Pergen names: Difference between revisions

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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, ...}. These commas are called **notational commas**. 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.

For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.

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. One possibility is highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. 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 properly as C Ev G.

Some combinations of periods and generators are duplicates of other pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. Some combinations are impossible. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.

The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.

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#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

Removing the ups and downs from an enharmonic interval makes a "bare" conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic 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 indicate that the enharmonic interval vanishes. 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 above table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic, thus the edo implies the enharmonic.

The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. __The enharmonic interval's number of ups or downs is equal to the height__. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

[//Question: what if there are highs and lows?//]

Not all enharmonics work with all pergens. The possible enharmonics can be deduced from the pergen as follows:

For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-factor of the period, the enharmonic interval's 3-factor is Mx, and the implied edo is |Mx|. For example, for {P8/2, P5}, the implied edo is |2x|, i.e., an even number. Possible bare enharmonics are M2, d2, and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8/4, P5}, the implied edo is a multiple of 4, and only d2 is possible.

For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are P4, P5, P11, P12, or some other voicing of the 5th. Thus the multi-gen's 3-factor is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-factor of the generator, the bare enharmonic's 3-factor is Ny ± 1, and the implied edo is |Ny ± 1|. For example, for {P8, P4/2}, the implied edo is 2y ± 1, thus it must be an odd number, which rules out 2, 12 and 26, and thus M2, d2 and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8, P12/5}, the implied edo is 5y ± 1.

For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd.

Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. In addition, ups/downs can be exchanged for highs/lows, making 6 possibilities.

For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a triple-dim 4th. The octave genchain would be C - E#^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Abbv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C. Seeing the same pitch represented as both E and A is rather disconcerting. For this reason, enharmonics that are unisons or 2nds are preferred.

For {**P8, multi-gen/N}**, the implied edo = Ny ± 1

For **{P8/M, multi-gen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,

or, the 8ve's implied edo = Mx and the multi-gen's implied edo = Ny ± T

The main table lists all possible notations for each pergen, following the above rules. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.

[//Question: how to find the notation for multi-comma tempers?//]

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen names&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;

&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&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, ...}. These commas are called &lt;strong&gt;notational commas&lt;/strong&gt;. 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;

For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &amp;gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.&lt;br /&gt;

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. One possibility is highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. 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 properly as C Ev G.&lt;br /&gt;

Some combinations of periods and generators are duplicates of other pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. Some combinations are impossible. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.&lt;br /&gt;

The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.&lt;br /&gt;

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#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.&lt;br /&gt;

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

         &lt;td style="text-align: center;"&gt;P4/2 = ^M2 = vm3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;vvm2&lt;br /&gt;

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

         &lt;td style="text-align: center;"&gt;C - D^=Ebv - F&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;{P8, P5/2}&lt;br /&gt;

Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic 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 indicate that the enharmonic interval vanishes. 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 above table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic, thus the edo implies the enharmonic.&lt;br /&gt;

The LCM of the pergen's two splitting fractions 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. &lt;u&gt;The enharmonic interval's number of ups or downs is equal to the height&lt;/u&gt;. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.&lt;br /&gt;

[&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;br /&gt;

Not all enharmonics work with all pergens. The possible enharmonics can be deduced from the pergen as follows:&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-factor of the period, the enharmonic interval's 3-factor is Mx, and the implied edo is |Mx|. For example, for {P8/2, P5}, the implied edo is |2x|, i.e., an even number. Possible bare enharmonics are M2, d2, and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8/4, P5}, the implied edo is a multiple of 4, and only d2 is possible.&lt;br /&gt;

For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are P4, P5, P11, P12, or some other voicing of the 5th. Thus the multi-gen's 3-factor is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-factor of the generator, the bare enharmonic's 3-factor is Ny ± 1, and the implied edo is |Ny ± 1|. For example, for {P8, P4/2}, the implied edo is 2y ± 1, thus it must be an odd number, which rules out 2, 12 and 26, and thus M2, d2 and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8, P12/5}, the implied edo is 5y ± 1.&lt;br /&gt;

For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd.&lt;br /&gt;

Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. In addition, ups/downs can be exchanged for highs/lows, making 6 possibilities.&lt;br /&gt;

For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a triple-dim 4th. The octave genchain would be C - E#^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Abbv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C. Seeing the same pitch represented as both E and A is rather disconcerting. For this reason, enharmonics that are unisons or 2nds are preferred.&lt;br /&gt;

For {&lt;strong&gt;P8, multi-gen/N}&lt;/strong&gt;, the implied edo = Ny ± 1&lt;br /&gt;

For &lt;strong&gt;{P8/M, multi-gen/N}&lt;/strong&gt;, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,&lt;br /&gt;

or, the 8ve's implied edo = Mx and the multi-gen's implied edo = Ny ± T&lt;br /&gt;