Kite's Genchain mode numbering: Difference between revisions
Wikispaces>TallKite **Imported revision 593224218 - Original comment: ** |
Wikispaces>TallKite **Imported revision 593233744 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-25 07:51:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593233744</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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[[xenharmonic/MOSScales|MOS scales]] are formed from a segment of the [[xenharmonic/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. | [[xenharmonic/MOSScales|MOS scales]] are formed from a segment of the [[xenharmonic/periods and generators|generator-chain]], or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. | ||
For example, here are all the modes of Meantone[7], using ~3/2 as the generator: | For example, here are all the modes of Meantone [7], using ~3/2 as the generator: | ||
|| old scale name || new scale name || sL pattern || example on white keys || genchain || | || old scale name || new scale name || sL pattern || example on white keys || genchain || | ||
|| Lydian || 1st Meantone [7] || LLLs LLs || F G A B C D E F || __**F**__ C G D A E B || | || Lydian || 1st Meantone [7] || LLLs LLs || F G A B C D E F || __**F**__ C G D A E B || | ||
| Line 34: | Line 34: | ||
|| Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | || Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | ||
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are several disadvantages of only using chroma-positive generators, see | The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are several disadvantages of only using chroma-positive generators, see the critique of UDP at the bottom of this page. | ||
Pentatonic meantone scales: | Pentatonic meantone scales: | ||
| Line 45: | Line 45: | ||
Chromatic meantone scales. | Chromatic meantone scales. | ||
|| scale name || sL pattern (assumes | || scale name || sL pattern (assumes | ||
~3/2 < 700¢) || example in C || genchain || | ~3/2 < 700¢) || example in C || genchain || | ||
|| 1st Meantone [12] || sLsL sLL sLsLL || C C# D D# E E# F# G G# A A# B C || __**C**__ G D A E B F# C# G# D# A# E# || | || 1st Meantone [12] || sLsL sLL sLsLL || C C# D D# E E# F# G G# A A# B C || __**C**__ G D A E B F# C# G# D# A# E# || | ||
| Line 140: | Line 140: | ||
F --- C --- G --- D --- A --- E --- B | 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, | But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, Srutal [10] might look like this: | ||
F^3 --- C^4 --- G^4 --- D^5 --- A^5 | F^3 --- C^4 --- G^4 --- D^5 --- A^5 | ||
C3 ---- G3 ----- D4 ---- A4 ---- E5 | C3 ---- G3 ----- D4 ---- A4 ---- E5 | ||
| Line 166: | Line 166: | ||
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. | ||
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. | If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names: | ||
F^ -- G --- G^ -- A --- A^ | F^ -- G --- G^ -- A --- A^ | ||
C --- C^ -- D --- D^ -- E | C --- C^ -- D --- D^ -- E | ||
Another alternative is to use [[Kite's color notation|color notation]]. The | Another alternative is to use [[Kite's color notation|color notation]]. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]: | ||
yF# --- yC# --- yG# --- yD# --- yA# | yF# --- yC# --- yG# --- yD# --- yA# | ||
| Line 179: | Line 179: | ||
gGb --- gDb --- gAb --- gEb --- gBb | gGb --- gDb --- gAb --- gEb --- gBb | ||
All five | All five Srutal [10] modes, using ups and downs. Every other scale note has an up. | ||
|| scale name || sL pattern || example in C || 1st genchain || 2nd genchain || | || scale name || sL pattern || example in C || 1st genchain || 2nd genchain || | ||
|| 1st | || 1st Srutal [10] || ssssL-ssssL || C C^ D D^ E F^ G G^ A A^ C || __**C**__ G D A E || F^ C^ G^ D^ A^ || | ||
|| 2nd | || 2nd Srutal [10] || sssLs-sssLs || C C^ D D^ F F^ G G^ A Bb^ C || F __**C**__ G D A || Bb^ F^ C^ G^ D^ || | ||
|| 3rd | || 3rd Srutal [10] || ssLss-ssLss || C C^ D Eb^ F F^ G G^ Bb Bb^ C || Bb F __**C**__ G D || Eb^ Bb^ F^ C^ G^ || | ||
|| 4th | || 4th Srutal [10] || sLsss-sLsss || C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C || Eb Bb F __**C**__ G || Ab^ Eb^ Bb^ F^ C^ || | ||
|| 5th | || 5th Srutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ || | ||
| Line 200: | Line 200: | ||
C ---------- Db^^ | C ---------- Db^^ | ||
Av --------- Bb^ | Av --------- Bb^ | ||
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The name is 4-EDO+y [8]. | In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8]. | ||
ggGb ----- ggDb | ggGb ----- ggDb | ||
gEb ------- gBb | gEb ------- gBb | ||
| Line 233: | Line 233: | ||
A ------ C#v | A ------ C#v | ||
Using color notation. The name is 5-EDO+y. | Using color notation. The color name is 5-EDO+y. | ||
wF ------ yA | wF ------ yA | ||
wC ------ yE | wC ------ yE | ||
| Line 240: | Line 240: | ||
wA ------ yC# | wA ------ yC# | ||
Both Blackwood modes, using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4": | |||
|| scale name || sL pattern || example in C || genchains || | || scale name || sL pattern || example in C || genchains || | ||
|| 1st Blackwood [10] || Ls-Ls-Ls-Ls-Ls || C C#v D Ev F F#v G Av A Bv C ||= __**C**__-Ev, D-F#v, F-Av, G-Bv, A-C#v || | || 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 || | ||
| Line 248: | Line 248: | ||
=[[#Rank-2 scales that are neither MOS nor MODMOS]]**__Other rank-2 scales__**= | =[[#Rank-2 scales that are neither MOS nor MODMOS]]**__Other rank-2 scales__**= | ||
Some scales have too many or too few notes to be MOS or MODMOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets | Some scales have too many or too few notes to be MOS or MODMOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets could perhaps be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}. | ||
However chromatic alterations create genchains with gaps that are very difficult to name. These scales must be named as MOS scales with notes added or removed, using "add" and "no", analogous to chord names. As with MODMOS scales, there is often more than one name for a scale. | However chromatic alterations create genchains with gaps that are very difficult to name. These scales must be named as MOS scales with notes added or removed, using "add" and "no", analogous to chord names. As with MODMOS scales, there is often more than one name for a scale. | ||
| Line 278: | Line 278: | ||
The sML notation requires X = extra-large for various intervals. | The sML notation requires X = extra-large for various intervals. | ||
Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4. | Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4. | ||
| Line 288: | Line 286: | ||
=[[#Numbering considerations]]__Non-heptatonic Scales__= | =[[#Numbering considerations]]__Non-heptatonic Scales__= | ||
As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what "# | As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context. | ||
If the scale is written using heptatonically using 7 note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone [5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is bering used. | |||
The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthwards side is sharp (assuming a fifth < 720¢). Use "+" for fifthwards and "-" for fourthwards. | |||
Using J K L M N for note names, and arbitrarily centering the genchain on L, we get this genchain: | |||
...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5... | |||
...-K -N -L -J -M K N L J M +K +N +L +J +M ++K... | |||
and these standard modes: | |||
L 1st Meantone [5] = L M +N J +K L | |||
L 2nd Meantone [5] = L M N J +K L | |||
L 3rd Meantone [5] = L M N J K L | |||
L 4th Meantone [5] = L -M N J K L | |||
L 5th Meantone [5] = L -M N -J K L | |||
The A C B D F E G A scale becomes L M -M N J +K K L, which has 3 possible names: | |||
L 3rd Meantone [5] add -2, +5 | |||
L 2nd Meantone [5] add -2, -5 | |||
L 4th Meantone [5] add +2, +5 | |||
Using the numbers 1-5 both as note names and as scale degrees, we get this genchain: | |||
...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5... | ...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5... | ||
...-5 -3 -1 -4 -2 5 3 1 4 2 +5 +3 +1 +4 +2 ++5... | ...-5 -3 -1 -4 -2 5 3 1 4 2 +5 +3 +1 +4 +2 ++5... | ||
| Line 401: | Line 417: | ||
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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.<br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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.<br /> | ||
<br /> | <br /> | ||
For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br /> | For example, here are all the modes of Meantone [7], using ~3/2 as the generator:<br /> | ||
| Line 608: | Line 624: | ||
<br /> | <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 (see below). <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> There are several disadvantages of only using chroma-positive generators, see | The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below). <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> There are several disadvantages of only using chroma-positive generators, see the critique of UDP at the bottom of this page.<br /> | ||
<br /> | <br /> | ||
Pentatonic meantone scales:<br /> | Pentatonic meantone scales:<br /> | ||
| Line 696: | Line 712: | ||
<td>scale name<br /> | <td>scale name<br /> | ||
</td> | </td> | ||
<td>sL pattern (assumes <br /> | <td>sL pattern (assumes<br /> | ||
~3/2 &lt; 700¢)<br /> | ~3/2 &lt; 700¢)<br /> | ||
</td> | </td> | ||
| Line 1,160: | Line 1,176: | ||
F --- C --- G --- D --- A --- E --- B<br /> | F --- C --- G --- D --- A --- E --- B<br /> | ||
<br /> | <br /> | ||
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, | But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, Srutal [10] might look like this:<br /> | ||
F^3 --- C^4 --- G^4 --- D^5 --- A^5<br /> | F^3 --- C^4 --- G^4 --- D^5 --- A^5<br /> | ||
C3 ---- G3 ----- D4 ---- A4 ---- E5<br /> | C3 ---- G3 ----- D4 ---- A4 ---- E5<br /> | ||
| Line 1,186: | Line 1,202: | ||
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 /> | 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 /> | <br /> | ||
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. | If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:<br /> | ||
F^ -- G --- G^ -- A --- A^<br /> | F^ -- G --- G^ -- A --- A^<br /> | ||
C --- C^ -- D --- D^ -- E<br /> | C --- C^ -- D --- D^ -- E<br /> | ||
<br /> | <br /> | ||
Another alternative is to use <a class="wiki_link" href="/Kite%27s%20color%20notation">color notation</a>. The | Another alternative is to use <a class="wiki_link" href="/Kite%27s%20color%20notation">color notation</a>. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:<br /> | ||
<br /> | <br /> | ||
yF# --- yC# --- yG# --- yD# --- yA#<br /> | yF# --- yC# --- yG# --- yD# --- yA#<br /> | ||
| Line 1,199: | Line 1,215: | ||
gGb --- gDb --- gAb --- gEb --- gBb<br /> | gGb --- gDb --- gAb --- gEb --- gBb<br /> | ||
<br /> | <br /> | ||
All five | All five Srutal [10] modes, using ups and downs. Every other scale note has an up.<br /> | ||
| Line 1,216: | Line 1,232: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>1st | <td>1st Srutal [10]<br /> | ||
</td> | </td> | ||
<td>ssssL-ssssL<br /> | <td>ssssL-ssssL<br /> | ||
| Line 1,228: | Line 1,244: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>2nd | <td>2nd Srutal [10]<br /> | ||
</td> | </td> | ||
<td>sssLs-sssLs<br /> | <td>sssLs-sssLs<br /> | ||
| Line 1,240: | Line 1,256: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>3rd | <td>3rd Srutal [10]<br /> | ||
</td> | </td> | ||
<td>ssLss-ssLss<br /> | <td>ssLss-ssLss<br /> | ||
| Line 1,252: | Line 1,268: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>4th | <td>4th Srutal [10]<br /> | ||
</td> | </td> | ||
<td>sLsss-sLsss<br /> | <td>sLsss-sLsss<br /> | ||
| Line 1,264: | Line 1,280: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>5th | <td>5th Srutal [10]<br /> | ||
</td> | </td> | ||
<td>Lssss-Lssss<br /> | <td>Lssss-Lssss<br /> | ||
| Line 1,291: | Line 1,307: | ||
C ---------- Db^^<br /> | C ---------- Db^^<br /> | ||
Av --------- Bb^<br /> | Av --------- Bb^<br /> | ||
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The name is 4-EDO+y [8].<br /> | In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8].<br /> | ||
ggGb ----- ggDb<br /> | ggGb ----- ggDb<br /> | ||
gEb ------- gBb<br /> | gEb ------- gBb<br /> | ||
| Line 1,374: | Line 1,390: | ||
A ------ C#v<br /> | A ------ C#v<br /> | ||
<br /> | <br /> | ||
Using color notation. The name is 5-EDO+y.<br /> | Using color notation. The color name is 5-EDO+y.<br /> | ||
wF ------ yA<br /> | wF ------ yA<br /> | ||
wC ------ yE<br /> | wC ------ yE<br /> | ||
| Line 1,381: | Line 1,397: | ||
wA ------ yC#<br /> | wA ------ yC#<br /> | ||
<br /> | <br /> | ||
Both Blackwood modes, using ups and downs to mean &quot;raised/lowered by 2/5 of an octave minus ~5/4&quot;:<br /> | |||
| Line 1,421: | Line 1,437: | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Other rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:22:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank-2 scales that are neither MOS nor MODMOS&quot; title=&quot;Anchor: Rank-2 scales that are neither MOS nor MODMOS&quot;/&gt; --><a name="Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextAnchorRule:22 --><strong><u>Other rank-2 scales</u></strong></h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Other rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:22:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank-2 scales that are neither MOS nor MODMOS&quot; title=&quot;Anchor: Rank-2 scales that are neither MOS nor MODMOS&quot;/&gt; --><a name="Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextAnchorRule:22 --><strong><u>Other rank-2 scales</u></strong></h1> | ||
<br /> | <br /> | ||
Some scales have too many or too few notes to be MOS or MODMOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets | Some scales have too many or too few notes to be MOS or MODMOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. Curly brackets could perhaps be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}.<br /> | ||
<br /> | <br /> | ||
However chromatic alterations create genchains with gaps that are very difficult to name. These scales must be named as MOS scales with notes added or removed, using &quot;add&quot; and &quot;no&quot;, analogous to chord names. As with MODMOS scales, there is often more than one name for a scale.<br /> | However chromatic alterations create genchains with gaps that are very difficult to name. These scales must be named as MOS scales with notes added or removed, using &quot;add&quot; and &quot;no&quot;, analogous to chord names. As with MODMOS scales, there is often more than one name for a scale.<br /> | ||
| Line 1,662: | Line 1,678: | ||
<ul><li>In the 3rd row, &quot;add b4&quot; means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th.</li></ul><br /> | <ul><li>In the 3rd row, &quot;add b4&quot; means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th.</li></ul><br /> | ||
The sML notation requires X = extra-large for various intervals.<br /> | The sML notation requires X = extra-large for various intervals.<br /> | ||
<br /> | <br /> | ||
Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4.<br /> | Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4.<br /> | ||
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As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what &quot;# | As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what something like &quot;#5&quot; means in a pentatonic or hexatonic context.<br /> | ||
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If the scale is written using heptatonically using 7 note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone [5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is bering used.<br /> | |||
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The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthwards side is sharp (assuming a fifth &lt; 720¢). Use &quot;+&quot; for fifthwards and &quot;-&quot; for fourthwards. <br /> | |||
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Using J K L M N for note names, and arbitrarily centering the genchain on L, we get this genchain:<br /> | |||
...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5...<br /> | |||
...-K -N -L -J -M K N L J M +K +N +L +J +M ++K...<br /> | |||
and these standard modes:<br /> | |||
L 1st Meantone [5] = L M +N J +K L<br /> | |||
L 2nd Meantone [5] = L M N J +K L<br /> | |||
L 3rd Meantone [5] = L M N J K L<br /> | |||
L 4th Meantone [5] = L -M N J K L<br /> | |||
L 5th Meantone [5] = L -M N -J K L<br /> | |||
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The A C B D F E G A scale becomes L M -M N J +K K L, which has 3 possible names:<br /> | |||
L 3rd Meantone [5] add -2, +5<br /> | |||
L 2nd Meantone [5] add -2, -5<br /> | |||
L 4th Meantone [5] add +2, +5<br /> | |||
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Using the numbers 1-5 both as note names and as scale degrees, we get this genchain:<br /> | |||
...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5...<br /> | ...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5...<br /> | ||
...-5 -3 -1 -4 -2 5 3 1 4 2 +5 +3 +1 +4 +2 ++5...<br /> | ...-5 -3 -1 -4 -2 5 3 1 4 2 +5 +3 +1 +4 +2 ++5...<br /> | ||