Pergen names: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-21 16:21:13 UTC</tt>.<br>

: The original revision id was <tt>622139791</tt>.<br>

Pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. And certain rank-2 temperaments require another additional pair, highs and lows, written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation. [//This part needs clarification//]

||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; = C# ||= C - Dv - Eb^ - F ||=   ||= 13b, 14*, 15, 21*,

||= {P8/5, P5} ||= P8/5 = ||= vm2 ||=  ||=  ||=  ||=  ||

||= {P8, P5/5} || P5/5 = ^m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;1 ||= ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= C - Db^ - Ebb^ -

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, the first enharmonic interval is ^^d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to make the enharmonic interval vanish. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the previous table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens are upped or downed as if the 5th were just.

[//Question: What to do if the edo's 5th falls in the sweet spot?//]  

The enharmonic's number of ups or downs is the LCM of the pergen's two splitting fractions. This number is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The implied edo is always a multiple of the height. Thus a half-anything pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. And a quarter-octave pergen must imply 12-edo, and its enharmonic must be ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 or v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2.

Pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. And certain rank-2 temperaments require another additional pair, highs and lows, written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.&lt;br /&gt;

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.&lt;br /&gt;

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation. [&lt;em&gt;This part needs clarification&lt;/em&gt;]&lt;br /&gt;

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

         &lt;td style="text-align: center;"&gt;vm2&lt;br /&gt;

         &lt;td&gt;P5/5 = ^m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;1&lt;br /&gt;

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, the first enharmonic interval is ^^d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to make the enharmonic interval vanish. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the previous table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens are upped or downed as if the 5th were just.&lt;br /&gt;

[&lt;em&gt;Question: What to do if the edo's 5th falls in the sweet spot?&lt;/em&gt;] &lt;br /&gt;

The enharmonic's number of ups or downs is the LCM of the pergen's two splitting fractions. This number is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The implied edo is always a multiple of the height. Thus a half-anything pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. And a quarter-octave pergen must imply 12-edo, and its enharmonic must be ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 or v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2.&lt;br /&gt;