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Kite's thoughts on pergens: Difference between revisions

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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. Both fractions are 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 '''multigen'''. The 3-limit multigen 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. Both fractions are 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 '''multigen'''. The 3-limit multigen 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 spoken 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, a pun on "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 spoken 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 aka Yoyo temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyo is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozo, a pun on "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 a few dozen 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 (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and Downs Notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Sagugu and Injera aka Gu & Biruyo sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and Downs Notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.


The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime &gt; 3 (a '''higher prime'''), e.g. 81/80 or 135/128. 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'''.
The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime &gt; 3 (a '''higher prime'''), e.g. 81/80 or 135/128. 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 final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.
The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.


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, '''lifts and drops''', written / and \. Dv\ is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.
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, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.


One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.
One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.


Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layo is unsplit and can be notated conventionally. But 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. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layo is unsplit and can be notated conventionally. But 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 vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.


Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
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The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.
The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.


<span style="display: block; text-align: center;">C - G implies ...Eb Bb F C G D A E B F# C#...</span><span style="display: block; text-align: center;">C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D...</span>
<span style="display: block; text-align: center;">C - G implies ...Eb Bb F C G D A E B F# C#...</span><span style="display: block; text-align: center;">C -</span> <span style="display: block; text-align: center;">^</span><span style="display: block; text-align: center;">Eb=</span><span style="display: block; text-align: center;">v</span><span style="display: block; text-align: center;">E - G implies ...F --</span> <span style="display: block; text-align: center;">^</span><span style="display: block; text-align: center;">Ab=</span><span style="display: block; text-align: center;">v</span><span style="display: block; text-align: center;">A -- C --</span> <span style="display: block; text-align: center;">^</span><span style="display: block; text-align: center;">Eb=</span><span style="display: block; text-align: center;">v</span><span style="display: block; text-align: center;">E -- G --</span> <span style="display: block; text-align: center;">^</span><span style="display: block; text-align: center;">Bb=</span><span style="display: block; text-align: center;">v</span><span style="display: block; text-align: center;">B -- D...</span>


If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.
If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- vF#=^Gb -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.


{| class="wikitable" style="text-align:center;"  
{| class="wikitable" style="text-align:center;"  
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| | ^^d2 (if 5th
| | ^^d2 (if 5th
&gt; 700¢
&gt; 700¢
| | C^^ = B#
| | ^^C = B#
| | P8/2 = vA4 = ^d5
| | P8/2 = vA4 = ^d5
| | C - F#v=Gb^ - C
| | C - vF#=^Gb - C
| | Srutal aka Sagugu
| | Srutal aka Sagugu
^1 = 81/80
^1 = 81/80
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&lt; 700¢)
&lt; 700¢)
| | C^^ = Db
| | ^^C = Db
| | P8/2 = ^A4 = vd5
| | P8/2 = ^A4 = vd5
| | C - F#^=Gbv - C
| | C - ^F#=vGb - C
| | Injera aka Gu & Biruyo
| | Injera aka Gu & Biruyo


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| | "
| | "
| | vvM2
| | vvM2
| | C^^ = D
| | ^^C = D
| | P8/2 = ^4 = vP5
| | P8/2 = ^4 = vP5
| | C - F^=Gv - C
| | C - ^F=vG - C
| | Thotho, if 13/8 = M6
| | Thotho, if 13/8 = M6


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half-4th
half-4th
| | vvm2
| | vvm2
| | C^^ = Db
| | ^^C = Db
| | P4/2 = ^M2 = vm3
| | P4/2 = ^M2 = vm3
| | C - D^=Ebv - F
| | C - ^D=vEb - F
| | Semaphore aka Zozo
| | Semaphore aka Zozo


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| | "
| | "
| | ^^dd2
| | ^^dd2
| | C^^ = B##
| | ^^C = B##
| | P4/2 = vA2 = ^d3
| | P4/2 = vA2 = ^d3
| | C - D#v=Ebb^ - F
| | C - vD#=^Ebb - F
| | Lala-yoyo
| | Lala-yoyo


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half-5th
half-5th
| | vvA1
| | vvA1
| | C^^ = C#
| | ^^C = C#
| | P5/2 = ^m3 = vM3
| | P5/2 = ^m3 = vM3
| | C - Eb^=Ev - G
| | C - ^Eb=vE - G
| | Mohajira aka Lulu
| | Mohajira aka Lulu


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vv\\M2
vv\\M2
| | C// = Db
| | //C = Db


C^^ = C#
^^C = C#


C^^// = D
^^//C = D
| | P4/2 = /M2 = \m3
| | P4/2 = /M2 = \m3


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= ^/4 = v\P5
= ^/4 = v\P5
| | C - D/=Eb\ - F,
| | C - /D=\Eb - F,


C - Eb^=Ev - G,
C - ^Eb=vE - G,


C - F#v/=Gb^\ - C,
C - v/F#=^\Gb - C,


C - F^/=Gv\ - C
C - ^/F=v\G - C
| | Zozo &amp; Lulu
| | Zozo &amp; Lulu


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vv\\A1
vv\\A1
| | C^^ = B#
| | ^^ C= B#


C// = Db
//C = Db


C^^// = C#
^^//C = C#
| | P8/2 = vA4 = ^d5
| | P8/2 = vA4 = ^d5


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P5/2 = ^/m3 = v\M3
P5/2 = ^/m3 = v\M3
| | C - F#v=Gb^ - C,
| | C - vF#=^Gb - C,


C - D/=Eb\ - F,
C - /D=\Eb - F,


C - Eb^/=Ev\ - G
C - ^/Eb=v\E - G
| | Sagugu &amp; Zozo
| | Sagugu &amp; Zozo


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^^\\m2
^^\\m2
| | C^^ = B#
| | ^^C = B#


C// = C#
//C = C#


C^^\\ = B
^^\\C = B
| | P8/2 = vA4 = ^d5
| | P8/2 = vA4 = ^d5


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P4/2 =v/M2 = ^\m3
P4/2 =v/M2 = ^\m3
| | C - F#v=Gb^ - C,
| | C - vF#=^Gb - C,


C - Eb/=E\ - G,
C - /Eb=\E - G,


C - Dv/=Eb^\ - F
C - v/D=^\Eb - F
| | Sagugu & Lulu
| | Sagugu & Lulu


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third-8ve
third-8ve
| | ^<span style="vertical-align: super;">3</span>d2
| | ^<span style="vertical-align: super;">3</span>d2
| | C^<span style="vertical-align: super;">3 </span> = B#
| | ^<span style="vertical-align: super;">3</span>C = B#
| | P8/3 = vM3 = ^^d4
| | P8/3 = vM3 = ^^d4
| | C - Ev - Ab^ - C
| | C - vE - ^Ab - C
| | Augmented aka Trigu
| | Augmented aka Trigu


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third-4th
third-4th
| | v<span style="vertical-align: super;">3</span>A1
| | v<span style="vertical-align: super;">3</span>A1
| | C^<span style="vertical-align: super;">3</span> = C#
| | ^<span style="vertical-align: super;">3</span>C = C#
| | P4/3 = vM2 = ^^m2
| | P4/3 = vM2 = ^^m2
| | C - Dv - Eb^ - F
| | C - vD - ^Eb - F
| | Porcupine aka Triyo
| | Porcupine aka Triyo


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third-5th
third-5th
| | v<span style="vertical-align: super;">3</span>m2
| | v<span style="vertical-align: super;">3</span>m2
| | C^<span style="vertical-align: super;">3 </span> = Db
| | ^<span style="vertical-align: super;">3</span>C = Db
| | P5/3 = ^M2 = vvm3
| | P5/3 = ^M2 = vvm3
| | C - D^ - Fv - G
| | C - ^D - vF - G
| | Slendric aka Latrizo
| | Slendric aka Latrizo


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third-11th
third-11th
| | ^<span style="vertical-align: super;">3</span>dd2
| | ^<span style="vertical-align: super;">3</span>dd2
| | C^<span style="vertical-align: super;">3</span> = B##
| | ^<span style="vertical-align: super;">3</span>C = B##
| | P11/3 = vA4 = ^^dd5
| | P11/3 = vA4 = ^^dd5
| | C - F#v - Cb^ - F
| | C - vF# - ^Cb - F
| | Satrilu, if 11/8 = A4
| | Satrilu, if 11/8 = A4


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| | "
| | "
| | v<span style="vertical-align: super;">3</span>M2
| | v<span style="vertical-align: super;">3</span>M2
| | C^<span style="vertical-align: super;">3 </span>= D
| | ^<span style="vertical-align: super;">3</span>C = D
| | P11/3 = ^4 = vv5
| | P11/3 = ^4 = vv5
| | C - F^ - Cv - F
| | C - ^F - vC - F
| | Satrilu, if 11/8 = P4
| | Satrilu, if 11/8 = P4


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third-8ve, half-4th
third-8ve, half-4th
| | v<span style="vertical-align: super;">6</span>A2
| | v<span style="vertical-align: super;">6</span>A2
| | C^<span style="vertical-align: super;">6</span> = D#
| | ^<span style="vertical-align: super;">6</span>C = D#
| | P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4
| | P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4


P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3
P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3
| | C - Eb^^ - Avv - C
| | C - ^^Eb - vvA - C


C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F
C - ^<span style="vertical-align: super;">3</span>Db=v<span style="vertical-align: super;">3</span>E - F
| | Tribilo, if 11/8 = P4
| | Tribilo, if 11/8 = P4


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\\m2
\\m2
| | C^<span style="vertical-align: super;">3</span> = B#
| | ^<span style="vertical-align: super;">3</span>C = B#


C// = Db
//C = Db
| | P8/3 = vM3 = ^^d4
| | P8/3 = vM3 = ^^d4


P4/2 = /M2 = \m3
P4/2 = /M2 = \m3
| | C - Ev - Ab^ - C
| | C - vE - ^Ab - C


C - D/=Eb\ - F
C - /D=\Eb - F
| | Triforce aka Trigu & Zozo
| | Triforce aka Trigu & Zozo


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\\A1
\\A1
| | C^<span style="vertical-align: super;">3</span> = B#
| | ^<span style="vertical-align: super;">3</span>C = B#


C// = C#
//C = C#
| | P8/3 = vM3 = ^^d4
| | P8/3 = vM3 = ^^d4


P5/2 = /m3 = \M3
P5/2 = /m3 = \M3
| | C - Ev - Ab^ - C
| | C - vE - ^Ab - C


C - Eb/=E\ - G
C - /Eb=\E - G
| | Satribizo
| | Satribizo


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\<span style="vertical-align: super;">3</span>A1
\<span style="vertical-align: super;">3</span>A1
| | C^^ = Dbb
| | ^^C = Dbb


C/<span style="vertical-align: super;">3</span> = C#
/<span style="vertical-align: super;">3</span>C = C#
| | P8/2 = vA4 = ^d5
| | P8/2 = vA4 = ^d5


P4/3 = \M2 = //m2
P4/3 = \M2 = //m2
| | C - F#v=Gb^ - C
| | C - vF#=^Gb - C


C - D\ - Eb/ - F
C - \D - /Eb - F
| | Latribiru
| | Latribiru


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half-8ve, third-5th
half-8ve, third-5th
| | ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2
| | ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2
| | C^<span style="vertical-align: super;">6</span> = B#<span style="vertical-align: super;">3</span>
| | ^<span style="vertical-align: super;">6</span>C = B#<span style="vertical-align: super;">3</span>
| | P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5
| | P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5


P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3
P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3
| | C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C
| | C - v<span style="vertical-align: super;">3</span>F<span style="vertical-align: super;">x</span>=^<span style="vertical-align: super;">3</span>Gbb C


C - D#vv - Fb^^ - G
C - vvD# - ^^Fb - G
| | Lartribiyo
| | Lartribiyo


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| | ^^d2,
| | ^^d2,


\\\m2
\<sup>3</sup>m2
| | C^^ = B#
| | ^^C = B#


C/// = Db
/<sup>3</sup>C = Db
| | P8/2 = vA4 = ^d5
| | P8/2 = vA4 = ^d5


P5/3 = /M2 = \\m3
P5/3 = /M2 = \\m3
| | C - F#v=Gb^ - C
| | C - vF#=^Gb - C


C - D/ - F\ - G
C - /D - \F - G
| | Lemba aka Latrizo & Biruyo
| | Lemba aka Latrizo & Biruyo


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half-8ve, third-11th
half-8ve, third-11th
| | v<span style="vertical-align: super;">6</span>M2
| | v<span style="vertical-align: super;">6</span>M2
| | C^<span style="vertical-align: super;">6</span> = D
| | ^<span style="vertical-align: super;">6</span>C = D
| | P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5
| | P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5


P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5
P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5
| | C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C
| | C - ^<span style="vertical-align: super;">3</span>F=v<span style="vertical-align: super;">3</span>G - C


C - F^^ - Cvv - F
C - ^^F - vvC - F
| | Latribilo, if 11/8 = P4
| | Latribilo, if 11/8 = P4


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\<span style="vertical-align: super;">3</span>A1
\<span style="vertical-align: super;">3</span>A1
| | C^<span style="vertical-align: super;">3 </span> = Dbb
| | ^<span style="vertical-align: super;">3</span>C = Dbb


C/3 = C#
/<sup>3</sup>C = C#
| | P8/3 = ^M3 = vvd4
| | P8/3 = ^M3 = vvd4


Line 743: Line 743:


P5/3 = v/M2
P5/3 = v/M2
| | C - E^ - Abv - C
| | C - ^E - vAb - C


C - D\ - Eb/ - F
C - \D - /Eb - F


C - Dv/ - F^\ - G
C - v/D - ^\F - G
| | Triyo &amp; Triru
| | Triyo &amp; Triru


Line 759: Line 759:


\<span style="vertical-align: super;">3</span>m2
\<span style="vertical-align: super;">3</span>m2
| | C^<span style="vertical-align: super;">3 </span> = B#
| | ^<span style="vertical-align: super;">3</span>C = B#


C/<span style="vertical-align: super;">3</span> = Db
/<span style="vertical-align: super;">3</span>C = Db
| | P8/3 = vM3 = ^^d4
| | P8/3 = vM3 = ^^d4


Line 767: Line 767:


P4/3 = v\M2
P4/3 = v\M2
| | C - Ev - Ab^ - C
| | C - vE - ^Ab - C


C - D/ - F\ - G
C - /D - \F - G


C - Dv\ - Eb^/ - F
C - v\D - ^/Eb - F
| | Trigu &amp; Latrizo
| | Trigu &amp; Latrizo


Line 783: Line 783:


\<span style="vertical-align: super;">3</span>m2
\<span style="vertical-align: super;">3</span>m2
| | C^<span style="vertical-align: super;">3 </span> = C#
| | ^<span style="vertical-align: super;">3</span>C = C#


C/3 = Db
/<sup>3</sup>C = Db
| | P4/3 = vM2 = ^^m2
| | P4/3 = vM2 = ^^m2


Line 791: Line 791:


P8/3 = v/M3
P8/3 = v/M3
| | C - Dv - Eb^ - F
| | C - vD - ^Eb - F


C - D/ - F\ - G
C - /D - \F - G


C - Ev/ - Ab^\ - C
C - v/E - ^\Ab - C
| | Triyo &amp; Latrizo
| | Triyo &amp; Latrizo


Line 813: Line 813:
| | (P8/4, P5)
| | (P8/4, P5)
| | ^<span style="vertical-align: super;">4</span>d2
| | ^<span style="vertical-align: super;">4</span>d2
| | C^<span style="vertical-align: super;">4</span> = B#
| | ^<span style="vertical-align: super;">4</span>C = B#
| | P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2
| | P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2
| | C Ebv Gbvv=F#^^ A^ C
| | C vEb vvGb=^^F# ^A C
| | Diminished aka Trigu
| | Diminished aka Trigu
|-
|-
Line 821: Line 821:
| | (P8, P4/4)
| | (P8, P4/4)
| | ^<span style="vertical-align: super;">4</span>dd2
| | ^<span style="vertical-align: super;">4</span>dd2
| | C^<span style="vertical-align: super;">4</span> = B##
| | ^<span style="vertical-align: super;">4</span>C = B##
| | P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1
| | P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1
| | C Db^ Ebb^^=D#vv Ev F
| | C ^Db ^^Ebb=vvD# vE F
| | Negri aka Laquadyo
| | Negri aka Laquadyo
|-
|-
Line 829: Line 829:
| | (P8, P5/4)
| | (P8, P5/4)
| | v<span style="vertical-align: super;">4</span>A1
| | v<span style="vertical-align: super;">4</span>A1
| | C^<span style="vertical-align: super;">4</span> = C#
| | ^<span style="vertical-align: super;">4</span>C = C#
| | P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2
| | P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2
| | C Dv Evv=Eb^^ F^ G
| | C vD vvE=^^Eb ^F G
| | Tetracot aka Saquadyo
| | Tetracot aka Saquadyo
|-
|-
Line 837: Line 837:
| | (P8, P11/4)
| | (P8, P11/4)
| | v<span style="vertical-align: super;">4</span>dd3
| | v<span style="vertical-align: super;">4</span>dd3
| | C^<span style="vertical-align: super;">4</span> = Eb<span style="vertical-align: super;">3</span>
| | ^<span style="vertical-align: super;">4</span>C = Eb<span style="vertical-align: super;">3</span>
| | P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5
| | P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5
| | C E^ G#^^ Dbv F
| | C ^E ^^G# vDb F
| | Squares aka Laquadru
| | Squares aka Laquadru
|-
|-
Line 845: Line 845:
| | (P8, P12/4)
| | (P8, P12/4)
| | v<span style="vertical-align: super;">4</span>m2
| | v<span style="vertical-align: super;">4</span>m2
| | C^<span style="vertical-align: super;">4</span> = Db
| | ^<span style="vertical-align: super;">4</span>C = Db
| | P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3
| | P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3
| | C Fv Bbvv=A^^ D^ G
| | C vF vvBb=^^A ^D G
| | Vulture aka Sasa-quadyo
| | Vulture aka Sasa-quadyo
|-
|-
Line 1,001: Line 1,001:
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyo) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyo) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:


P4/3: C - Dv - Eb^ - F
P4/3: C - vD - ^Eb - F


A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split)
A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split)


m7/3: C - Eb^ - Gv - Bb (also m7/6: C - Dv - Eb^ - F - Gv - Ab^ - Bb)
m7/3: C - ^Eb - vG - Bb (also m7/6: C - vD - ^Eb - F - vG - ^Ab - Bb)


M7/3: C - Ev - G^ - B
M7/3: C - vE - ^G - B


m10/3: C - F - Bb - Eb (also already split) (m10/9 also occurs)
m10/3: C - F - Bb - Eb (also already split) (m10/9 also occurs)


M10/3: C - F^ - Bv - E
M10/3: C - ^F - vB - E


Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:
Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:


^m3/2: C - Dv - Eb^ (^m3 = 6/5)
^m3/2: C - vD - ^Eb (^m3 = 6/5)


^m6/5: C - Dv - Eb^ - F - Gv - Ab^ (^m6 = 8/5)
^m6/5: C - vD - ^Eb - F - vG - ^Ab (^m6 = 8/5)


vm9/4: C - Eb^ - Gv - Bb - Db^ (vm9 = 32/15)
vm9/4: C - ^Eb - vG - Bb - ^Db (vm9 = 32/15)


vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
vM7/2: C - ^F - vB (vM7 = 15/8, probably more harmonious than M7 = 243/128)


More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps.
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps.


For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.
For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.
Line 1,188: Line 1,188:


<span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span>
<span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span>
<span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span>
<span style="display: block; text-align: center;">C -- ^^D=v<span style="vertical-align: super;">3</span>Eb -- vF -- ^G -- ^<span style="vertical-align: super;">3</span>A=vvBb -- C</span>


Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5<span style="">⋅</span>m2 = [5,3] - 5<span style="">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span style="">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span style="">⋅</span>G - 2<span style="">⋅</span>E, G must be ^^m2. The genchain is:
Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5<span style="">⋅</span>m2 = [5,3] - 5<span style="">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span style="">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span style="">⋅</span>G - 2<span style="">⋅</span>E, G must be ^^m2. The genchain is:


<span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^ -- Dv -- Eb^ -- Evv -- F</span>
<span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- ^^Db -- vD -- ^Eb -- vvE -- F</span>


To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.
To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.
Line 1,199: Line 1,199:


<span style="display: block; text-align: center;">P1- - ^<span style="vertical-align: super;">4</span>M2=v<span style="vertical-align: super;">6</span>m3 -- vvP4 -- ^^P5 -- ^<span style="vertical-align: super;">6</span>M6=v<span style="vertical-align: super;">4</span>m7 -- P8</span>
<span style="display: block; text-align: center;">P1- - ^<span style="vertical-align: super;">4</span>M2=v<span style="vertical-align: super;">6</span>m3 -- vvP4 -- ^^P5 -- ^<span style="vertical-align: super;">6</span>M6=v<span style="vertical-align: super;">4</span>m7 -- P8</span>
<span style="display: block; text-align: center;">C -- D^<span style="vertical-align: super;">4</span>=Ebv<span style="vertical-align: super;">6</span> -- Fvv -- G^^ -- A^<span style="vertical-align: super;">6</span>=Bbv<span style="vertical-align: super;">4</span> -- C</span>
<span style="display: block; text-align: center;">C -- ^<span style="vertical-align: super;">4</span>D=v<span style="vertical-align: super;">6</span>Eb -- vvF -- ^^G -- ^<span style="vertical-align: super;">6</span>A=v<span style="vertical-align: super;">4</span>Bb -- C</span>
<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">5</span>M2=v<span style="vertical-align: super;">5</span>m3 -- P4</span>
<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">5</span>M2=v<span style="vertical-align: super;">5</span>m3 -- P4</span>
<span style="display: block; text-align: center;">C -- D^<span style="vertical-align: super;">5</span>=Ebv<span style="vertical-align: super;">5</span> -- F</span>
<span style="display: block; text-align: center;">C -- ^<span style="vertical-align: super;">5</span>D=v<span style="vertical-align: super;">5</span>Eb -- F</span>


To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.
To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.
Line 1,210: Line 1,210:


<span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span>
<span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span>
<span style="display: block; text-align: center;">C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F</span>
<span style="display: block; text-align: center;">C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F</span>


One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling. Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = //d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^//d4.
One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling. Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = //d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^//d4.


<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — P11
<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — \E — /Ab — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — P11
<span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span>
<span style="display: block; text-align: center;">C — ^\E — ^^/Ab=vv\A — v/Db — F</span>


It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.
It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.
Line 1,231: Line 1,231:


<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- P8
<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- P8
<span style="display: block; text-align: center;">C -- Eb^<span style="vertical-align: super;">4</span> -- Av<span style="vertical-align: super;">4</span> -- C
<span style="display: block; text-align: center;">C -- ^<span style="vertical-align: super;">4</span>Eb -- v<span style="vertical-align: super;">4</span>A -- C
<span style="display: block; text-align: center;">P1 -- v<span style="vertical-align: super;">3</span>M2 -- v<span style="vertical-align: super;">6</span>M3=^<span style="vertical-align: super;">6</span>m2 -- ^<span style="vertical-align: super;">3</span>m3 -- P4
<span style="display: block; text-align: center;">P1 -- v<span style="vertical-align: super;">3</span>M2 -- v<span style="vertical-align: super;">6</span>M3=^<span style="vertical-align: super;">6</span>m2 -- ^<span style="vertical-align: super;">3</span>m3 -- P4
<span style="display: block; text-align: center;">C -- Dv<span style="vertical-align: super;">3</span> -- Ev<span style="vertical-align: super;">6</span>=Db^<span style="vertical-align: super;">6</span> -- Eb^<span style="vertical-align: super;">3</span> -- F
<span style="display: block; text-align: center;">C -- v<span style="vertical-align: super;">3</span>D -- v<span style="vertical-align: super;">6</span>E=^<span style="vertical-align: super;">6</span>Db -- ^<span style="vertical-align: super;">3</span>Eb -- F


Because G is a M2 and E is an A2, the equivalent generator G - E is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that Dv<span style="vertical-align: super;">3</span> -- Db^<span style="vertical-align: super;">6</span> is ascending. Double-pair notation may be preferable. This makes P = vM3, E = ^3d2, G = /m2, and E' = /4dd2.
Because G is a M2 and E is an A2, the equivalent generator G - E is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v<span style="vertical-align: super;">3</span>D -- ^<span style="vertical-align: super;">6</span>Db is ascending. Double-pair notation may be preferable. This makes P = vM3, E = ^3d2, G = /m2, and E' = /4dd2.


<span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8
<span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8
<span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C
<span style="display: block; text-align: center;">C -- vE -- ^Ab -- C
<span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
<span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
<span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span>
<span style="display: block; text-align: center;">C -- /Db -- //Ebb=\\D# -- \E -- F</span>


Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic.
Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic.
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To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).
To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).


For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone.
For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone.


Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to E, or the multi-E if the count is &gt; 1. For example, consider Semaphore aka Zozo (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider Lala-yoyo, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to E, or the multi-E if the count is &gt; 1. For example, consider Semaphore aka Zozo (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyo, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.


Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.
Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.
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For example, Satrilu tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.
For example, Satrilu tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.


Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, Liese aka Gu & Trizogu (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and E = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- B^ -- F.
Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, Liese aka Gu & Trizogu (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and E = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.


This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.
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Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups_and_Downs_Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups_and_Downs_Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.


In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.
In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.


Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Sagugu & Ru (2.3.5.7 with 2048/2025 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = Cv,7.
Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Sagugu & Ru (2.3.5.7 with 2048/2025 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.


A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyo (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.
A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyo (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.


Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.
Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.


Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A Porcupine aka Triyo (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.
Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyo (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.


Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with E = v<sup>3</sup>A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.
Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with E = v<sup>3</sup>A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.


<u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3) C^<sup>3</sup> = C#</span></u>
<u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^<sup>3</sup>C = C#</span></u>


[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]
[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]
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| | C Eb G
| | C Eb G
| | ^M3
| | ^M3
| | C E^ G
| | C ^E G
| | ^A1
| | ^A1
| | 80/81 = d1
| | 80/81 = d1
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| | C E# G
| | C E# G
| | ^M3
| | ^M3
| | C E^ G
| | C ^E G
| | vA1
| | vA1
| | 80/81 = A1
| | 80/81 = A1
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| | C Fb G
| | C Fb G
| | vM3
| | vM3
| | C Ev G
| | C vE G
| | ^d2
| | ^d2
| | 81/80 = -d2
| | 81/80 = -d2
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| | C D## G
| | C D## G
| | vM3
| | vM3
| | C Ev G
| | C vE G
| | vd2
| | vd2
| | 81/80 = d2
| | 81/80 = d2
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| | C F G
| | C F G
| | vM3
| | vM3
| | C Ev G
| | C vE G
| | ^m2
| | ^m2
| | 81/80 = -m2
| | 81/80 = -m2
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| | C D# G
| | C D# G
| | vM3
| | vM3
| | C Ev G
| | C vE G
| | vm2
| | vm2
| | 81/80 = m2
| | 81/80 = m2
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For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).
For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).


A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Ru aka Archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.
A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Ru aka Archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.


==Notating rank-3 pergens==
==Notating rank-3 pergens==
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| | ^^\\\dd3
| | ^^\\\dd3
|}
|}
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.


There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyo is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyo is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.


With this standardization, the enharmonic can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.
With this standardization, the enharmonic can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.
Line 1,609: Line 1,609:
This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyo doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.
This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyo doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.


Unlike the previous examples, Demeter aka Trizo-agugu's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.
Unlike the previous examples, Demeter aka Trizo-agugu's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.


There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.
There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.
Line 1,788: Line 1,788:
| | E = m2
| | E = m2
| | D E=F G A B=C D
| | D E=F G A B=C D
| | D F#v=Gv Bvv...
| | D vF#=vG vvB...
| | 81/80 = 16/15
| | 81/80 = 16/15
| | ---
| | ---
Line 1,797: Line 1,797:
| | E = A1
| | E = A1
| | D E F G A B C D
| | D E F G A B C D
| | D F^ A^^...
| | D ^F ^^A...
| | 80/81 = 135/128
| | 80/81 = 135/128
| | ---
| | ---
Line 1,805: Line 1,805:
| | rank-2 10-edo
| | rank-2 10-edo
| | E = m2, E' = vvA1 = vvM2
| | E = m2, E' = vvA1 = vvM2
| | D D^=Ev E=F F^=Gv G...
| | D ^D=vE E=F ^F=vG G...
| | D F#\=G\ B\\...
| | D \F#=\G \\B...
| | (see below)
| | (see below)
| | 81/80
| | 81/80
Line 1,815: Line 1,815:
| | E = d2
| | E = d2
| | D D#=Eb E F F#=Gb...
| | D D#=Eb E F F#=Gb...
| | D G^ C^^
| | D ^G ^^C
| | 33/32
| | 33/32
| | ---
| | ---
Line 1,824: Line 1,824:
| | "
| | "
| | "
| | "
| | D G#v=Abv Dvv...
| | D vG#=vAb vvD...
| | 729/704
| | 729/704
| | ---
| | ---
Line 1,832: Line 1,832:
| | rank-2 17-edo
| | rank-2 17-edo
| | E = dd3, E' = vm2 = vvA1
| | E = dd3, E' = vm2 = vvA1
| | D D^=Eb D#=Ev E F...
| | D ^D=Eb D#=vE E F...
| | D F#\ A#\\=Bv\\...
| | D \F# \\A#=v\\B...
| | 256/243
| | 256/243
| | 81/80
| | 81/80
Line 1,863: Line 1,863:
| | fifth-8ve
| | fifth-8ve
| | E = v<span style="vertical-align: super;">5</span>m2
| | E = v<span style="vertical-align: super;">5</span>m2
| | D E^^ Gv A^ Cvv D
| | D ^^E vG ^A vvC D
| | C G D A E...
| | C G D A E...
| | 49/48
| | 49/48
Line 1,877: Line 1,877:
| | 64/63
| | 64/63
|}
|}
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C ^^D vF ^G vvBb C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.


==Notating non-8ve and no-5ths pergens==
==Notating non-8ve and no-5ths pergens==
Line 1,885: Line 1,885:
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.


But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.
But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.


Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.
Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.
Line 2,080: Line 2,080:


C2 --- G2<br>
C2 --- G2<br>
F#v1 C#v2<br>
vF#1 vC#2<br>
C1 --- G1
C1 --- G1


A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.
A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and vC#2 bisects it. vG#2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.


C2 --- G2 --- D3 --- A3<br>
C2 --- G2 --- D3 --- A3<br>
F#v1 C#v2 G#v2 D#v3<br>
vF#1 vC#2 vG#2 vD#3<br>
C1 --- G1 --- D2 --- A2
C1 --- G1 --- D2 --- A2


Splitting the 5th adds notes to the horizontal edges of the square:
Splitting the 5th adds notes to the horizontal edges of the square:


C2 Ev2 G2<br>
C2 vE2 G2<br>
| . . . . . . |<br>
| . . . . . . |<br>
C1 Ev1 G1
C1 vE1 G1


Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.
Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.


C3 Ev3 G3<br>
C3 vE3 G3<br>
| . . . . . . |<br>
| . . . . . . |<br>
C2 Ev2 G2<br>
C2 vE2 G2<br>
| . . . . . . |<br>
| . . . . . . |<br>
C1 Ev1 G1
C1 vE1 G1


From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.
From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.


C2 ---- G2<br>
C2 ---- G2<br>
| . A^1 . |<br>
| . ^A1 . |<br>
C1 ---- G1
C1 ---- G1


A^1 also bisects the P12 from C1 to G2.
^A1 also bisects the P12 from C1 to G2.


Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.
Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.
Line 3,949: Line 3,949:
{| class="wikitable" style="text-align:center;"  
{| class="wikitable" style="text-align:center;"  
|-
|-
| | D#v
| | vD#
| | E^
| | ^E
| | F#
| | F#
| | G#v
| | vG#
| | A^
| | ^A
| | B
| | B
| | C#v
| | vC#
| | D^
| | ^D
|-
|-
| | D^
| | ^D
| | E
| | E
| | F#v
| | vF#
| | G^
| | ^G
| | A
| | A
| | Bv
| | vB
| | C^
| | ^C
| | D
| | D
|-
|-
| | D
| | D
| | Ev
| | vE
| | F^
| | ^F
| | G
| | G
| | Av
| | vA
| | B^
| | ^B
| | C
| | C
| | Dv
| | vD
|-
|-
| | Dv
| | vD
| | Eb^
| | ^Eb
| | F
| | F
| | Gv
| | vG
| | Ab^
| | ^Ab
| | Bb
| | Bb
| | Cv
| | vC
| | Db^
| | ^Db
|}
|}


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