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Comparison of mode notation systems: Difference between revisions

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**Imported revision 593118992 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>2016-09-22 23:46:17 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-23 01:30:45 UTC</tt>.<br>
: The original revision id was <tt>593118992</tt>.<br>
: The original revision id was <tt>593121782</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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:  
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:  
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
They would follow a more regular pattern if using octotonic fourth-based notation:
The modes would follow a more regular pattern if using octotonic fourth-based notation:
A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A
A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.
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|| 6th Porcupine [7] || sLss sss || C Dv Ev F^ G Av Bb^ C || Ev F^ G Av Bb^ __**C**__ Dv ||
|| 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**__ ||
|| 7th Porcupine [7] || Lsss sss || C D Ev F^ G Av Bb^ C || D Ev F^ G Av Bb^ __**C**__ ||
 
Again, the modes would follow a more regular pattern if using the appropriate notation, in this case 2nd-based:
C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.


==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**==  
==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**==  
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==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**==  
==[[#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:
 
F^ -- C^ -- G^ -- D^ -- A^
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
C --- G ---- D --- A --- E
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3
 
When the period is an octave, the genweb octave-reduces to a single horizontal genchain:
F --- C --- G --- D --- A --- E --- B
 
But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, shrutal [10] might look like this:
F^3 --- C^4 --- G^4 --- D^5 --- A^5
C3 ---- G3 ----- D4 ---- A4 ---- E5
F^2 --- C^3 --- G^3 --- D^4 --- A^4
C2 ---- G2 ----- D3 ---- A3 ---- E3
F^1 --- C^2 --- G^2 --- D^3 --- A^3
C1 ---- G1 ----- D2 ---- A2 ---- E2
 
which octave-reduces to two genchains:
F^ --- C^ --- G^ --- D^ --- A^
C ---- G ----- D ---- A ---- E


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


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


It would also be valid to exchange the two rows:
It would also be valid to exchange the two rows:
C ---- G ---- D --- A --- E
C ----- G ----- D ---- A ---- E
Gv -- Dv -- Av -- Ev -- Bv
Gv --- Dv --- Av --- Ev --- Bv
 
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.
Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
C3 - G3 - D4 - A4 - E5
C2 - G2 - D3 - A3 - E4
C1 - G1 - D2 - A2 - E3
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.
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.
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wC ---- wG ---- wD ---- wA ---- wE
wC ---- wG ---- wD ---- wA ---- wE


Here y means "~81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. With Tg5 not Ty4 as the period:
As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. With Tg5 not Ty4 as the period:
wC ---- wG ---- wD ---- wA ----- wE
wC ---- wG ---- wD ----- wA ---- wE
gGb --- gDb --- gAb --- gEb --- gBb
gGb --- gDb --- gAb --- gEb --- gBb


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


Using ~16/15 as the generator yields the same scales and mode numbers:
Using ~25/24 as the generator yields the same scales and mode numbers:
Gb^^ ----- G
Gb^^ ----- G
Eb^ ------- Ev
Eb^ ------- Ev
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wC -------- wG
wC -------- wG
yA --------- yE
yA --------- yE
Both Diminished [8] modes, using ups and downs:
Both Diminished [8] modes, using ups and downs:
|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain ||
|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain ||
|| 1st Diminished[ 8] || sLsL sLsL || C Db^^ Eb^ Ev Gb^^ G Av Bb^ C ||= __**C**__ G || Eb^ Bb^ || Gb^^ Db^^ || Av Ev ||
|| 1st Diminished[ 8] || sLsL sLsL || C Db^^ Eb^ Ev Gb^^ G Av Bb^ C ||= __**C**__ G || Eb^ Bb^ || Gb^^ Db^^ || Av Ev ||
|| 2nd Diminished [8] || LsLs LsLs || C D Eb^ F F# Ab^ A B C ||= F __**C**__ || Ab^ Eb^ || B F# || D A ||
|| 2nd Diminished [8] || LsLs LsLs || C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C ||= F __**C**__ || Ab^ Eb^ || Cb^^ Gb^^ || Dv Av ||
 


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


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¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.
Ups and downs could indicate the generator instead of the period:
|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || 5th chain ||
F ------ Av
|| 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 ||
D ------ F#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 ||
C ------ Ev
A ------ C#v
G ------ Bv
 
Assuming octave equivalence, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.
F ------ Av
C ------ Ev
G ------ Bv
D ------ F#v
A ------ C#v
 
Using color notation:
wF ------ yA
wC ------ yE
wG ------ yB
wD ------ yF#
wA ------ yC#
 
Using ups and downs to mean "raised/lowered by ~81/80":
|| scale name || Ls pattern || example in C || genchains ||
|| 1st Blackwood [10] || Ls Ls Ls Ls Ls || C C#v D Ev F F#v G Av A Bv C ||= __**C**__-Ev, D-F#v, F-Av, G-Bv, A-C#v ||
|| 2nd Blackwood [10] || sL sL sL sL sL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^-__**C**__, Bb^-D, C^-E, Eb^-G, F^-A ||




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**__2nd method__:**
**__2nd method__:**
As with MODMOS scales.
As with MODMOS scales.
C 2nd Meantone [8]
C 2nd Meantone [8] b7
A 3rd Meantone [9] b3 b6
Alterations are heptatonic because Meantone [7] is the largest MOS contained in Meantone [8] or Meantone [9].


F 1st Meantone [6] no6
|| scale || genchain || name || name ||
A 5th Meantone [7] no4 no7
|| C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone [7] add #4 || C 2nd Meantone [8] ||
||= " ||= " || C 1st Meantone [7] add b4 ||  ||
|| C D E F F# G A Bb C || Bb F __**C**__ G D A E * F# || C 3rd Meantone [7] add #4 || C 2nd Meantone [8] b7 ||
|| A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone [7] add #4, #7 || A 3rd Meantone [9] ||
||= " ||= " || A 2nd Meantone [7] add #4, b7 ||  ||
|| A B C D D# E F G G# A || F C G D __**A**__ E B * * G# D# || A 5th Meantone [7] add #4, #7 || A 3rd Meantone [9] b3 b6 ||
|| F G A C D E F || __**F**__ C G D A E ||  || F 1st Meantone [6] ||
|| F G A C E F || __**F**__ C G * A E ||  || F 1st Meantone [6] no4 ||
|| G A B D E F# G || __**G**__ D A E B F# ||  || G 1st Meantone [6] ||
|| G A C D E F# G || C __**G**__ D A E * F# ||  || G 3rd Meantone [6] #7 ||
|| A B C E F A || F C * * __**A**__ E B || A 5th Meantone [7] no4 no7 ||  ||
In the 2nd example, "b4" means a 4th flattened relative to the 4th in 1st
 
Alterations are heptatonic because the notation is heptatonic. The notation is heptatonic because Meantone [7] is the largest MOS contained in Meantone [8] or Meantone [9].


G A B D E F# G, with genchain __**G**__ D A E B F# = G 1st Meantone [6]
G A C D E F# G, with genchain C __**G**__ D A E * F# = G 3rd Meantone [6] #7


==[[#How to name rank-2 scales-Non-MOS scales]]__Explanation / Rationale__==  
==[[#How to name rank-2 scales-Non-MOS scales]]__Explanation / Rationale__==  
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The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation: &lt;br /&gt;
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation: &lt;br /&gt;
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C&lt;br /&gt;
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C&lt;br /&gt;
They would follow a more regular pattern if using octotonic fourth-based notation:&lt;br /&gt;
The modes would follow a more regular pattern if using octotonic fourth-based notation:&lt;br /&gt;
A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A&lt;br /&gt;
A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A&lt;br /&gt;
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.&lt;br /&gt;
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.&lt;br /&gt;
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&lt;/table&gt;
&lt;/table&gt;


&lt;br /&gt;
Again, the modes would follow a more regular pattern if using the appropriate notation, in this case 2nd-based:&lt;br /&gt;
C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C&lt;br /&gt;
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.&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="Kite Giedraitis method-MODMOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:47:&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:47 --&gt;&lt;strong&gt;&lt;u&gt;MODMOS scales&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Kite Giedraitis method-MODMOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:47:&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:47 --&gt;&lt;strong&gt;&lt;u&gt;MODMOS scales&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Kite Giedraitis method-Fractional-octave periods"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:48:&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:48 --&gt;&lt;strong&gt;&lt;u&gt;Fractional-octave periods&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Kite Giedraitis method-Fractional-octave periods"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:48:&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:48 --&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;
  &lt;br /&gt;
F^ -- C^ -- G^ -- D^ -- A^&lt;br /&gt;
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &amp;quot;genweb&amp;quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. &lt;br /&gt;
C --- G ---- D --- A --- E&lt;br /&gt;
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5&lt;br /&gt;
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4&lt;br /&gt;
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3&lt;br /&gt;
&lt;br /&gt;
When the period is an octave, the genweb octave-reduces to a single horizontal genchain: &lt;br /&gt;
F --- C --- G --- D --- A --- E --- B&lt;br /&gt;
&lt;br /&gt;
But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, shrutal [10] might look like this:&lt;br /&gt;
F^3 --- C^4 --- G^4 --- D^5 --- A^5&lt;br /&gt;
C3 ---- G3 ----- D4 ---- A4 ---- E5&lt;br /&gt;
F^2 --- C^3 --- G^3 --- D^4 --- A^4&lt;br /&gt;
C2 ---- G2 ----- D3 ---- A3 ---- E3&lt;br /&gt;
F^1 --- C^2 --- G^2 --- D^3 --- A^3&lt;br /&gt;
C1 ---- G1 ----- D2 ---- A2 ---- E2&lt;br /&gt;
&lt;br /&gt;
which octave-reduces to two genchains:&lt;br /&gt;
F^ --- C^ --- G^ --- D^ --- A^&lt;br /&gt;
C ---- G ----- D ---- A ---- E&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of &amp;quot;up&amp;quot; here is &amp;quot;a half-octave minus a fourth&amp;quot;, with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.&lt;br /&gt;
Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of &amp;quot;up&amp;quot; here is &amp;quot;a half-octave minus a fourth&amp;quot;, with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.&lt;br /&gt;
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.&lt;br /&gt;
Gv -- Dv -- Av -- Ev -- Bv&lt;br /&gt;
Gv --- Dv --- Av --- Ev --- Bv&lt;br /&gt;
C ---- G ---- D --- A --- E&lt;br /&gt;
C ----- G ----- D ---- A ---- E&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It would also be valid to exchange the two rows:&lt;br /&gt;
It would also be valid to exchange the two rows:&lt;br /&gt;
C ---- G ---- D --- A --- E&lt;br /&gt;
C ----- G ----- D ---- A ---- E&lt;br /&gt;
Gv -- Dv -- Av -- Ev -- Bv&lt;br /&gt;
Gv --- Dv --- Av --- Ev --- Bv&lt;br /&gt;
&lt;br /&gt;
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.&lt;br /&gt;
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.&lt;br /&gt;
&lt;br /&gt;
Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &amp;quot;genweb&amp;quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. &lt;br /&gt;
C3 - G3 - D4 - A4 - E5&lt;br /&gt;
C2 - G2 - D3 - A3 - E4&lt;br /&gt;
C1 - G1 - D2 - A2 - E3&lt;br /&gt;
&lt;br /&gt;
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;
&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;
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;
Line 1,171: Line 1,233:
wC ---- wG ---- wD ---- wA ---- wE&lt;br /&gt;
wC ---- wG ---- wD ---- wA ---- wE&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Here y means &amp;quot;~81/80 below w&amp;quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. With Tg5 not Ty4 as the period:&lt;br /&gt;
As always, y means &amp;quot;81/80 below w&amp;quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. With Tg5 not Ty4 as the period:&lt;br /&gt;
wC ---- wG ---- wD ---- wA ----- wE&lt;br /&gt;
wC ---- wG ---- wD ----- wA ---- wE&lt;br /&gt;
gGb --- gDb --- gAb --- gEb --- gBb&lt;br /&gt;
gGb --- gDb --- gAb --- gEb --- gBb&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Line 1,255: Line 1,317:
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The Diminished [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains.&lt;br /&gt;
The Diminished [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.&lt;br /&gt;
Gb^^ ----- Db^^&lt;br /&gt;
Gb^^ ----- Db^^&lt;br /&gt;
Eb^ ------- Bb^&lt;br /&gt;
Eb^ ------- Bb^&lt;br /&gt;
C ---------- G&lt;br /&gt;
C ---------- G&lt;br /&gt;
Av --------- Ev&lt;br /&gt;
Av --------- Ev&lt;br /&gt;
The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢.&lt;br /&gt;
The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. &amp;quot;Up&amp;quot; means &amp;quot;a quarter-octave minus a ~32/27&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Using ~16/15 as the generator yields the same scales and mode numbers:&lt;br /&gt;
Using ~25/24 as the generator yields the same scales and mode numbers:&lt;br /&gt;
Gb^^ ----- G&lt;br /&gt;
Gb^^ ----- G&lt;br /&gt;
Eb^ ------- Ev&lt;br /&gt;
Eb^ ------- Ev&lt;br /&gt;
Line 1,272: Line 1,334:
wC -------- wG&lt;br /&gt;
wC -------- wG&lt;br /&gt;
yA --------- yE&lt;br /&gt;
yA --------- yE&lt;br /&gt;
&lt;br /&gt;
Both Diminished [8] modes, using ups and downs:&lt;br /&gt;
Both Diminished [8] modes, using ups and downs:&lt;br /&gt;


Line 1,313: Line 1,376:
         &lt;td&gt;LsLs LsLs&lt;br /&gt;
         &lt;td&gt;LsLs LsLs&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;C D Eb^ F F# Ab^ A B C&lt;br /&gt;
         &lt;td&gt;C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
Line 1,319: Line 1,382:
         &lt;td&gt;Ab^ Eb^&lt;br /&gt;
         &lt;td&gt;Ab^ Eb^&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;B F#&lt;br /&gt;
         &lt;td&gt;Cb^^ Gb^^&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;D A&lt;br /&gt;
         &lt;td&gt;Dv Av&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 1,328: Line 1,391:
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.&lt;br /&gt;
There are only two Blackwood [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different &amp;quot;height&amp;quot;:&lt;br /&gt;
E^^ ------- G#^^&lt;br /&gt;
D^ -------- F#^&lt;br /&gt;
C ---------- E&lt;br /&gt;
Bbv ------- Fv&lt;br /&gt;
Gvv ------- Dvv&lt;br /&gt;
&lt;br /&gt;
Ups and downs could indicate the generator instead of the period:&lt;br /&gt;
F ------ Av&lt;br /&gt;
D ------ F#v&lt;br /&gt;
C ------ Ev&lt;br /&gt;
A ------ C#v&lt;br /&gt;
G ------ Bv&lt;br /&gt;
&lt;br /&gt;
Assuming octave equivalence, the lattice rows can be reordered to make a &amp;quot;pseudo-period&amp;quot; of 3\5 = ~3/2.&lt;br /&gt;
F ------ Av&lt;br /&gt;
C ------ Ev&lt;br /&gt;
G ------ Bv&lt;br /&gt;
D ------ F#v&lt;br /&gt;
A ------ C#v&lt;br /&gt;
&lt;br /&gt;
Using color notation:&lt;br /&gt;
wF ------ yA&lt;br /&gt;
wC ------ yE&lt;br /&gt;
wG ------ yB&lt;br /&gt;
wD ------ yF#&lt;br /&gt;
wA ------ yC#&lt;br /&gt;
&lt;br /&gt;
Using ups and downs to mean &amp;quot;raised/lowered by ~81/80&amp;quot;:&lt;br /&gt;




Line 1,339: Line 1,430:
         &lt;td&gt;example in C&lt;br /&gt;
         &lt;td&gt;example in C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1st chain&lt;br /&gt;
         &lt;td&gt;genchains&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;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 1,353: Line 1,436:
         &lt;td&gt;1st Blackwood [10]&lt;br /&gt;
         &lt;td&gt;1st Blackwood [10]&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;LsLsLs LsLs&lt;br /&gt;
         &lt;td&gt;Ls Ls Ls Ls Ls&lt;br /&gt;
&lt;/td&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;C C#v D Ev F F#v G Av A Bv C&lt;br /&gt;
&lt;/td&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 style="text-align: center;"&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;-Ev, D-F#v, F-Av, G-Bv, A-C#v&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;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 1,371: Line 1,446:
         &lt;td&gt;2nd Blackwood [10]&lt;br /&gt;
         &lt;td&gt;2nd Blackwood [10]&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;sLsLsL sLsL&lt;br /&gt;
         &lt;td&gt;sL sL sL sL sL&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;C C^ D Eb^ E F^ G Ab^ A Bb^ C&lt;br /&gt;
         &lt;td&gt;C C^ D Eb^ E F^ G Ab^ A Bb^ C&lt;br /&gt;
&lt;/td&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 style="text-align: center;"&gt;Ab^-&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;, Bb^-D, C^-E, Eb^-G, F^-A&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;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 1,414: Line 1,481:
&lt;strong&gt;&lt;u&gt;2nd method&lt;/u&gt;:&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;2nd method&lt;/u&gt;:&lt;/strong&gt;&lt;br /&gt;
As with MODMOS scales.&lt;br /&gt;
As with MODMOS scales.&lt;br /&gt;
C 2nd Meantone [8]&lt;br /&gt;
C 2nd Meantone [8] b7&lt;br /&gt;
A 3rd Meantone [9] b3 b6&lt;br /&gt;
Alterations are heptatonic because Meantone [7] is the largest MOS contained in Meantone [8] or Meantone [9].&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
F 1st Meantone [6] no6&lt;br /&gt;
 
A 5th Meantone [7] no4 no7&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;genchain&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;name&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;name&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;C D E F 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 F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C 2nd Meantone [7] add #4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C 2nd Meantone [8]&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&gt;C 1st Meantone [7] add b4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;C D E F F# G A 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 A E * F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C 3rd Meantone [7] add #4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C 2nd Meantone [8] b7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;A B C# D D# E F# G G# A&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G D &lt;u&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;/u&gt; E B F# C# G# D#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A 3rd Meantone [7] add #4, #7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A 3rd Meantone [9]&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&gt;A 2nd Meantone [7] add #4, b7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;A B C D D# E F G 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 * * G# D#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A 5th Meantone [7] add #4, #7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A 3rd Meantone [9] b3 b6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;F G A 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&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F 1st Meantone [6]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;F G A 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 G * A E&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F 1st Meantone [6] no4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;G A B D E F# G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E B F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G 1st Meantone [6]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;G A C D E F# G&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C &lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E * F#&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G 3rd Meantone [6] #7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&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;A 5th Meantone [7] no4 no7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
In the 2nd example, &amp;quot;b4&amp;quot; means a 4th flattened relative to the 4th in 1st&lt;br /&gt;
&lt;br /&gt;
Alterations are heptatonic because the notation is heptatonic. The notation is heptatonic because Meantone [7] is the largest MOS contained in Meantone [8] or Meantone [9].&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
G A B D E F# G, with genchain &lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E B F# = G 1st Meantone [6]&lt;br /&gt;
G A C D E F# G, with genchain C &lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E * F# = G 3rd Meantone [6] #7&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Kite Giedraitis method-Explanation / Rationale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:50:&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:50 --&gt;&lt;u&gt;Explanation / Rationale&lt;/u&gt;&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Kite Giedraitis method-Explanation / Rationale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:50:&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:50 --&gt;&lt;u&gt;Explanation / Rationale&lt;/u&gt;&lt;/h2&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Request for admins"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Request for admins&lt;/h1&gt;
  &lt;br /&gt;
  &lt;br /&gt;
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