Comparison of mode notation systems: Difference between revisions
Wikispaces>TallKite **Imported revision 593109490 - 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 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-22 21:06:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593112466</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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C --- G ---- D --- A --- E | 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. | 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. | ||
Multiple genchains occur because rank-2 really | 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 | |||
C ---- G ---- D --- A --- E | |||
It would also be valid to exchange the two rows: | |||
C ---- G ---- D --- A --- E | |||
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. | |||
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. 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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C --- C^ -- D --- D^ -- E | C --- C^ -- D --- D^ -- E | ||
All five Shrutal [10] modes | Another alternative is to use color notation. The shrutal comma is 2048/2025 = sgg2. This makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. | ||
yF# --- yC# --- yG# --- yD# --- yA# | |||
wC ---- wG ---- wD ---- wA ---- wE | |||
TyF# = TgGb because the interval between them, sgg2, is tempered out. With Tg5 not Ty4 as the period: | |||
wC ---- wG ---- wD ---- wA ----- wE | |||
gGb --- gDb --- gAb --- gEb --- gBb | |||
All five Shrutal [10] modes, using ups and downs. Every other scale note has an up. | |||
|| scale name || Ls pattern || example in C || 1st genchain || 2nd genchain || | || 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^ || | || 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^ || | ||
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|| 4th Shrutal [10] || sLsss-sLsss || C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C || Eb Bb F __**C**__ G || Ab^ Eb^ Bb^ F^ C^ || | || 4th Shrutal [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 Shrutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ || | || 5th Shrutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ || | ||
The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains. | The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains. | ||
Gb^^ ----- Db^^ | |||
Eb^ ------- Bb^ | |||
C ---------- G | |||
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¢. | |||
Using ~16/15 as the generator yields the same scales and mode numbers: | |||
Gb^^ ----- G | |||
Eb^ ------- Ev | |||
C ---------- Db^^ | |||
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. | |||
ggGb ----- ggDb | |||
gEb ------- gBb | |||
wC -------- wG | |||
yA --------- yE | |||
Both Diminished 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 | || 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 D Eb F F# Ab A B C ||= F __**C**__ || Ab Eb || B F# || D A || | ||
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. | 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. | ||
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C --- G ---- D --- A --- E<br /> | C --- G ---- D --- A --- E<br /> | ||
<br /> | <br /> | ||
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. It would be | 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 &quot;up&quot; here is &quot;a half-octave minus a fourth&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.<br /> | ||
Gv -- Dv -- Av -- Ev -- Bv | <br /> | ||
C ---- G ---- D --- A --- E | It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.<br /> | ||
Gv -- Dv -- Av -- Ev -- Bv<br /> | |||
C ---- G ---- D --- A --- E<br /> | |||
<br /> | <br /> | ||
Multiple genchains occur because rank-2 really | It would also be valid to exchange the two rows:<br /> | ||
C ---- G ---- D --- A --- E<br /> | |||
Gv -- Dv -- Av -- Ev -- Bv<br /> | |||
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.<br /> | |||
<br /> | |||
Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &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 /> | <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 /> | 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 /> | ||
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C --- C^ -- D --- D^ -- E<br /> | C --- C^ -- D --- D^ -- E<br /> | ||
<br /> | <br /> | ||
All five Shrutal [10] modes | Another alternative is to use color notation. The shrutal comma is 2048/2025 = sgg2. This makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb.<br /> | ||
<br /> | |||
yF# --- yC# --- yG# --- yD# --- yA#<br /> | |||
wC ---- wG ---- wD ---- wA ---- wE<br /> | |||
<br /> | |||
TyF# = TgGb because the interval between them, sgg2, is tempered out. With Tg5 not Ty4 as the period:<br /> | |||
wC ---- wG ---- wD ---- wA ----- wE<br /> | |||
gGb --- gDb --- gAb --- gEb --- gBb<br /> | |||
<br /> | |||
All five Shrutal [10] modes, using ups and downs. Every other scale note has an up.<br /> | |||
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</table> | </table> | ||
<br /> | |||
<br /> | <br /> | ||
The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains.<br /> | The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains.<br /> | ||
Gb^^ ----- Db^^<br /> | |||
Eb^ ------- Bb^<br /> | |||
C ---------- G<br /> | |||
Av --------- Ev<br /> | |||
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¢.<br /> | |||
<br /> | <br /> | ||
Using ~16/15 as the generator yields the same scales and mode numbers:<br /> | |||
Gb^^ ----- G<br /> | |||
Eb^ ------- Ev<br /> | |||
C ---------- Db^^<br /> | |||
Av --------- Bb^<br /> | |||
<br /> | <br /> | ||
<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.<br /> | ||
ggGb ----- ggDb<br /> | |||
<br /> | gEb ------- gBb<br /> | ||
wC -------- wG<br /> | |||
yA --------- yE<br /> | |||
Both Diminished modes, using ups and downs:<br /> | |||
| Line 1,228: | Line 1,289: | ||
<td>sLsL sLsL<br /> | <td>sLsL sLsL<br /> | ||
</td> | </td> | ||
<td>C Db Eb | <td>C Db^^ Eb^ Ev Gb^^ G Av Bb^ C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><u><strong>C</strong></u> G<br /> | <td style="text-align: center;"><u><strong>C</strong></u> G<br /> | ||
</td> | </td> | ||
<td>Eb Bb<br /> | <td>Eb^ Bb^<br /> | ||
</td> | </td> | ||
<td>Gb Db<br /> | <td>Gb^^ Db^^<br /> | ||
</td> | </td> | ||
<td> | <td>Av Ev<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,257: | Line 1,318: | ||
</table> | </table> | ||
<br /> | |||
<br /> | <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¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.<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¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.<br /> | ||
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