Pergen names: Difference between revisions

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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), then the period can be expressed as the octave 2/1, or some fraction of an octave. The generator can usually be expressed as some 3-limit interval, or some fraction of one. The fraction is always of the form 1/N, in other words, the octave or the 3-limit interval is **split** into N parts. An interval which is split into multiple generators is called a **multi-gen**.

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one "W" per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.

[//Question: at what point is// "//W" actually needed?//]

For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate higher primes. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).

Examples: Marvel (2.3.5.7 and 225/224, or reddish yellow) is {P8, P5, ^1} = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is {P8/2, P5, g1} = half-octave with green. Triple bluish (1029/1000) is {P8, P11/3, ^1} = third-eleventh with ups.

A rank-4 temperament has a pergen of four intervals. 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.

The preferred pergen for untempered just intonation is 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 monzo must be 1 or -1. These commas are called **notational commas**. They are not necessarily tempered out, but they determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is universal agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. Note that the choice of 11's notational comma affects the mapping of other 11-limit commas.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.

The genchain shown is a short section of the full genchain.  

C - G implies ...Eb Bb F C G D A E B F# C#...  

C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...  

Each enharmonic interval implies a different notation. If every pergen could use every enharmonic, there would be an overwhelming choice of notations! Fortunately, not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic.  

For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-exponent of the period, the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|.  

For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are some voicing of the 5th, and the multi-gen's 3-exponent is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-exponent of the generator, the bare enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|.  

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), then the period can be expressed as the octave 2/1, or some fraction of an octave. The generator can usually be expressed as some 3-limit interval, or some fraction of one. The fraction is always of the form 1/N, in other words, the octave or the 3-limit interval is &lt;strong&gt;split&lt;/strong&gt; into N parts. An interval which is split into multiple generators is called a &lt;strong&gt;multi-gen&lt;/strong&gt;.&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;{P8/2, P4/2}&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;half-octave, half-fourth&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;25/24, 49/48&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;decimal&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;deep yellow and deep blue&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;yy&amp;amp;bbT&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;{P8/4, P5}&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;quarter-octave&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;(3,4,-4)&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;diminished&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;quadruple green&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;g&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;T&lt;br /&gt;

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.&lt;br /&gt;

[&lt;em&gt;Question: at what point is&lt;/em&gt; &amp;quot;&lt;em&gt;W&amp;quot; actually needed?&lt;/em&gt;]&lt;br /&gt;

For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;Color notation &lt;/a&gt;is used to indicate higher primes. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).&lt;br /&gt;

Examples: Marvel (2.3.5.7 and 225/224, or reddish yellow) is {P8, P5, ^1} = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is {P8/2, P5, g1} = half-octave with green. Triple bluish (1029/1000) is {P8, P11/3, ^1} = third-eleventh with ups.&lt;br /&gt;

A rank-4 temperament has a pergen of four intervals. 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.&lt;br /&gt;

The preferred pergen for untempered just intonation is 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 monzo must be 1 or -1. These commas are called &lt;strong&gt;notational commas&lt;/strong&gt;. They are not necessarily tempered out, but they determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is universal agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. Note that the choice of 11's notational comma affects the mapping of other 11-limit commas.&lt;br /&gt;

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.&lt;br /&gt;

The genchain shown is a short section of the full genchain. &lt;br /&gt;

C - G implies ...Eb Bb F C G D A E B F# C#... &lt;br /&gt;

C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... &lt;br /&gt;

Each enharmonic interval implies a different notation. If every pergen could use every enharmonic, there would be an overwhelming choice of notations! Fortunately, not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic. &lt;br /&gt;

For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-exponent of the period, the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. &lt;br /&gt;

For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are some voicing of the 5th, and the multi-gen's 3-exponent is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-exponent of the generator, the bare enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. &lt;br /&gt;