br />
An even larger problem is that 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 &quot;Dominant 8|3&quot; could mean either &quot;4th Dominant[12]&quot; or &quot;9th Dominant[12]&quot;. Something similar happens with Meantone[19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone[19] ambiguous. Another example is Dicot[7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore[5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.<br />
<br />
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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and Mode Numbers notation uses the more intuitive one-based counting.<br />
<br />
<br />
<br />
<br />
Admins: Please delete the blank page at <!-- ws:start:WikiTextUrlRule:1522:http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position --><a href="http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position">http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position</a><!-- ws:end:WikiTextUrlRule:1522 --></body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2016-04-25 01:45:09 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-25 02:09:04 UTC</tt>.<br>
-: The original revision id was <tt>581086549</tt>.<br>
+: The original revision id was <tt>581087571</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 84: / Line 84: @@
 ||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7,-6] ||
 || Double harmonic minor || A B C D# E F G# A || F C * * __**A**__ E B * * G# D# || F C G D __**A**__ E B || 5th Meantone[7,+3,+4] ||
-||   ||   ||   || __**A**__ E B F# C# G# D# || 1st Meantone[7,-4,-5] ||
+||= " ||= " ||= " || __**A**__ E B F# C# G# D# || 1st Meantone[7,-4,-5] ||
-|| Double harmonic major || A Bb C# D E F G# A || Bb F * * D A E * * C# G# || Bb F C G D __**A**__ E || 6th Meantone[7,+3,+4] ||
+|| Double harmonic major || A Bb C# D E F G# A || Bb F * * D __**A**__ E * * C# G# || Bb F C G D __**A**__ E || 6th Meantone[7,+3,+4] ||
-||   ||   ||   || D __**A**__ E B F# C# G# || 2nd Meantone[7,-4,-5] ||
+||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone[7,-4,-5] ||
+|| &lt;span class="mw-redirect"&gt;Hungarian gypsy &lt;/span&gt;minor || A B C D# E F G A || F C G * __**A**__ E B * * * D# || F C G D __**A**__ E B || 5th Meantone[7,+4] ||
+|| Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || Bb F C G D __**A**__ E || 6th Meantone[7,+3] ||
 || Japanese pentatonic || A B C E F A || F C * * __**A**__ E B || __**A**__ E B F# C# || 1st Meantone[5,-4,-5] ||
 || (a mode of the above) || F A B C E F || __**F**__ C * * A E B || Ab Eb Bb __**F**__ C || 4th Meantone[5,-4,-5] ||
@@ Line 92: / Line 94: @@
-//Things to consider://
+//A point to consider://
-The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that if the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that if a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In other words, in this context, melodic minor is better described as 5th Meantone[7,+1,+3] than as 4th Meantone[7,+2].
+The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.
-//Perhaps use [7,^3] in place of [7,+3]?//
-//Old method: Japanese pentatonic was "1st Meantone[5] b3 b6". 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".//
-//Old method: The F mode of Japanese pentatonic was "4th Meantone[5] #2 #3 #6". 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.//
@@ Line 855: / Line 851: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&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# G# D#&lt;br /&gt;
@@ Line 871: / Line 867: @@
          &lt;td&gt;A Bb C# D E F G# A&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Bb F * * D A E * * C# G#&lt;br /&gt;
+         &lt;td&gt;Bb F * * D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E * * C# G#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Bb F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E&lt;br /&gt;
@@ Line 879: / Line 875: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&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,-4,-5]&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span class="mw-redirect"&gt;Hungarian gypsy &lt;/span&gt;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 G * &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B * * * D#&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,+4]&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Phrygian dominant&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A Bb C# D E F G A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Bb F * G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E * * C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Bb F C G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6th Meantone[7,+3]&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 919: / Line 939: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;Things to consider:&lt;/em&gt;&lt;br /&gt;
+&lt;em&gt;A point to consider:&lt;/em&gt;&lt;br /&gt;
-&lt;br /&gt;
-The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that if the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that if a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In other words, in this context, melodic minor is better described as 5th Meantone[7,+1,+3] than as 4th Meantone[7,+2].&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;Perhaps use [7,^3] in place of [7,+3]?&lt;/em&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;Old method: Japanese pentatonic was &amp;quot;1st Meantone[5] b3 b6&amp;quot;. 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;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;Old method: The F mode of Japanese pentatonic was &amp;quot;4th Meantone[5] #2 #3 #6&amp;quot;. 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;/em&gt;&lt;br /&gt;
+The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is 1st Meantone[7,-4,-5] and the double harmonic major is 6th Meantone[7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone[7] to melodic minor = 5th Meantone[7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,309: / Line 1,323: @@
 &lt;br /&gt;
 &lt;br /&gt;
-Admins: Please delete the blank page at &lt;!-- ws:start:WikiTextUrlRule:1492: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:1492 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+Admins: Please delete the blank page at &lt;!-- ws:start:WikiTextUrlRule:1522: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:1522 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Comparison of mode notation systems: Difference between revisions

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