Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 08:25:21 UTC</tt>.

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: The original revision id was <tt>~~624778785~~</tt>.

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For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament 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.

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]].

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]].

The largest category contains all single-comma temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a **higher prime**), e.g. 81/80 or 64/63. 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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**The rank-2 pergen from the [(x, y) (0, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x <= n <= x**

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

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Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one ~~multi-gen~~ equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.

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Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] is a JI-agnostic ~~way to~~ name the Porcupine [7] scale.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-fourth heptatonic is a JI-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, the MOS scales of any two temperaments that have the same pergen tend to be the same.

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments.

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.

For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 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 and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.

For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 and g implies dicot, using j and a implies mohajira, but using ^ and 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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.

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==Supplemental materials==

This PDF shows the full lattice for the first 15 pergens, up through the third-something block.

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-something block.

This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.

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For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament 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.

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>.

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>.

The largest category contains all single-comma temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime &gt; 3 (a higher prime), e.g. 81/80 or 64/63. 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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The rank-2 pergen from the [(x, y) (0, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x &lt;= n &lt;= x

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

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Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one ~~multi-gen~~ equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.

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Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] is a JI-agnostic ~~way to~~ name the Porcupine [7] scale.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-fourth heptatonic is a JI-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, the MOS scales of any two temperaments that have the same pergen tend to be the same.

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments.

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.

For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. Certain rank-2 temperaments require another additional pair, highs and lows, written / and \. Dv\ is down-low D, and /5 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 and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.

For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. Certain rank-2 temperaments require another additional pair, highs and lows, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 and g implies dicot, using j and a implies mohajira, but using ^ and 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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.

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<h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a>Supplemental materials</h2>

This PDF shows the full lattice for the first 15 pergens, up through the third-something block.

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-something block.

This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.

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<h2 id="toc16"><a name="Further Discussion-Misc notes"></a>Misc notes</h2>

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a>

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a>

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2018-01-12 08:25:21 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 09:16:10 UTC</tt>.<br>
-: The original revision id was <tt>624778785</tt>.<br>
+: The original revision id was <tt>624782391</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>
@@ Line 17: / Line 17: @@
 For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament 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.
-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]].
+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]].
 The largest category contains all single-comma temperaments with a comma of the form 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P or 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;, where P is a prime &gt; 3 (a **higher prime**), e.g. 81/80 or 64/63. 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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 &lt;span style="display: block; text-align: center;"&gt;**&lt;span style="font-size: 110%;"&gt;The rank-2 pergen from the [(x, y) (0, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x &lt;= n &lt;= x&lt;/span&gt;**
 &lt;/span&gt;
+A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.
 For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).
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 Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.
-Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.
+Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.
 The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.
@@ Line 115: / Line 117: @@
 Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.
-Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] is a JI-agnostic way to name the Porcupine [7] scale.
+Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-fourth heptatonic is a JI-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, the MOS scales of any two temperaments that have the same pergen tend to be the same.
-The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments.
+The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.
-For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 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 and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.
+For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 and g implies dicot, using j and a implies mohajira, but using ^ and 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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.
@@ Line 459: / Line 461: @@
 ==Supplemental materials==
-This PDF shows the full lattice for the first 15 pergens, up through the third-something block.
+This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-something block.
 This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.
@@ Line 554: / Line 556: @@
 For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &amp;quot;semi-fourth&amp;quot;, is of course half-fourth.&lt;br /&gt;
 &lt;br /&gt;
-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 &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;.&lt;br /&gt;
+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 &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. See the notation guide below, under &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials"&gt;Supplemental materials&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 The largest category contains all single-comma temperaments with a comma of the form 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P or 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;, where P is a prime &amp;gt; 3 (a &lt;strong&gt;higher prime&lt;/strong&gt;), e.g. 81/80 or 64/63. 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 &lt;strong&gt;unsplit&lt;/strong&gt;.&lt;br /&gt;
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 &lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;&lt;span style="font-size: 110%;"&gt;The rank-2 pergen from the [(x, y) (0, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x &amp;lt;= n &amp;lt;= x&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/span&gt;&lt;br /&gt;
+A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.&lt;br /&gt;
+&lt;br /&gt;
 For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &amp;lt;= n &amp;lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).&lt;br /&gt;
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@@ Line 972: / Line 976: @@
 Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.&lt;br /&gt;
 &lt;br /&gt;
-Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a &lt;u&gt;double&lt;/u&gt; octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.&lt;br /&gt;
+Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a &lt;u&gt;double&lt;/u&gt; octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.&lt;br /&gt;
 &lt;br /&gt;
 The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.&lt;br /&gt;
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 Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.&lt;br /&gt;
 &lt;br /&gt;
-Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] is a JI-agnostic way to name the Porcupine [7] scale.&lt;br /&gt;
+Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-fourth heptatonic is a JI-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, the MOS scales of any two temperaments that have the same pergen tend to be the same.&lt;br /&gt;
 &lt;br /&gt;
-The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. &lt;br /&gt;
+The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.&lt;br /&gt;
 &lt;br /&gt;
-For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other 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;. Certain rank-2 temperaments require another additional pair, &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \. Dv\ is down-low D, and /5 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 and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.&lt;br /&gt;
+For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other 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;. Certain rank-2 temperaments require another additional pair, &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \. Dv\ is down-low D, and /5 is a high-fifth. 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 and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.&lt;br /&gt;
 &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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Supplemental materials"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;Supplemental materials&lt;/h2&gt;
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-This PDF shows the full lattice for the first 15 pergens, up through the third-something block.&lt;br /&gt;
+This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-something block.&lt;br /&gt;
 &lt;br /&gt;
 This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Misc notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Misc notes&lt;/h2&gt;
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-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3100:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3100 --&gt;&lt;br /&gt;
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3103:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3103 --&gt;&lt;br /&gt;
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 Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:&lt;br /&gt;