Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626589313 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626596551 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-19 03:38:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626596551</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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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc. | If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc. | ||
For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its 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, (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 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 fewer than perhaps a hundred 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]]. | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps a hundred 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]]. | ||
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||= 20 ||= (P8, P12/4) ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 ||= C Fv Bbvv=A^^ D^ G ||= vulture || | ||= 20 ||= (P8, P12/4) ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 ||= C Fv Bbvv=A^^ D^ G ||= vulture || | ||
||= ||= etc. ||= ||= ||= ||= ||= || | ||= ||= etc. ||= ||= ||= ||= ||= || | ||
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so | The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably isn't all that complex. | ||
==Tipping points== | ==Tipping points== | ||
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__**Staff notation**__ | __**Staff notation**__ | ||
Highs and lows can be added to the score just like ups and downs can. They precede the note. Scores for melody instruments can have them above or below the staff. | 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. | ||
__**Combining pergens**__ | __**Combining pergens**__ | ||
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* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | * (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | ||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension. | ||
__**Expanding gedras to 5-limit**__ | __**Expanding gedras to 5-limit**__ | ||
Gedras can be expanded to 5-limit | Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]: | ||
k = 12a + 19b + 28c + 34d | k = 12a + 19b + 28c + 34d | ||
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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multi-gen</strong>. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.<br /> | If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multi-gen</strong>. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.<br /> | ||
<br /> | <br /> | ||
For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its 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, (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 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 fewer than perhaps a hundred 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 /> | Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps a hundred 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 /> | ||
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</table> | </table> | ||
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so | The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably isn't all that complex.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:65:&lt;h2&gt; --><h2 id="toc4"><a name="Applications-Tipping points"></a><!-- ws:end:WikiTextHeadingRule:65 -->Tipping points</h2> | <!-- ws:start:WikiTextHeadingRule:65:&lt;h2&gt; --><h2 id="toc4"><a name="Applications-Tipping points"></a><!-- ws:end:WikiTextHeadingRule:65 -->Tipping points</h2> | ||
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<u><strong>Staff notation</strong></u><br /> | <u><strong>Staff notation</strong></u><br /> | ||
<br /> | <br /> | ||
Highs and lows can be added to the score just like ups and downs can. They precede the note. Scores for melody instruments can have them above or below the staff.<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.<br /> | ||
<br /> | <br /> | ||
<u><strong>Combining pergens</strong></u><br /> | <u><strong>Combining pergens</strong></u><br /> | ||
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General rules for combining pergens:<br /> | General rules for combining pergens:<br /> | ||
<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li></ul><br /> | <ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li></ul><br /> | ||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious<br /> | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.<br /> | ||
<br /> | <br /> | ||
<u><strong>Expanding gedras to 5-limit</strong></u><br /> | <u><strong>Expanding gedras to 5-limit</strong></u><br /> | ||
<br /> | <br /> | ||
Gedras can be expanded to 5-limit | Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:<br /> | ||
<br /> | <br /> | ||
k = 12a + 19b + 28c + 34d<br /> | k = 12a + 19b + 28c + 34d<br /> | ||