Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624544941 - 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> | ||
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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. | 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. | 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. | ||
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A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3). | A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3). | ||
Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4*M3 = [1,2] = dd3. But by using double-pair notation, we can get enharmonics that are 2nds. First we find P11/2, which equals two generators: P11/2 = 2*G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2*m6 = [1,0] = A1, so E = vvA1 and 2*G = ^m6 or vM6. Next we find G = (2*G)/2, using either ^m6 or vM6. | Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4*M3 = [1,2] = dd3. But by using double-pair notation, we can get enharmonics that are 2nds. First we find P11/2, which equals two generators: P11/2 = 2*G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2*m6 = [1,0] = A1, so E = vvA1 and 2*G = ^m6 or vM6. Next we find G = (2*G)/2, using either ^m6 or vM6. | ||
//But ^m6 has an up sign, and there's no such thing as half an up. The answer is to double all ups and downs: P11/2 = ^^m6 or vvM6, and E = v<span style="vertical-align: super;">4</span>A1. The bare generator is ^^m6/2 = ^^[8,5]/2 = ^[4,2] = ^M3, and the bare enharmonic is ^^m6 - 2*(^M3) = [0,1] = d2. For the second enharmonic, we use the second pair of accidentals: E' = //d2, and G = ^\M3 or ^/d4. Here is the genchain: | //But ^m6 has an up sign, and there's no such thing as half an up. The answer is to double all ups and downs: P11/2 = ^^m6 or vvM6, and E = v<span style="vertical-align: super;">4</span>A1. The bare generator is ^^m6/2 = ^^[8,5]/2 = ^[4,2] = ^M3, and the bare enharmonic is ^^m6 - 2*(^M3) = [0,1] = d2. For the second enharmonic, we use the second pair of accidentals: E' =// d2, and G = ^\M3 or ^/d4. Here is the genchain: | ||
P1 -- ^\M3=^/d4 -- ^^m6=vvM6 -- v\A8=v/m9 -- P11 | |||
C -- E^\=Fb^/ -- Ab^^=Avv -- C#v\=Dbv/ -- F | |||
Using vvM6/2 for 2*G gives a different but equally valid notation: vvM6/2 = vv[9,5]/2 = v[4,2] = vM3, and vvM6 - 2*(vM3) = [1,1] = m2, and E' = \\m2 and G = v/M3 or v\4. | |||
P1 -- v/M3=v\4 -- vvM6=^^m6 -- ^/8=^\m9 -- P11 | |||
C -- Ev/=Fv\ -- Avv=Ab^^ -- C^/=Db^\ -- F | |||
<span style="display: block; text-align: center;"> | |||
</span><span style="display: block; text-align: center;"> | </span><span style="display: block; text-align: center;"> | ||
<span style="display: block; text-align: left;">The bare generator is m6/2 = [8,5]/2 = [4,2] = M3, and the bare enharmonic is P11 - 4*M3 = [17,10] - [16,8] = [1,2] = dd3. For the second enharmonic, we use the second pair of accidentals: E' = \\\\dd3, and G = /M3. E = vvA1, so ^1 = 50¢ + 3.5c. E' = \\\\dd3, so /1 = 25¢ - 4.25c. E + E' = vv\\\\d3 = 2 * v\\m2 = 0¢. E' - E = [0,2] = 2 * ^\\d2. Here is the genchain:</span> | |||
<span style="display: block; text-align: left;">The bare generator is m6/2 = [8,5]/2 = [4,2] = M3, and the bare enharmonic is P11 - 4*M3 = [17,10] - [16,8] = [1,2] = dd3. For the second enharmonic, we use the second pair of accidentals: E' = \\\\dd3, and G = /M3. E = vvA1, so ^1 = 50¢ + 3.5c. E' = \\\\dd3, so /1 = 25¢ - 4.25c. E + E' = vv\\\\d3 = 2 * v\\m2 = 0¢. E' - E = [0,2] = 2 * ^\\d2. Here is the genchain:</span | |||
<span style="display: block; text-align: center;">P1 -- \M3=/d4 -- ^m6=vM6 -- \A8=/m9 -- P11C -- E/=Fv\ -- Ab^=Av -- C^/=Db\ -- F</span> | |||
<span style="display: block; text-align: center;">P1 -- \M3=/d4 -- ^m6=vM6 -- \A8=/m9 -- | |||
<span style="display: block; text-align: left;">Using vM6/2 for 2*G gives a different but equally valid notation: M6/2 = [9,5]/2 = [4,2] = M3, and M6 - 2*(M3) = [1,1] = m2, and E' = \\m2 and G = /M3 or \4.</span> | <span style="display: block; text-align: left;">Using vM6/2 for 2*G gives a different but equally valid notation: M6/2 = [9,5]/2 = [4,2] = M3, and M6 - 2*(M3) = [1,1] = m2, and E' = \\m2 and G = /M3 or \4.</span> | ||
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C -- E/=F\ -- Av=Ab^ -- C/=Db\ -- F | C -- E/=F\ -- Av=Ab^ -- C/=Db\ -- F | ||
</span> | </span> | ||
E' = \\m2, so /1 = 50¢ - 2.5c. But P11 = 1700¢ - c, so /1 should be 425¢ - c/4 - (400¢ + 4c) = 25¢ - 4.25c = (100¢ - 17c)/4 = dd3/4, and E' = \\\\dd3. E/ = Gbb\\\. G# //= Bbb\\ = Av = Ab^. So \\vm2 = 0¢.// | |||
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The comma equals xE and/or yE'. | //The comma equals xE and/or yE'.// | ||
If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE", but not always the smallest E" | //If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE", but not always the smallest E"// | ||
bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2. | //bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2.// | ||
(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F. | //(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.// | ||
These commas are called **notational commas**. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. | //These commas are called **notational commas**. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.// | ||
An alternate enharmonic will arise if the notational comma changes. For example, 11's notational comma can be either 33/32, with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4. The keyspan of all 11-limit intervals will reflect this choice of notational comma. For (P8, P5/2), G ~ 350¢. If G = 11/9, the (vanishing, not notational) comma is P5 - 2*G = 243/242. For the first notational comma, 11/9 is a m3, and the comma is an A1. For the 2nd, 11/9= M3, and the comma is a d1. | //An alternate enharmonic will arise if the notational comma changes. For example, 11's notational comma can be either 33/32, with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4. The keyspan of all 11-limit intervals will reflect this choice of notational comma. For (P8, P5/2), G ~ 350¢. If G = 11/9, the (vanishing, not notational) comma is P5 - 2*G = 243/242. For the first notational comma, 11/9 is a m3, and the comma is an A1. For the 2nd, 11/9= M3, and the comma is a d1.// | ||
For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves. | //For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.// | ||
(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions. | //(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.// | ||
==Alternate keyspans and stepspans== | ==Alternate keyspans and stepspans== | ||
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==Chord names== | ==Chord names== | ||
Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows 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 highs and lows 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-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.) | In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.) | ||
Given a specific temperament, the full mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8*P - 2*G = 4*P8 - 2*P5 = WWm7, and 7/4 = m7. 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 = C7(v3). | Given a specific temperament, the full mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8*P - 2*G = 4*P8 - 2*P5 = WWm7, and 7/4 = m7. 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 = C7(v3). | ||
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 (2.3.5.7 with 81/80 and 50/49, or rryy&gT) is also half-octave. 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 0) (2 1) (0 4) (1 4)]. The square mapping (the first two columns) are the same, hence the pergen is the same, but the other columns are different, hence 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 (2.3.5.7 with 81/80 and 50/49, or rryy&gT) is also half-octave. 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 0) (2 1) (0 4) (1 4)]. The square mapping (the first two columns) are the same, hence the pergen is the same, but the other columns are different, hence 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. | ||
== == | == == | ||
==Combining pergens== | ==Combining pergens== | ||
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How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinite possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7. | How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinite possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7. | ||
How many edos support a given pergen? | How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. | ||
Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). | Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen. | ||
/ | |||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. | ||
||||~ pergen ||~ supporting edos (12-31 only) || | ||||~ pergen ||~ supporting edos (12-31 only) || | ||
||= (P8, P5) ||= unsplit ||= 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*, | ||= (P8, P5) ||= unsplit ||= 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*, | ||
21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31 || | 21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31 || | ||
||~ halves ||~ ||~ || | ||~ halves ||~ ||~ || | ||
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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. <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.<br /> | ||
<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 /> | 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 /> | ||
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A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br /> | A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br /> | ||
<br /> | <br /> | ||
Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4*M3 = [1,2] = dd3. But by using double-pair notation, we can get enharmonics that are 2nds. First we find P11/2, which equals two generators: P11/2 = 2*G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2*m6 = [1,0] = A1, so E = vvA1 and 2*G = ^m6 or vM6. Next we find G = (2*G)/2, using either ^m6 or vM6. <br /> | Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4*M3 = [1,2] = dd3. But by using double-pair notation, we can get enharmonics that are 2nds. First we find P11/2, which equals two generators: P11/2 = 2*G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2*m6 = [1,0] = A1, so E = vvA1 and 2*G = ^m6 or vM6. Next we find G = (2*G)/2, using either ^m6 or vM6.<br /> | ||
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<em>But ^m6 has an up sign, and there's no such thing as half an up. The answer is to double all ups and downs: P11/2 = ^^m6 or vvM6, and E = v<span style="vertical-align: super;">4</span>A1. The bare generator is ^^m6/2 = ^^[8,5]/2 = ^[4,2] = ^M3, and the bare enharmonic is ^^m6 - 2*(^M3) = [0,1] = d2. For the second enharmonic, we use the second pair of accidentals: E' =</em> d2, and G = ^\M3 or ^/d4. Here is the genchain:<br /> | |||
P1 -- ^\M3=^/d4 -- ^^m6=vvM6 -- v\A8=v/m9 -- P11<br /> | |||
C -- E^\=Fb^/ -- Ab^^=Avv -- C#v\=Dbv/ -- F<br /> | |||
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Using vvM6/2 for 2*G gives a different but equally valid notation: vvM6/2 = vv[9,5]/2 = v[4,2] = vM3, and vvM6 - 2*(vM3) = [1,1] = m2, and E' = \\m2 and G = v/M3 or v\4.<br /> | |||
P1 -- v/M3=v\4 -- vvM6=^^m6 -- ^/8=^\m9 -- P11<br /> | |||
C -- Ev/=Fv\ -- Avv=Ab^^ -- C^/=Db^\ -- F<br /> | |||
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</span><span style="display: block; text-align: center;"><br /> | </span><span style="display: block; text-align: center;"><br /> | ||
<span style="display: block; text-align: left;">The bare generator is m6/2 = [8,5]/2 = [4,2] = M3, and the bare enharmonic is P11 - 4*M3 = [17,10] - [16,8] = [1,2] = dd3. For the second enharmonic, we use the second pair of accidentals: E' = \\\\dd3, and G = /M3. E = vvA1, so ^1 = 50¢ + 3.5c. E' = \\\\dd3, so /1 = 25¢ - 4.25c. E + E' = vv\\\\d3 = 2 * v\\m2 = 0¢. E' - E = [0,2] = 2 * ^\\d2. Here is the genchain:</span><br /> | |||
<span style="display: block; text-align: left;">The bare generator is m6/2 = [8,5]/2 = [4,2] = M3, and the bare enharmonic is P11 - 4*M3 = [17,10] - [16,8] = [1,2] = dd3. For the second enharmonic, we use the second pair of accidentals: E' = \\\\dd3, and G = /M3. E = vvA1, so ^1 = 50¢ + 3.5c. E' = \\\\dd3, so /1 = 25¢ - 4.25c. E + E' = vv\\\\d3 = 2 * v\\m2 = 0¢. E' - E = [0,2] = 2 * ^\\d2. Here is the genchain:</span | <br /> | ||
<span style="display: block; text-align: center;">P1 -- \M3=/d4 -- ^m6=vM6 -- \A8=/m9 -- P11C -- E/=Fv\ -- Ab^=Av -- C^/=Db\ -- F</span><br /> | |||
<span style="display: block; text-align: center;">P1 -- \M3=/d4 -- ^m6=vM6 -- \A8=/m9 -- | |||
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<span style="display: block; text-align: left;">Using vM6/2 for 2*G gives a different but equally valid notation: M6/2 = [9,5]/2 = [4,2] = M3, and M6 - 2*(M3) = [1,1] = m2, and E' = \\m2 and G = /M3 or \4.</span><br /> | <span style="display: block; text-align: left;">Using vM6/2 for 2*G gives a different but equally valid notation: M6/2 = [9,5]/2 = [4,2] = M3, and M6 - 2*(M3) = [1,1] = m2, and E' = \\m2 and G = /M3 or \4.</span><br /> | ||
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C -- E/=F\ -- Av=Ab^ -- C/=Db\ -- F<br /> | C -- E/=F\ -- Av=Ab^ -- C/=Db\ -- F<br /> | ||
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E' = \\m2, so /1 = 50¢ - 2.5c. But P11 = 1700¢ - c, so /1 should be 425¢ - c/4 - (400¢ + 4c) = 25¢ - 4.25c = (100¢ - 17c)/4 = dd3/4, and E' = \\\\dd3. E/ = Gbb\\\. G# <em>= Bbb\\ = Av = Ab^. So \\vm2 = 0¢.</em><br /> | |||
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The comma equals xE and/or yE'.<br /> | <em>The comma equals xE and/or yE'.</em><br /> | ||
If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE&quot;, but not always the smallest E&quot;<br /> | <em>If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE&quot;, but not always the smallest E&quot;</em><br /> | ||
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bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2.<br /> | <em>bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2.</em><br /> | ||
(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.<br /> | <em>(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.</em><br /> | ||
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These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.<br /> | <em>These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.</em><br /> | ||
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An alternate enharmonic will arise if the notational comma changes. For example, 11's notational comma can be either 33/32, with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4. The keyspan of all 11-limit intervals will reflect this choice of notational comma. For (P8, P5/2), G ~ 350¢. If G = 11/9, the (vanishing, not notational) comma is P5 - 2*G = 243/242. For the first notational comma, 11/9 is a m3, and the comma is an A1. For the 2nd, 11/9= M3, and the comma is a d1.<br /> | <em>An alternate enharmonic will arise if the notational comma changes. For example, 11's notational comma can be either 33/32, with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4. The keyspan of all 11-limit intervals will reflect this choice of notational comma. For (P8, P5/2), G ~ 350¢. If G = 11/9, the (vanishing, not notational) comma is P5 - 2*G = 243/242. For the first notational comma, 11/9 is a m3, and the comma is an A1. For the 2nd, 11/9= M3, and the comma is a d1.</em><br /> | ||
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For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.<br /> | <em>For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.</em><br /> | ||
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(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | <em>(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.</em><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:51:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Alternate keyspans and stepspans"></a><!-- ws:end:WikiTextHeadingRule:51 -->Alternate keyspans and stepspans</h2> | <!-- ws:start:WikiTextHeadingRule:51:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Alternate keyspans and stepspans"></a><!-- ws:end:WikiTextHeadingRule:51 -->Alternate keyspans and stepspans</h2> | ||
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Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> 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. <br /> | Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> 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.<br /> | ||
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In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)<br /> | In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)<br /> | ||
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Given a specific temperament, the full mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8*P - 2*G = 4*P8 - 2*P5 = WWm7, and 7/4 = m7. 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 = C7(v3). <br /> | Given a specific temperament, the full mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8*P - 2*G = 4*P8 - 2*P5 = WWm7, and 7/4 = m7. 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 = C7(v3).<br /> | ||
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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 (2.3.5.7 with 81/80 and 50/49, or rryy&amp;gT) is also half-octave. 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 0) (2 1) (0 4) (1 4)]. The square mapping (the first two columns) are the same, hence the pergen is the same, but the other columns are different, hence 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.<br /> | 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 (2.3.5.7 with 81/80 and 50/49, or rryy&amp;gT) is also half-octave. 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 0) (2 1) (0 4) (1 4)]. The square mapping (the first two columns) are the same, hence the pergen is the same, but the other columns are different, hence 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:57:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:57 -->Combining pergens</h2> | <!-- ws:start:WikiTextHeadingRule:57:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:57 -->Combining pergens</h2> | ||
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How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinite possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.<br /> | How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinite possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.<br /> | ||
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How many edos support a given pergen? | How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br /> | ||
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Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). | Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br /> | ||
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This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br /> | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br /> | ||
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<td style="text-align: center;">unsplit<br /> | <td style="text-align: center;">unsplit<br /> | ||
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<td style="text-align: center;">12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*, <br /> | <td style="text-align: center;">12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,<br /> | ||
21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31<br /> | 21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31<br /> | ||
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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule: | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule:3082:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3082 --><br /> | ||
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