Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626598157 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626601993 - 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>2018-02-19 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-19 06:50:00 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626601993</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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For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth. | For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth. | ||
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using [[Ups and Downs Notation|ups and downs]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]]. | ||
The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime > 3 (a **higher prime**), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**. | The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime > 3 (a **higher prime**), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**. | ||
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=__Derivation__= | =__Derivation__= | ||
For any comma | For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts. | ||
In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth. | In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth. | ||
For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, | For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, ^1), the same as Blackwood. | ||
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5, | To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents. | ||
For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n. | For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n. | ||
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< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red | < 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red | ||
^1 = 64/63 || | ^1 = 64/63 || | ||
||= ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= | ||= ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 169/162 | ||
^1 = | ^1 = 27/26 || | ||
||= 3 ||= (P8, P4/2) | ||= 3 ||= (P8, P4/2) | ||
half-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | half-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | ||
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C - Eb^=Ev - G, | C - Eb^=Ev - G, | ||
C - F#v/=Gb^\ - C, | C - F#v/=Gb^\ - C, | ||
C - F^/=Gv\ - C ||= 49/48 & | C - F^/=Gv\ - C ||= 49/48 & 243/242 | ||
^1 = 33/32 | ^1 = 33/32 | ||
/1 = 64/63 || | /1 = 64/63 || | ||
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P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C, | P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C, | ||
C - Eb/=E\ - G, | C - Eb/=E\ - G, | ||
C - Dv/=Eb^\ - F ||= 2048/2025 & | C - Dv/=Eb^\ - F ||= 2048/2025 & 243/242 | ||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 33/32 || | /1 = 33/32 || | ||
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For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.<br /> | For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.<br /> | ||
<br /> | <br /> | ||
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. See the notation guide below, under <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a>.<br /> | ||
<br /> | <br /> | ||
The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime &gt; 3 (a <strong>higher prime</strong>), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called <strong>unsplit</strong>.<br /> | The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime &gt; 3 (a <strong>higher prime</strong>), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called <strong>unsplit</strong>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:61:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:61 --><u>Derivation</u></h1> | <!-- ws:start:WikiTextHeadingRule:61:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:61 --><u>Derivation</u></h1> | ||
<br /> | <br /> | ||
For any comma | For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br /> | ||
<br /> | <br /> | ||
In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br /> | In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br /> | ||
<br /> | <br /> | ||
For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, | For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, ^1), the same as Blackwood.<br /> | ||
<br /> | <br /> | ||
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5, | To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.<br /> | ||
<br /> | <br /> | ||
For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.<br /> | For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.<br /> | ||
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<td style="text-align: center;">C - F^=Gv - C<br /> | <td style="text-align: center;">C - F^=Gv - C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">169/162<br /> | ||
^1 = | ^1 = 27/26<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,741: | Line 1,741: | ||
C - F^/=Gv\ - C<br /> | C - F^/=Gv\ - C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">49/48 &amp; | <td style="text-align: center;">49/48 &amp; 243/242<br /> | ||
^1 = 33/32<br /> | ^1 = 33/32<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
| Line 1,793: | Line 1,793: | ||
C - Dv/=Eb^\ - F<br /> | C - Dv/=Eb^\ - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2048/2025 &amp; | <td style="text-align: center;">2048/2025 &amp; 243/242<br /> | ||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
/1 = 33/32<br /> | /1 = 33/32<br /> | ||