Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626589053 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626589313 - 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>2018-02-18 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-18 21:07:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626589313</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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Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. | Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. | ||
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a notation guide. It even allows every pergen to be numbered. | Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF|notation guide]]. It even allows every pergen to be numbered. | ||
The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation. | The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation. | ||
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||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\<span style="vertical-align: super;">4</span>dd3 || | ||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\<span style="vertical-align: super;">4</span>dd3 || | ||
||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 || | ||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 || | ||
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. | There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. | ||
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Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator. Examples: | Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator. Examples: | ||
||~ temperament ||~ ||~ pergen ||~ spoken ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ||~ /1 || | ||~ temperament ||~ ||~ pergen ||~ spoken ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ||~ /1 || | ||
||= Blackwood ||= 5edo+y ||= (P8/5, ^1) ||= fifth-8ve no-threes ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 ||= --- || | ||= Blackwood ||= 5edo+y ||= (P8/5, ^1) ||= fifth-8ve no-threes ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 ||= --- || | ||
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(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.) | (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.) | ||
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. But in no-threes pergens, | In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. But in no-threes pergens, | ||
edo-subset notations | edo-subset notations | ||
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==Notating tunings with an arbitrary generator== | ==Notating tunings with an arbitrary generator== | ||
Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is | Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it barely includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢. | ||
The next table lists all the ranges for all multigens up to seventh-splits. | The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns. | ||
||||~ __primary choice__ ||||||||~ __secondary choices__ || | ||||~ __primary choice__ ||||||||~ __secondary choices__ || | ||
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||= (P8/4, P4/4) ||= quarter-everything ||= 8 = 4L 4s * ||= 12 = 8L 4s * ||= ||= || || || | ||= (P8/4, P4/4) ||= quarter-everything ||= 8 = 4L 4s * ||= 12 = 8L 4s * ||= ||= || || || | ||
A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. | A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. | ||
||~ MOS scale ||||~ primary example ||~ secondary examples || | ||~ MOS scale ||||~ primary example ||~ secondary examples || | ||
||~ Pentatonic ||~ ||~ ||~ || | ||~ Pentatonic ||~ ||~ ||~ || | ||
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||~ Heptatonic ||~ ||~ ||~ || | ||~ Heptatonic ||~ ||~ ||~ || | ||
||= 1L 6s ||= (P8, P4/3) [7] ||= third-4th heptatonic ||< quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th || | ||= 1L 6s ||= (P8, P4/3) [7] ||= third-4th heptatonic ||< quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th || | ||
||= 2L 5s ||= (P8, P11/3) [7] ||= third-11th heptatonic ||< fifth- | ||= 2L 5s ||= (P8, P11/3) [7] ||= third-11th heptatonic ||< fifth-double-wide-4th, sixth-double-wide-5th || | ||
||= 3L 4s ||= (P8, P5/2) [7] ||= half-5th heptatonic ||< fifth-12th || | ||= 3L 4s ||= (P8, P5/2) [7] ||= half-5th heptatonic ||< fifth-12th || | ||
||= 4L 3s ||= (P8, P11/5) [7] ||= fifth-11th heptatonic ||< sixth-12th || | ||= 4L 3s ||= (P8, P11/5) [7] ||= fifth-11th heptatonic ||< sixth-12th || | ||
||= 5L 2s ||= (P8, P5) [7] ||= unsplit heptatonic ||< sixth- | ||= 5L 2s ||= (P8, P5) [7] ||= unsplit heptatonic ||< sixth-double-wide-4th || | ||
||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||< || | ||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||< || | ||
||~ Octotonic ||~ ||~ ||~ || | ||~ Octotonic ||~ ||~ ||~ || | ||
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Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.<br /> | Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.<br /> | ||
<br /> | <br /> | ||
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a notation guide. It even allows every pergen to be numbered.<br /> | Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF">notation guide</a>. It even allows every pergen to be numbered.<br /> | ||
<br /> | <br /> | ||
The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation.<br /> | The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation.<br /> | ||
| Line 3,180: | Line 3,178: | ||
</table> | </table> | ||
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = <!-- ws:start:WikiTextRawRule:054:``//`` -->//<!-- ws:end:WikiTextRawRule:054 -->d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.<br /> | There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = <!-- ws:start:WikiTextRawRule:054:``//`` -->//<!-- ws:end:WikiTextRawRule:054 -->d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator. Examples:<br /> | Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator. Examples:<br /> | ||
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(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)<br /> | (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)<br /> | ||
<br /> | <br /> | ||
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. But in no-threes pergens, <br /> | In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. But in no-threes pergens,<br /> | ||
<br /> | <br /> | ||
edo-subset notations<br /> | edo-subset notations<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:93:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a><!-- ws:end:WikiTextHeadingRule:93 -->Notating tunings with an arbitrary generator</h2> | <!-- ws:start:WikiTextHeadingRule:93:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a><!-- ws:end:WikiTextHeadingRule:93 -->Notating tunings with an arbitrary generator</h2> | ||
<br /> | <br /> | ||
Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is | Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it barely includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.<br /> | ||
<br /> | <br /> | ||
The next table lists all the ranges for all multigens up to seventh-splits. | The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G &gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.<br /> | A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.<br /> | ||
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<td style="text-align: center;">third-11th heptatonic<br /> | <td style="text-align: center;">third-11th heptatonic<br /> | ||
</td> | </td> | ||
<td style="text-align: left;">fifth- | <td style="text-align: left;">fifth-double-wide-4th, sixth-double-wide-5th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">unsplit heptatonic<br /> | <td style="text-align: center;">unsplit heptatonic<br /> | ||
</td> | </td> | ||
<td style="text-align: left;">sixth- | <td style="text-align: left;">sixth-double-wide-4th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:6817:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:6817 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:6818:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><!-- ws:end:WikiTextUrlRule:6818 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||