Comparison of mode notation systems: Difference between revisions
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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 20:17:42 UTC</tt>.<br> | ||
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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: | Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this: | ||
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. It would be just as valid to write the half-octave not as an up-fourth but as a down-fifth. It's also valid to exchange the two rows: | |||
Gv -- Dv -- Av -- Ev -- Bv or C ---- G ---- D --- A --- E | |||
A --- | C ---- G ---- D --- A --- E Gv -- Dv -- Av -- Ev -- Bv | ||
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. | 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. | ||
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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. Shrutal'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 | 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. Shrutal'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^ | |||
C --- C^ -- D --- D^ -- E | |||
All five Shrutal [10] modes: | All five Shrutal [10] modes: | ||
|| 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 | || 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 | || 2nd Shrutal [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 Shrutal [10] || ssLss-ssLss || C C | || 3rd Shrutal [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 Shrutal [10] || sLsss-sLsss || C 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 | || 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. | ||
C --- G | |||
|| 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 E Gb G A Bb C ||= __**C**__ G || Eb Bb || Gb Db || A E || | || 1st Diminished[8] || sLsL sLsL || C Db Eb E Gb G A Bb C ||= __**C**__ G || Eb Bb || Gb Db || A E || | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- ws:start:WikiTextAnchorRule:54:&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:54 --><strong><u>Fractional-octave periods</u></strong></h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- ws:start:WikiTextAnchorRule:54:&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:54 --><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 /> | Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br /> | ||
F^ -- C^ -- G^ -- D^ -- A^<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 just as valid to write the half-octave not as an up-fourth but as a down-fifth. It's also valid to exchange the two rows:<br /> | |||
Gv -- Dv -- Av -- Ev -- Bv or C ---- G ---- D --- A --- E<br /> | |||
A --- | C ---- G ---- D --- A --- E Gv -- Dv -- Av -- Ev -- Bv<br /> | ||
<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 /> | 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 /> | ||
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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. Shrutal'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 | 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. Shrutal'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 /> | |||
C --- C^ -- D --- D^ -- E<br /> | |||
<br /> | <br /> | ||
All five Shrutal [10] modes:<br /> | All five Shrutal [10] modes:<br /> | ||
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<td>ssssL-ssssL<br /> | <td>ssssL-ssssL<br /> | ||
</td> | </td> | ||
<td>C C | <td>C C^ D D^ E F^ G G^ A A^ C<br /> | ||
</td> | </td> | ||
<td><u><strong>C</strong></u> G D A E<br /> | <td><u><strong>C</strong></u> G D A E<br /> | ||
</td> | </td> | ||
<td>F | <td>F^ C^ G^ D^ A^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>sssLs-sssLs<br /> | <td>sssLs-sssLs<br /> | ||
</td> | </td> | ||
<td>C C | <td>C C^ D D^ F F^ G G^ A Bb^ C<br /> | ||
</td> | </td> | ||
<td>F <u><strong>C</strong></u> G D A<br /> | <td>F <u><strong>C</strong></u> G D A<br /> | ||
</td> | </td> | ||
<td> | <td>Bb^ F^ C^ G^ D^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>ssLss-ssLss<br /> | <td>ssLss-ssLss<br /> | ||
</td> | </td> | ||
<td>C C | <td>C C^ D Eb^ F F^ G G^ Bb Bb^ C<br /> | ||
</td> | </td> | ||
<td>Bb F <u><strong>C</strong></u> G D<br /> | <td>Bb F <u><strong>C</strong></u> G D<br /> | ||
</td> | </td> | ||
<td> | <td>Eb^ Bb^ F^ C^ G^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>sLsss-sLsss<br /> | <td>sLsss-sLsss<br /> | ||
</td> | </td> | ||
<td>C C | <td>C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C<br /> | ||
</td> | </td> | ||
<td>Eb Bb F <u><strong>C</strong></u> G<br /> | <td>Eb Bb F <u><strong>C</strong></u> G<br /> | ||
</td> | </td> | ||
<td> | <td>Ab^ Eb^ Bb^ F^ C^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>Lssss-Lssss<br /> | <td>Lssss-Lssss<br /> | ||
</td> | </td> | ||
<td>C | <td>C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C<br /> | ||
</td> | </td> | ||
<td>Ab Eb Bb F <u><strong>C</strong></u><br /> | <td>Ab Eb Bb F <u><strong>C</strong></u><br /> | ||
</td> | </td> | ||
<td> | <td>Db^ Ab^ Eb^ Bb^ F^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<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 /> | ||
<br /> | |||
<br /> | |||
<br /> | |||
C --- G<br /> | |||
<br /> | |||
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