Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 622022613 - Original comment: ** |
Wikispaces>TallKite **Imported revision 622065493 - 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>2017-11-20 09: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-20 20:09:55 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>622065493</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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||= {P8/2, P4/2} ||= half-octave, half-fourth ||= 25/24 & 49/48 ||= decimal ||= deep yellow and deep blue ||= yy&bbT || | ||= {P8/2, P4/2} ||= half-octave, half-fourth ||= 25/24 & 49/48 ||= decimal ||= deep yellow and deep blue ||= yy&bbT || | ||
||= {P8/4, P5} ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g<span style="vertical-align: super;">4</span>T || | ||= {P8/4, P5} ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g<span style="vertical-align: super;">4</span>T || | ||
The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For | The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 9/4, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one "W" per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc. Thus magic is {P8, P12/5} = fifth-twelfth. | ||
For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4). | For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4). | ||
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=__Applications__= | =__Applications__= | ||
Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair | Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair, highs and lows, written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation. | ||
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G. | Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G. | ||
Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. | Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor. | ||
The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one. | The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one. | ||
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||= {P8/3, P4/3} ||= ||= ||= ||= ||= ||= 15, 21, 30* || | ||= {P8/3, P4/3} ||= ||= ||= ||= ||= ||= 15, 21, 30* || | ||
||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/4, P5} ||= ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span> | ||= {P8/4, P5} ||= ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 | ||
(^/v may reverse) ||= v<span style="vertical-align: super;">4</span>d2 ||= C Ebv | (^/v may reverse) ||= v<span style="vertical-align: super;">4</span>d2 ||= C Ebv Gbvv A^ C ||= diminished, | ||
^1 = 81/80 ||= 12, 16, 20, 24*, 28 || | |||
||= {P8, P4/4} ||= ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C Db^ Ebb^^ Ev F ||= ||= || | ||= {P8, P4/4} ||= ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C Db^ Ebb^^ Ev F ||= ||= || | ||
||= {P8, P5/4} ||= ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= v<span style="vertical-align: super;">4</span>A1 || C Dv Evv F^ G^ ||= ||= || | ||= {P8, P5/4} ||= ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= v<span style="vertical-align: super;">4</span>A1 || C Dv Evv F^ G^ ||= ||= || | ||
||= {P8, P11/4} ||= ||= | ||= {P8, P11/4} ||= || P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= v<span style="vertical-align: super;">4</span>dd3 ||= C E^ G#^^ Dbv F ||= ||= || | ||
||= {P8, P12/4} ||= ||= | ||= {P8, P12/4} ||= || P12/4 = vP4 = ^<span style="vertical-align: super;">3</span>M3 ||= v<span style="vertical-align: super;">4</span>m2 ||= C Fv Bbvv D^ G ||= ||= || | ||
||= {P8/4, P4/2} ||= ||= ||= ||= ||= ||= || | ||= {P8/4, P4/2} ||= ||= ||= ||= ||= ||= || | ||
||= {P8/2, P4/4} ||= ||= ||= ||= ||= ||= || | ||= {P8/2, P4/4} ||= ||= ||= ||= ||= ||= || | ||
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</table> | </table> | ||
The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For | The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 9/4, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc. Thus magic is {P8, P12/5} = fifth-twelfth.<br /> | ||
<br /> | <br /> | ||
For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation </a>is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).<br /> | For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation </a>is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Applications</u></h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Applications</u></h1> | ||
<br /> | <br /> | ||
Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair | Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair, highs and lows, written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.<br /> | ||
<br /> | <br /> | ||
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.<br /> | Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.<br /> | ||
<br /> | <br /> | ||
Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. | Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.<br /> | ||
<br /> | <br /> | ||
The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.<br /> | The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.<br /> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span> | <td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2<br /> | ||
(^/v may reverse)<br /> | (^/v may reverse)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>d2<br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>d2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C Ebv | <td style="text-align: center;">C Ebv Gbvv A^ C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">diminished,<br /> | ||
^1 = 81/80<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">12, 16, 20, 24*, 28<br /> | <td style="text-align: center;">12, 16, 20, 24*, 28<br /> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style=" | <td>P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>dd3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C E^ G#^^ Dbv F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style=" | <td>P12/4 = vP4 = ^<span style="vertical-align: super;">3</span>M3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">v<span style="vertical-align: super;">4</span>m2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C Fv Bbvv D^ G<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||