Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624381327 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624535539 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-08 00:01:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624535539</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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 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. | 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 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. | ||
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. Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, grouped into blocks by the size of the larger splitting fraction, and | 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. | ||
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to quarter-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens. | |||
The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic. | The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic. | ||
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C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E... | C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E... | ||
If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one. | If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one. | ||
The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination. | The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination. | ||
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lence(s) ||~ split | lence(s) ||~ split | ||
interval(s) ||~ perchain(s) and | interval(s) ||~ perchain(s) and | ||
genchains(s) ||~ examples | genchains(s) ||~ examples || | ||
||= (P8, P5) | ||= (P8, P5) | ||
unsplit ||= none ||= none ||= none ||= C - G ||= meantone, | unsplit ||= none ||= none ||= none ||= C - G ||= meantone, | ||
schismic | schismic || | ||
||~ halves ||~ ||~ ||~ ||~ ||~ || | |||
||~ halves | |||
||= (P8/2, P5) | ||= (P8/2, P5) | ||
half-octave ||= ^^d2 (if 5th | half-octave ||= ^^d2 (if 5th | ||
``>`` 700¢ ||= C^^ = B# ||= P8/2 = vA4 = ^d5 ||= C - F#v=Gb^ - C ||= srutal | ``>`` 700¢ ||= C^^ = B# ||= P8/2 = vA4 = ^d5 ||= C - F#v=Gb^ - C ||= srutal | ||
^1 = 81/80 | ^1 = 81/80 || | ||
||= " ||= vvd2 (if 5th | ||= " ||= vvd2 (if 5th | ||
< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red | < 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red | ||
^1 = 64/63 | ^1 = 64/63 || | ||
||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^ | ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 128/121 | ||
^1 = 33/32 | ^1 = 33/32 || | ||
||= (P8, P4/2) | ||= (P8, P4/2) | ||
half-fourth ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | half-fourth ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore | ||
^1 = 64/63 | ^1 = 64/63 || | ||
||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) || | |||
||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) | |||
||= (P8, P5/2) | ||= (P8, P5/2) | ||
half-fifth ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira | half-fifth ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira | ||
^1 = 33/32 | ^1 = 33/32 || | ||
||= (P8/2, P4/2) | ||= (P8/2, P4/2) | ||
half- | half- | ||
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P5/2 = ^m3 = vM3 | P5/2 = ^m3 = vM3 | ||
P8/2 = v/A4 = ^\d5 | P8/2 = v/A4 = ^\d5 | ||
``=`` ^/ | ``=`` ^/4 ``=`` v\P5 ||= C - D/=Eb\ - F, | ||
C - Eb^=Ev - G, | C - Eb^=Ev - G, | ||
C - F#v/=Gb^\ - C, | C - F#v/=Gb^\ - C, | ||
C - F^/=Gv\ - C ||= 49/48 & 128/121 | C - F^/=Gv\ - C ||= 49/48 & 128/121 | ||
^1 = 33/32 | ^1 = 33/32 | ||
/1 = 64/63 | /1 = 64/63 || | ||
||= " ||= ^^d2, | ||= " ||= ^^d2, | ||
\\m2, | \\m2, | ||
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P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C, | P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C, | ||
C - D/=Eb\ - F, | C - D/=Eb\ - F, | ||
C - Eb^/=Ev\ - G ||= 2048/2025 | C - Eb^/=Ev\ - G ||= 2048/2025 & 49/48 | ||
& 49/48 | |||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 64/63 | /1 = 64/63 || | ||
||= " ||= ^^d2, | ||= " ||= ^^d2, | ||
\\A1, | \\A1, | ||
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P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C, | P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C, | ||
C - Eb/=E\ - G, | C - Eb/=E\ - G, | ||
C - Dv/=Eb^\ - F ||= 2048/2025 | C - Dv/=Eb^\ - F ||= 2048/2025 & 128/121 | ||
& 128/121 | |||
^1 = 81/80 | ^1 = 81/80 | ||
/1 = 33/32 | /1 = 33/32 || | ||
||~ thirds | ||~ thirds ||~ ||~ ||~ ||~ ||~ || | ||
||= (P8/3, P5) | ||= (P8/3, P5) | ||
third-octave ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented | third-octave ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented || | ||
||= (P8, P4/3) | ||= (P8, P4/3) | ||
third-fourth ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = ^^m2 = vM2 ||= C - Dv - Eb^ - F ||= porcupine | third-fourth ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = ^^m2 = vM2 ||= C - Dv - Eb^ - F ||= porcupine || | ||
||= (P8, P5/3) | ||= (P8, P5/3) | ||
third-fifth ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric | third-fifth ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric || | ||
||= (P8, P11/3) | ||= (P8, P11/3) | ||
third-11th ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F | third-11th ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= || | ||
||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^ | ||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vvP5 ||= C - F^ - Cv - F ||= || | ||
||= (P8/3, P4/2) | ||= (P8/3, P4/2) | ||
third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 | third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 | ||
P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C | P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C | ||
C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= | C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= || | ||
||= (P8/3, P5/2) | ||= (P8/3, P5/2) | ||
third-8ve, half-5th ||= v<span style="vertical-align: super;">6</span>m3 ||= C^<span style="vertical-align: super;">6</span> ``=`` Eb ||= ||= ||= | third-8ve, half-5th ||= v<span style="vertical-align: super;">6</span>m3 ||= C^<span style="vertical-align: super;">6</span> ``=`` Eb ||= ||= ||= || | ||
||= (P8/2, P4/3) | ||= (P8/2, P4/3) | ||
half-8ve, third-4th ||= v<span style="vertical-align: super;">6</span>d4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb ||= ||= ||= | half-8ve, third-4th ||= v<span style="vertical-align: super;">6</span>d4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb ||= ||= ||= || | ||
||= " ||= v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb3 | ||= " ||= v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb3 ||= ||= ||= || | ||
||= (P8/2, P5/3) | ||= (P8/2, P5/3) | ||
half-8ve, | half-8ve, | ||
third-5th ||= ^<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> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5 | third-5th ||= ^<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> ||= 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 ||= 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 | 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 - D#vv - Fb^^ - G ||= | C - D#vv - Fb^^ - G ||= || | ||
||= " ||= ^^d2, | ||= " ||= ^^d2, | ||
\\\m2 ||= C^^ = B# | \\\m2 ||= C^^ = B# | ||
C``///`` = Db ||= P8/2 = vA4 = ^d5 | C``///`` = Db ||= P8/2 = vA4 = ^d5 | ||
P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C | P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C | ||
C - /D - \F - G ||= | C - /D - \F - G ||= || | ||
||= (P8/2, P11/3) | ||= (P8/2, P11/3) | ||
half-8ve, | half-8ve, | ||
third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D | third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= ||= ||= || | ||
||= (P8/3, P4/3) | ||= (P8/3, P4/3) | ||
third- | third- | ||
everything ||= ||= ||= ||= ||= | everything ||= ||= ||= ||= ||= || | ||
||~ quarters | ||~ quarters ||~ ||~ ||~ ||~ ||~ || | ||
||= {P8/4, P5} ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv A^ C ||= diminished, | ||= {P8/4, P5} ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished, | ||
^1 = 81/80 | ^1 = 81/80 || | ||
||= {P8, P4/4} ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^ Ev F | ||= {P8, P4/4} ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= || | ||
||= {P8, P5/4} ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv F^ G ||= | ||= {P8, P5/4} ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot || | ||
||= {P8, P11/4} ||= 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> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F | ||= {P8, P11/4} ||= 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> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F ||= || | ||
||= {P8, P12/4} ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = | ||= {P8, P12/4} ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 ||= C Fv Bbvv=A^^ D^ G ||= || | ||
||= {P8/4, P4/2} | ||= {P8/4, P4/2} ||= ||= ||= ||= ||= || | ||
||= {P8/2, P4/4} | ||= {P8/2, P4/4} ||= ||= ||= ||= ||= || | ||
||= {P8/2, P5/4} | ||= {P8/2, P5/4} ||= ||= ||= ||= ||= || | ||
||= {P8/4, P4/3} | ||= {P8/4, P4/3} ||= ||= ||= ||= ||= || | ||
||= {P8/4, P5/3} | ||= {P8/4, P5/3} ||= ||= ||= ||= ||= || | ||
||= {P8/4, P11/3} | ||= {P8/4, P11/3} ||= ||= ||= ||= ||= || | ||
||= {P8/3, P4/4} | ||= {P8/3, P4/4} ||= ||= ||= ||= ||= || | ||
||= {P8/3, P5/4} | ||= {P8/3, P5/4} ||= ||= ||= ||= ||= || | ||
||= {P8/3, P11/4} | ||= {P8/3, P11/4} ||= ||= ||= ||= ||= || | ||
||= {P8/3, P12/4} | ||= {P8/3, P12/4} ||= ||= ||= ||= ||= || | ||
||= {P8/4, P4/4} | ||= {P8/4, P4/4} ||= ||= ||= ||= ||= || | ||
Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just. | Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just. | ||
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However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. | ||
==Pergens and EDOs== | |||
Pergens have much in common with edos. Pergens (rank-2 ones) assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but far fewer than a hundred have been explored. There are an infinite number of pergens, but far fewer than a hundred will suffice most composers. | |||
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. However (P8/3, P5) is supported. | |||
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! For example, 12edo supports (P8, P4/5), (P8, P11/17), (P8, WWP4/29), (P8, W<span style="vertical-align: super;">3</span>P4/41), etc. | |||
How many edos support a given pergen? Again, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and if k is M's N-edo keyspan, k by n. To be fully supported, N/m and k/n must be coprime. | |||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead. | |||
[13b and 18b - needs explanation] | |||
||||~ pergen ||~ supporting edos (12-31 only) || | |||
||= (P8, P5) ||= unsplit ||= 12, 13b, 14*, 15*, 16, 17, 18b*, 19, 20*, 21*, | |||
22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31 || | |||
||~ halves ||~ ||~ || | |||
||= (P8/2, P5) ||= half-octave ||= 12, 14, 16, 18b, 20*, 22, 24*, 26, 28*, 30* || | |||
||= (P8, P4/2) ||= half-fourth ||= 14, 15*, 18b*, 19, 20*, 23,24, 25*, 28*, 29, 30* || | |||
||= (P8, P5/2) ||= half-fifth ||= 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31 || | |||
||= (P8/2, P4/2) ||= half-everything ||= 14, 18b, 20*, 24, 28*, 30* || | |||
||~ thirds ||~ ||~ || | |||
||= (P8/3, P5) ||= third-octave ||= 12, 15, 18b*, 21, 24*, 27, 30* || | |||
||= (P8, P4/3) ||= third-fourth ||= 13b, 14*, 15, 21*, 22, 28*, 29, 30* || | |||
||= (P8, P5/3) ||= third-fifth ||= 15*, 16, 20*, 21, 25*, 26, 30*, 31 || | |||
||= (P8, P11/3) ||= third-11th ||= || | |||
||= (P8/3, P4/2) ||= third-8ve, half-4th ||= 15, 18b*, 24, 30 || | |||
||= (P8/3, P5/2) ||= third-8ve, half-5th ||= 18b, 24, 30 || | |||
||= (P8/2, P4/3) ||= half-8ve, third-4th ||= 14, 22, 28*, 30* || | |||
||= (P8/2, P5/3) ||= half-8ve, third-5th ||= 16, 20*, 26, 30* || | |||
||= (P8/2, P11/3) ||= half-8ve, third-11th ||= || | |||
||= (P8/3, P4/3) ||= third-everything ||= 15, 21, 30* || | |||
||~ quarters ||~ ||~ || | |||
||= {P8/4, P5} ||= ||= 12, 16, 20, 24*, 28 || | |||
||= {P8, P4/4} ||= ||= || | |||
||= {P8, P5/4} ||= ||= || | |||
||= {P8, P11/4} ||= ||= || | |||
||= {P8, P12/4} ||= ||= || | |||
||= {P8/4, P4/2} ||= ||= || | |||
||= {P8/2, P4/4} ||= ||= || | |||
||= {P8/2, P5/4} ||= ||= || | |||
||= {P8/4, P4/3} ||= ||= || | |||
||= {P8/4, P5/3} ||= ||= || | |||
||= {P8/4, P11/3} ||= ||= || | |||
||= {P8/3, P4/4} ||= ||= || | |||
||= {P8/3, P5/4} ||= ||= || | |||
||= {P8/3, P11/4} ||= ||= || | |||
||= {P8/3, P12/4} ||= ||= || | |||
||= {P8/4, P4/4} ||= ||= || | |||
==Misc notes== | |||
Pergens were discovered by Kite Giedraitis in 2017. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names | |||
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<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:31:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:31 --> </h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:31:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:31 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:61:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 2em;"><a href="#toc11"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div> | |||
<!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --></div> | |||
<!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextHeadingRule:33:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:33 --><u><strong>Definition</strong></u></h1> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
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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 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.<br /> | 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 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.<br /> | ||
<br /> | <br /> | ||
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. Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, grouped into blocks by the size of the larger splitting fraction, and | 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. <br /> | ||
<br /> | |||
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to quarter-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.<br /> | |||
<br /> | <br /> | ||
The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.<br /> | The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.<br /> | ||
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C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...<br /> | C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...<br /> | ||
If the octave is split, the table has a <strong>perchain</strong> (&quot;peer-chain&quot;, chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.<br /> | If the octave is split, the table has a <strong>perchain</strong> (&quot;peer-chain&quot;, chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.<br /> | ||
<br /> | <br /> | ||
The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.<br /> | The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.<br /> | ||
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</th> | </th> | ||
<th>examples<br /> | <th>examples<br /> | ||
</th> | </th> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">meantone,<br /> | <td style="text-align: center;">meantone,<br /> | ||
schismic | schismic<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<th>halves<br /> | <th>halves<br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
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<td style="text-align: center;">srutal<br /> | <td style="text-align: center;">srutal<br /> | ||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">large deep red<br /> | <td style="text-align: center;">large deep red<br /> | ||
^1 = 64/63<br /> | ^1 = 64/63<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">C^^ = D<br /> | <td style="text-align: center;">C^^ = D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P8/2 = ^ | <td style="text-align: center;">P8/2 = ^4 = vP5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C - F^=Gv - C<br /> | <td style="text-align: center;">C - F^=Gv - C<br /> | ||
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<td style="text-align: center;">128/121<br /> | <td style="text-align: center;">128/121<br /> | ||
^1 = 33/32<br /> | ^1 = 33/32<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">semaphore<br /> | <td style="text-align: center;">semaphore<br /> | ||
^1 = 64/63<br /> | ^1 = 64/63<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">(-22,-11,2)<br /> | <td style="text-align: center;">(-22,-11,2)<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">mohajira<br /> | <td style="text-align: center;">mohajira<br /> | ||
^1 = 33/32<br /> | ^1 = 33/32<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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P5/2 = ^m3 = vM3<br /> | P5/2 = ^m3 = vM3<br /> | ||
P8/2 = v/A4 = ^\d5<br /> | P8/2 = v/A4 = ^\d5<br /> | ||
<!-- ws:start:WikiTextRawRule:03:``=`` -->=<!-- ws:end:WikiTextRawRule:03 --> ^/ | <!-- ws:start:WikiTextRawRule:03:``=`` -->=<!-- ws:end:WikiTextRawRule:03 --> ^/4 <!-- ws:start:WikiTextRawRule:04:``=`` -->=<!-- ws:end:WikiTextRawRule:04 --> v\P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C - D/=Eb\ - F,<br /> | <td style="text-align: center;">C - D/=Eb\ - F,<br /> | ||
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^1 = 33/32<br /> | ^1 = 33/32<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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C - Eb^/=Ev\ - G<br /> | C - Eb^/=Ev\ - G<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2048/2025 | <td style="text-align: center;">2048/2025 &amp; 49/48<br /> | ||
&amp; 49/48<br /> | |||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
/1 = 64/63<br /> | /1 = 64/63<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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C - Dv/=Eb^\ - F<br /> | C - Dv/=Eb^\ - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2048/2025 | <td style="text-align: center;">2048/2025 &amp; 128/121<br /> | ||
&amp; 128/121<br /> | |||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
/1 = 33/32<br /> | /1 = 33/32<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<th>thirds<br /> | <th>thirds<br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">augmented<br /> | <td style="text-align: center;">augmented<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">porcupine<br /> | <td style="text-align: center;">porcupine<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">slendric<br /> | <td style="text-align: center;">slendric<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">C - F#v - Cb^ - F<br /> | <td style="text-align: center;">C - F#v - Cb^ - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">C^<span style="vertical-align: super;">3 </span><!-- ws:start:WikiTextRawRule:012:``=`` -->=<!-- ws:end:WikiTextRawRule:012 --> D<br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">3 </span><!-- ws:start:WikiTextRawRule:012:``=`` -->=<!-- ws:end:WikiTextRawRule:012 --> D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P11/3 = ^ | <td style="text-align: center;">P11/3 = ^4 = vvP5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C - F^ - Cv - F<br /> | <td style="text-align: center;">C - F^ - Cv - F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,474: | Line 1,462: | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> Fb3<br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> Fb3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,500: | Line 1,486: | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,520: | Line 1,504: | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,532: | Line 1,514: | ||
</td> | </td> | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> D<br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,556: | Line 1,536: | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<th>quarters<br /> | <th>quarters<br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
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<td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2<br /> | <td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C Ebv Gbvv A^ C<br /> | <td style="text-align: center;">C Ebv Gbvv=F#^^ A^ C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">diminished,<br /> | <td style="text-align: center;">diminished,<br /> | ||
^1 = 81/80<br /> | ^1 = 81/80<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,602: | Line 1,576: | ||
<td style="text-align: center;">P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1<br /> | <td style="text-align: center;">P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C Db^ Ebb^^ Ev F | <td style="text-align: center;">C Db^ Ebb^^=D#vv Ev F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,618: | Line 1,590: | ||
<td style="text-align: center;">P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2<br /> | <td style="text-align: center;">P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C Dv Evv F^ G | <td style="text-align: center;">C Dv Evv=Eb^^ F^ G<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">tetracot<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,635: | Line 1,605: | ||
</td> | </td> | ||
<td style="text-align: center;">C E^ G#^^ Dbv F<br /> | <td style="text-align: center;">C E^ G#^^ Dbv F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,648: | Line 1,616: | ||
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 --> Db<br /> | <td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 --> Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">P12/4 = | <td style="text-align: center;">P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C Fv Bbvv=A^^ D^ G<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,659: | Line 1,625: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/4, P4/2}<br /> | <td style="text-align: center;">{P8/4, P4/2}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,675: | Line 1,639: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/2, P4/4}<br /> | <td style="text-align: center;">{P8/2, P4/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,691: | Line 1,653: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/2, P5/4}<br /> | <td style="text-align: center;">{P8/2, P5/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,707: | Line 1,667: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/4, P4/3}<br /> | <td style="text-align: center;">{P8/4, P4/3}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,723: | Line 1,681: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/4, P5/3}<br /> | <td style="text-align: center;">{P8/4, P5/3}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,739: | Line 1,695: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/4, P11/3}<br /> | <td style="text-align: center;">{P8/4, P11/3}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,755: | Line 1,709: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P4/4}<br /> | <td style="text-align: center;">{P8/3, P4/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,771: | Line 1,723: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P5/4}<br /> | <td style="text-align: center;">{P8/3, P5/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,787: | Line 1,737: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P11/4}<br /> | <td style="text-align: center;">{P8/3, P11/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,803: | Line 1,751: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/3, P12/4}<br /> | <td style="text-align: center;">{P8/3, P12/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,819: | Line 1,765: | ||
<tr> | <tr> | ||
<td style="text-align: center;">{P8/4, P4/4}<br /> | <td style="text-align: center;">{P8/4, P4/4}<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,194: | Line 2,138: | ||
<ul><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li></ul><br /> | <ul><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li></ul><br /> | ||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br /> | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br /> | ||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:57:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:57 -->Pergens and EDOs</h2> | |||
<br /> | |||
Pergens have much in common with edos. Pergens (rank-2 ones) assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but far fewer than a hundred have been explored. There are an infinite number of pergens, but far fewer than a hundred will suffice most composers.<br /> | |||
<br /> | |||
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. However (P8/3, P5) is supported.<br /> | |||
<br /> | |||
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! For example, 12edo supports (P8, P4/5), (P8, P11/17), (P8, WWP4/29), (P8, W<span style="vertical-align: super;">3</span>P4/41), etc.<br /> | |||
<br /> | |||
How many edos support a given pergen? Again, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and if k is M's N-edo keyspan, k by n. To be fully supported, N/m and k/n must be coprime.<br /> | |||
<br /> | |||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.<br /> | |||
<br /> | |||
[13b and 18b - needs explanation]<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th colspan="2">pergen<br /> | |||
</th> | |||
<th>supporting edos (12-31 only)<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8, P5)<br /> | |||
</td> | |||
<td style="text-align: center;">unsplit<br /> | |||
</td> | |||
<td style="text-align: center;">12, 13b, 14*, 15*, 16, 17, 18b*, 19, 20*, 21*,<br /> | |||
22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>halves<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/2, P5)<br /> | |||
</td> | |||
<td style="text-align: center;">half-octave<br /> | |||
</td> | |||
<td style="text-align: center;">12, 14, 16, 18b, 20*, 22, 24*, 26, 28*, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8, P4/2)<br /> | |||
</td> | |||
<td style="text-align: center;">half-fourth<br /> | |||
</td> | |||
<td style="text-align: center;">14, 15*, 18b*, 19, 20*, 23,24, 25*, 28*, 29, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8, P5/2)<br /> | |||
</td> | |||
<td style="text-align: center;">half-fifth<br /> | |||
</td> | |||
<td style="text-align: center;">14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/2, P4/2)<br /> | |||
</td> | |||
<td style="text-align: center;">half-everything<br /> | |||
</td> | |||
<td style="text-align: center;">14, 18b, 20*, 24, 28*, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>thirds<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/3, P5)<br /> | |||
</td> | |||
<td style="text-align: center;">third-octave<br /> | |||
</td> | |||
<td style="text-align: center;">12, 15, 18b*, 21, 24*, 27, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8, P4/3)<br /> | |||
</td> | |||
<td style="text-align: center;">third-fourth<br /> | |||
</td> | |||
<td style="text-align: center;">13b, 14*, 15, 21*, 22, 28*, 29, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8, P5/3)<br /> | |||
</td> | |||
<td style="text-align: center;">third-fifth<br /> | |||
</td> | |||
<td style="text-align: center;">15*, 16, 20*, 21, 25*, 26, 30*, 31<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8, P11/3)<br /> | |||
</td> | |||
<td style="text-align: center;">third-11th<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/3, P4/2)<br /> | |||
</td> | |||
<td style="text-align: center;">third-8ve, half-4th<br /> | |||
</td> | |||
<td style="text-align: center;">15, 18b*, 24, 30<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/3, P5/2)<br /> | |||
</td> | |||
<td style="text-align: center;">third-8ve, half-5th<br /> | |||
</td> | |||
<td style="text-align: center;">18b, 24, 30<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/2, P4/3)<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve, third-4th<br /> | |||
</td> | |||
<td style="text-align: center;">14, 22, 28*, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/2, P5/3)<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve, third-5th<br /> | |||
</td> | |||
<td style="text-align: center;">16, 20*, 26, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/2, P11/3)<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve, third-11th<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/3, P4/3)<br /> | |||
</td> | |||
<td style="text-align: center;">third-everything<br /> | |||
</td> | |||
<td style="text-align: center;">15, 21, 30*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>quarters<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/4, P5}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;">12, 16, 20, 24*, 28<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8, P4/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8, P5/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8, P11/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8, P12/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/4, P4/2}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/2, P4/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/2, P5/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/4, P4/3}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/4, P5/3}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/4, P11/3}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/3, P4/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/3, P5/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/3, P11/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/3, P12/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">{P8/4, P4/4}<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:59:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:59 -->Misc notes</h2> | |||
<br /> | |||
Pergens were discovered by Kite Giedraitis in 2017. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule:3025:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3025 --><br /> | |||
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