Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 628679225 - 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>2018-04- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-04-17 23:40:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>628798773</tt>.<br> | ||
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | ||
==Chord names | ==Chord names, scale names, staff notation== | ||
Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the [[Ups and Downs Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling. | Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the [[Ups and Downs Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling. | ||
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A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7. | A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7. | ||
Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [ | Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. | ||
Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^. | Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^. | ||
Highs and lows can be added to the score just like ups and downs can. They precede the note head and any sharps or flats. Scores for melody instruments can optionally have them above or below the staff. This score uses ups and downs, and has chord names. | |||
__<span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3)</span>__ | |||
[[image:Mizarian Porcupine Overture.png width="800" height="692"]] | |||
==Tipping points and sweet spots== | ==Tipping points and sweet spots== | ||
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==Pergen squares== | ==Pergen squares== | ||
Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions. | Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions. | ||
For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods). | For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods). | ||
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gedra | gedra | ||
edomapping | edomapping | ||
__**Combining pergens**__ | __**Combining pergens**__ | ||
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__**Credits**__ | __**Credits**__ | ||
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:63:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:63 --> </h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:63:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:63 --> </h1> | ||
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<!-- ws:end:WikiTextTocRule:132 --><!-- ws:start:WikiTextTocRule:133: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | <!-- ws:end:WikiTextTocRule:132 --><!-- ws:start:WikiTextTocRule:133: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | ||
<!-- ws:end:WikiTextTocRule:133 --><!-- ws:start:WikiTextTocRule:134: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | <!-- ws:end:WikiTextTocRule:133 --><!-- ws:start:WikiTextTocRule:134: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule:134 --><!-- ws:start:WikiTextTocRule:135: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names | <!-- ws:end:WikiTextTocRule:134 --><!-- ws:start:WikiTextTocRule:135: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names, scale names, staff notation">Chord names, scale names, staff notation</a></div> | ||
<!-- ws:end:WikiTextTocRule:135 --><!-- ws:start:WikiTextTocRule:136: --><div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div> | <!-- ws:end:WikiTextTocRule:135 --><!-- ws:start:WikiTextTocRule:136: --><div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div> | ||
<!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div> | <!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div> | ||
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:89:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Chord names | <!-- ws:start:WikiTextHeadingRule:89:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Chord names, scale names, staff notation"></a><!-- ws:end:WikiTextHeadingRule:89 -->Chord names, scale names, staff notation</h2> | ||
<br /> | <br /> | ||
Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | ||
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A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.<br /> | A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.<br /> | ||
<br /> | <br /> | ||
Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [ | Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes.<br /> | ||
<br /> | <br /> | ||
Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.<br /> | Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.<br /> | ||
<br /> | |||
Highs and lows can be added to the score just like ups and downs can. They precede the note head and any sharps or flats. Scores for melody instruments can optionally have them above or below the staff. This score uses ups and downs, and has chord names.<br /> | |||
<br /> | |||
<u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3)</span></u><br /> | |||
<!-- ws:start:WikiTextLocalImageRule:4962:&lt;img src=&quot;/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 692px; width: 800px;&quot; /&gt; --><img src="/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png" alt="Mizarian Porcupine Overture.png" title="Mizarian Porcupine Overture.png" style="height: 692px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:4962 --><br /> | |||
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<!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a><!-- ws:end:WikiTextHeadingRule:91 -->Tipping points and sweet spots</h2> | <!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a><!-- ws:end:WikiTextHeadingRule:91 -->Tipping points and sweet spots</h2> | ||
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<!-- ws:start:WikiTextHeadingRule:101:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Pergen squares"></a><!-- ws:end:WikiTextHeadingRule:101 -->Pergen squares</h2> | <!-- ws:start:WikiTextHeadingRule:101:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Pergen squares"></a><!-- ws:end:WikiTextHeadingRule:101 -->Pergen squares</h2> | ||
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Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.<br /> | Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.<br /> | ||
<br /> | <br /> | ||
For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).<br /> | For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).<br /> | ||
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Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.<br /> | Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4963:&lt;img src=&quot;/file/view/pergen%20squares.png/627986281/pergen%20squares.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" /><!-- ws:end:WikiTextLocalImageRule:4963 --><br /> | ||
A similar chart could be made for all rank-3 pergens, using pergen cubes.<br /> | A similar chart could be made for all rank-3 pergens, using pergen cubes.<br /> | ||
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Screenshots of the first 2 pages:<br /> | Screenshots of the first 2 pages:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:115:&lt;h3&gt; --><h3 id="toc26"><a name="Further Discussion-Supplemental materials-alt-pergenLister"></a><!-- ws:end:WikiTextHeadingRule:115 -->alt-pergenLister</h3> | <!-- ws:start:WikiTextHeadingRule:115:&lt;h3&gt; --><h3 id="toc26"><a name="Further Discussion-Supplemental materials-alt-pergenLister"></a><!-- ws:end:WikiTextHeadingRule:115 -->alt-pergenLister</h3> | ||
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Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8237: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:8237 --><br /> | ||
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The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | ||
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Screenshots of the first 69 pergens:<br /> | Screenshots of the first 69 pergens:<br /> | ||
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The first 29 pergens supported by 12edo:<br /> | The first 29 pergens supported by 12edo:<br /> | ||
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Some of the pergens supported by 15edo. A red asterisk means partial support.<br /> | Some of the pergens supported by 15edo. A red asterisk means partial support.<br /> | ||
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Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.<br /> | Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.<br /> | ||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||
| Line 6,772: | Line 6,780: | ||
gedra<br /> | gedra<br /> | ||
edomapping<br /> | edomapping<br /> | ||
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<u><strong>Combining pergens</strong></u><br /> | <u><strong>Combining pergens</strong></u><br /> | ||
| Line 6,807: | Line 6,811: | ||
<u><strong>Credits</strong></u><br /> | <u><strong>Credits</strong></u><br /> | ||
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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</body></html></pre></div> | ||