Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626394499 - 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>2018-02-14 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-14 01:18:26 UTC</tt>.<br> | ||
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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher. | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher. | ||
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...) | In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...). | ||
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A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv. | A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv. | ||
==Notating rank-3 pergens== | ==Notating rank-3 pergens *== | ||
Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples: | Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples: | ||
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A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. | A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. | ||
All the previous rank-3 examples had the 2nd generator be a | All the previous rank-3 examples had the 2nd generator be a P1 comma, represented by an up. However, this isn't always possible, as the last example shows: | ||
||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic || | ||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ notation ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic || | ||
||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- || | ||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= single-pair ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- || | ||
||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= half-8ve with ups ||= /d5 = 7/5 ||= P5 ||= ^1 = 81/80 ||= ``//``d2 || | ||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= half-8ve with ups ||= double-pair ||= /d5 = 7/5 ||= P5 ||= ^1 = 81/80 ||= ``//``d2 || | ||
||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= third-11th with ups ||= P8 ||= \d5 = 7/5 ||= ^1 = 81/80 ||= ``\\\``dd3 || | ||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= third-11th with ups ||= double-pair ||= P8 ||= \d5 = 7/5 ||= ^1 = 81/80 ||= ``\\\``dd3 || | ||
||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= half-5th with ups ||= P8 ||= \d4 = 49/40 ||= ^1 = 64/63 ||= ``\\``dd3 || | ||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= half-5th with ups ||= double-pair ||= P8 ||= \d4 = 49/40 ||= ^1 = 64/63 ||= ``\\``dd3 || | ||
||= demeter ||= 686/675 ||= (P8, P5, | ||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 || | ||
Demeter | Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). | ||
fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢ | fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢ | ||
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<!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div> | <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div> | ||
<!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div> | <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens">Notating rank-3 pergens</a></div> | <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens *">Notating rank-3 pergens *</a></div> | ||
<!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens">Notating Blackwood-like pergens</a></div> | <!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens">Notating Blackwood-like pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | <!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | ||
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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.<br /> | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.<br /> | ||
<br /> | <br /> | ||
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...) | In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
| Line 2,699: | Line 2,699: | ||
A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.<br /> | A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens"></a><!-- ws:end:WikiTextHeadingRule:90 -->Notating rank-3 pergens</h2> | <!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens *"></a><!-- ws:end:WikiTextHeadingRule:90 -->Notating rank-3 pergens *</h2> | ||
<br /> | <br /> | ||
Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:<br /> | Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:<br /> | ||
| Line 2,891: | Line 2,891: | ||
A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.<br /> | A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.<br /> | ||
<br /> | <br /> | ||
All the previous rank-3 examples had the 2nd generator be a | All the previous rank-3 examples had the 2nd generator be a P1 comma, represented by an up. However, this isn't always possible, as the last example shows:<br /> | ||
| Line 2,903: | Line 2,903: | ||
</th> | </th> | ||
<th>spoken pergen<br /> | <th>spoken pergen<br /> | ||
</th> | |||
<th>notation<br /> | |||
</th> | </th> | ||
<th>period<br /> | <th>period<br /> | ||
| Line 2,921: | Line 2,923: | ||
</td> | </td> | ||
<td style="text-align: center;">unsplit with ups<br /> | <td style="text-align: center;">unsplit with ups<br /> | ||
</td> | |||
<td style="text-align: center;">single-pair<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">P8<br /> | <td style="text-align: center;">P8<br /> | ||
| Line 2,939: | Line 2,943: | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve with ups<br /> | <td style="text-align: center;">half-8ve with ups<br /> | ||
</td> | |||
<td style="text-align: center;">double-pair<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">/d5 = 7/5<br /> | <td style="text-align: center;">/d5 = 7/5<br /> | ||
| Line 2,957: | Line 2,963: | ||
</td> | </td> | ||
<td style="text-align: center;">third-11th with ups<br /> | <td style="text-align: center;">third-11th with ups<br /> | ||
</td> | |||
<td style="text-align: center;">double-pair<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">P8<br /> | <td style="text-align: center;">P8<br /> | ||
| Line 2,975: | Line 2,983: | ||
</td> | </td> | ||
<td style="text-align: center;">half-5th with ups<br /> | <td style="text-align: center;">half-5th with ups<br /> | ||
</td> | |||
<td style="text-align: center;">double-pair<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">P8<br /> | <td style="text-align: center;">P8<br /> | ||
| Line 2,990: | Line 3,000: | ||
<td style="text-align: center;">686/675<br /> | <td style="text-align: center;">686/675<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P8, P5, | <td style="text-align: center;">(P8, P5, vM3/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-downmajor-3rd<br /> | ||
</td> | |||
<td style="text-align: center;">double-pair<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">P8<br /> | <td style="text-align: center;">P8<br /> | ||
| Line 3,005: | Line 3,017: | ||
</table> | </table> | ||
Demeter | Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3).<br /> | ||
<br /> | <br /> | ||
fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢<br /> | fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢<br /> | ||
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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:5556: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:5556 --><br /> | ||
<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:5557: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:5557 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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<br /> | <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 /> | 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 /> | ||