Kite's thoughts on pergens: Difference between revisions
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Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc. | Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc. | ||
For example, Marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit. | For example, Marvel aka Ruyoyo (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit. | ||
More examples: Trizogu (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyo (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyo | More examples: Trizogu (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyo (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. | ||
A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher. | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher. | ||
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In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes. | In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes. | ||
For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below). | For example, consider Sawa & Ruyoyo (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below). | ||
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents. | To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents. | ||
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A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen. | A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen. | ||
For example, Porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3). | For example, Porcupine aka Triyo (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3). | ||
Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix: | Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozogu is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix: | ||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
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Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources. | Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources. | ||
Another obvious application is to | Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator. | ||
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyo, and the second one is Triyo & Ru. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly. | Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyo, and the second one is Triyo & Ru. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly. | ||
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The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3. | The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3. | ||
All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups_and_Downs_Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. Dv\ is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira and Dicot are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation. | All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups_and_Downs_Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. Dv\ is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation. | ||
One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible. | One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible. | ||
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below. | Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layo is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below. | ||
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. | Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5. | ||
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| | P8/2 = vA4 = ^d5 | | | P8/2 = vA4 = ^d5 | ||
| | C - F#v=Gb^ - C | | | C - F#v=Gb^ - C | ||
| | Srutal | | | Srutal aka Sagugu | ||
^1 = 81/80 | ^1 = 81/80 | ||
|- | |- | ||
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| | P8/2 = ^A4 = vd5 | | | P8/2 = ^A4 = vd5 | ||
| | C - F#^=Gbv - C | | | C - F#^=Gbv - C | ||
| | Injera | | | Injera aka Gu & Biruyo | ||
^1 = 64/63 | ^1 = 64/63 | ||
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| | P4/2 = ^M2 = vm3 | | | P4/2 = ^M2 = vm3 | ||
| | C - D^=Ebv - F | | | C - D^=Ebv - F | ||
| | Semaphore | | | Semaphore aka Zozo | ||
^1 = 64/63 | ^1 = 64/63 | ||
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| | P5/2 = ^m3 = vM3 | | | P5/2 = ^m3 = vM3 | ||
| | C - Eb^=Ev - G | | | C - Eb^=Ev - G | ||
| | Mohajira | | | Mohajira aka Lulu | ||
^1 = 33/32 | ^1 = 33/32 | ||
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| | P8/3 = vM3 = ^^d4 | | | P8/3 = vM3 = ^^d4 | ||
| | C - Ev - Ab^ - C | | | C - Ev - Ab^ - C | ||
| | Augmented | | | Augmented aka Trigu | ||
^1 = 81/80 | ^1 = 81/80 | ||
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| | P4/3 = vM2 = ^^m2 | | | P4/3 = vM2 = ^^m2 | ||
| | C - Dv - Eb^ - F | | | C - Dv - Eb^ - F | ||
| | Porcupine | | | Porcupine aka Triyo | ||
^1 = 81/80 | ^1 = 81/80 | ||
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| | P5/3 = ^M2 = vvm3 | | | P5/3 = ^M2 = vvm3 | ||
| | C - D^ - Fv - G | | | C - D^ - Fv - G | ||
| | Slendric | | | Slendric aka Latrizo | ||
^1 = 64/63 | ^1 = 64/63 | ||
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C - D/=Eb\ - F | C - D/=Eb\ - F | ||
| | Triforce | | | Triforce aka Trigu & Zozo | ||
^1 = 81/80, /1 = 64/63 | ^1 = 81/80, /1 = 64/63 | ||
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C - D/ - F\ - G | C - D/ - F\ - G | ||
| | Lemba | | | Lemba aka Latrizo & Biruyo | ||
^1 = (10,-6,1,-1), /1 = 64/63 | ^1 = (10,-6,1,-1), /1 = 64/63 | ||
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| | P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 | | | P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 | ||
| | C Ebv Gbvv=F#^^ A^ C | | | C Ebv Gbvv=F#^^ A^ C | ||
| | Diminished | | | Diminished aka Trigu | ||
|- | |- | ||
| | 17 | | | 17 | ||
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| | P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 | | | P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 | ||
| | C Db^ Ebb^^=D#vv Ev F | | | C Db^ Ebb^^=D#vv Ev F | ||
| | Negri | | | Negri aka Laquadyo | ||
|- | |- | ||
| | 18 | | | 18 | ||
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| | P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 | | | P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 | ||
| | C Dv Evv=Eb^^ F^ G | | | C Dv Evv=Eb^^ F^ G | ||
| | Tetracot | | | Tetracot aka Saquadyo | ||
|- | |- | ||
| | 19 | | | 19 | ||
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| | P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 | | | P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 | ||
| | C E^ G#^^ Dbv F | | | C E^ G#^^ Dbv F | ||
| | Squares | | | Squares aka Laquadru | ||
|- | |- | ||
| | 20 | | | 20 | ||
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| | P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 | | | P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 | ||
| | C Fv Bbvv=A^^ D^ G | | | C Fv Bbvv=A^^ D^ G | ||
| | Vulture | | | Vulture aka Sasa-quadyo | ||
|- | |- | ||
| | | | | | ||
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==Naming very large intervals== | ==Naming very large intervals== | ||
So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, | So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, adding an 8ve is indicated by "c" for '''compound''' (a conventional music theory term). Thus 10/3 = cM6 = compound major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5. | ||
==Secondary splits== | ==Secondary splits== | ||
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. | Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyo) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with: | ||
P4/3: C - Dv - Eb^ - F | P4/3: C - Dv - Eb^ - F | ||
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==Finding an example temperament== | ==Finding an example temperament== | ||
To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span style="">⋅</span>P and P8. If P is 6/5, the comma is 4<span style="">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4<span style="">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>, the | To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span style="">⋅</span>P and P8. If P is 6/5, the comma is 4<span style="">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4<span style="">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>, the Quadru temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. | ||
Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n<span style="">⋅x</span> gens = n<span style="">⋅</span>I = x<span style="">⋅</span>M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5<span style="">⋅</span>(7/6) = 2<span style="">⋅P5. Thus </span>2<span style="">⋅P</span>5 - 5<span style="">⋅</span>(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7<span style="">⋅</span>(11/9) = 2<span style="">⋅</span>P8, and the comma is (-2, -14, 0, 0, 7). | Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n<span style="">⋅x</span> gens = n<span style="">⋅</span>I = x<span style="">⋅</span>M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5<span style="">⋅</span>(7/6) = 2<span style="">⋅P5. Thus </span>2<span style="">⋅P</span>5 - 5<span style="">⋅</span>(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7<span style="">⋅</span>(11/9) = 2<span style="">⋅</span>P8, and the comma is (-2, -14, 0, 0, 7), Saseplo. | ||
If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20. | If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20. | ||
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The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup> · 3<sup>y</sup> · P<sup>±1</sup>, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for P = 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 are the currently employed commas and combinations of them. | The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup> · 3<sup>y</sup> · P<sup>±1</sup>, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for P = 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 are the currently employed commas and combinations of them. | ||
If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in | If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyo, where ^1 equals 64/63 minus 81/80. | ||
Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below. | Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below. | ||
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This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples: | This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples: | ||
* 15-edo: # = 240¢, ^ = 80¢ (^ = | * 15-edo: # = 240¢, ^ = 80¢ (^ = third-sharp) | ||
* 16-edo: # = -75¢ | * 16-edo: # = -75¢ | ||
* 17-edo: # = 141¢, ^ = 71¢ (^ = | * 17-edo: # = 141¢, ^ = 71¢ (^ = half-sharp) | ||
* 18b-edo: # = -133¢, ^ = 67¢ (^ = | * 18b-edo: # = -133¢, ^ = 67¢ (^ = half-sharp) | ||
* 19-edo: # = 63¢ | * 19-edo: # = 63¢ | ||
* 21-edo: ^ = 57¢ (if used, # = 0¢) | * 21-edo: ^ = 57¢ (if used, # = 0¢) | ||
* 22-edo: # = 164¢, ^ = 55¢ (^ = | * 22-edo: # = 164¢, ^ = 55¢ (^ = third-sharp) | ||
* quarter-comma Meantone: # = 76¢ | * quarter-comma Meantone: # = 76¢ | ||
* fifth-comma Meantone: # = 84¢ | * fifth-comma Meantone: # = 84¢ | ||
* third-comma Archy: # = 177¢ | * third-comma Archy aka Ru: # = 177¢ | ||
* eighth-comma Porcupine: # = 157¢, ^ = 52¢ (^ = | * eighth-comma Porcupine aka Triyo: # = 157¢, ^ = 52¢ (^ = third-sharp) | ||
* | * seventh-comma Srutal aka Sagugu & Zoquadyo: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025) | ||
* third-comma Injera: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80) | * third-comma Injera aka Gu & Biruyo: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80) | ||
* eighth-comma Hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243) | * eighth-comma Hedgehog aka Triyo & Biruyo: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243) | ||
The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different. | |||
==Finding a notation for a pergen== | ==Finding a notation for a pergen== | ||
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It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan. | It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan. | ||
Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 enharmonic. It might result from combining | Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 enharmonic. It might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozo), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \<span style="vertical-align: super;">3</span>A1 and G' = \M2 = //m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1. | ||
<span style="display: block; text-align: center;">P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4 | <span style="display: block; text-align: center;">P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4 | ||
<span style="display: block; text-align: center;">C — C^/=Dbv/ — Db//=D\ — D^=Ebv — Eb/=E\\ — E^/=Fv\ — F | <span style="display: block; text-align: center;">C — C^/=Dbv/ — Db//=D\ — D^=Ebv — Eb/=E\\ — E^/=Fv\ — F | ||
When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). | When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, E must be an A1, with any number of downs except 3. v<sup>5</sup>A1 creates m ^m v~ ^~ vM M. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all. | ||
==Alternate enharmonics== | ==Alternate enharmonics== | ||
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For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone. | For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone. | ||
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to E, or the multi-E if the count is > 1. For example, consider Semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider Lala-yoyo, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd. | Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to E, or the multi-E if the count is > 1. For example, consider Semaphore aka Zozo (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider Lala-yoyo, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd. | ||
Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | ||
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For example, Satrilu tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2. | For example, Satrilu tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2. | ||
Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, Liese (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and E = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- B^ -- F. | Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, Liese aka Gu & Trizogu (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and E = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- B^ -- F. | ||
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | ||
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In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo. | In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo. | ||
Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara (2.3.5.7 with | Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Sagugu & Ru (2.3.5.7 with 2048/2025 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = Cv,7. | ||
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) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7. | A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyo (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7. | ||
Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. | Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales. | ||
Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A Porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^. | Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A Porcupine aka Triyo (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^. | ||
Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with E = v<sup>3</sup>A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps. | Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with E = v<sup>3</sup>A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps. | ||
| Line 1,286: | Line 1,286: | ||
The tipping point depends on the choice of enharmonic. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an E of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on. | The tipping point depends on the choice of enharmonic. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an E of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on. | ||
The notation's tipping point is determined by the bare enharmonic, which is implied by the vanishing comma. For example, Porcupine's 250/243 comma is an A1 = (-11,7), which implies a bare E of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot's 25/24 comma is also an A1, and has the same tipping point. Semaphore's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list. | The notation's tipping point is determined by the bare enharmonic, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies a bare E of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyo's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozo's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list. | ||
Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping. | Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping. | ||
An example of a temperament that tips easily is Negri, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation. | An example of a temperament that tips easily is Negri aka Laquadyo, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation. | ||
Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyo comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th. | Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyo comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th. | ||
| Line 1,320: | Line 1,320: | ||
! | cents | ! | cents | ||
|- | |- | ||
| | Meantone | | | Meantone aka Gu | ||
| | 81/80 = P1 | | | 81/80 = P1 | ||
| | c = -3¢ to -5¢ | | | c = -3¢ to -5¢ | ||
| Line 1,331: | Line 1,331: | ||
| | --- | | | --- | ||
|- | |- | ||
| | Mavila | | | Mavila aka Layobi | ||
| | 135/128 = A1 | | | 135/128 = A1 | ||
| | c = -21¢ to -22¢ | | | c = -21¢ to -22¢ | ||
| Line 1,353: | Line 1,353: | ||
| | 100¢ + 7c = 26¢-30¢ | | | 100¢ + 7c = 26¢-30¢ | ||
|- | |- | ||
| | Schismic | | | Schismic aka Layo | ||
| | (-15,8,1) = -d2 | | | (-15,8,1) = -d2 | ||
| | c = 1.7¢ to 2.0¢ | | | c = 1.7¢ to 2.0¢ | ||
| Line 1,375: | Line 1,375: | ||
| | -12c = 10¢-15¢ | | | -12c = 10¢-15¢ | ||
|- | |- | ||
| | Father | | | Father aka Gubi | ||
| | 16/15 = m2 | | | 16/15 = m2 | ||
| | c = 56¢ to 58¢ | | | c = 56¢ to 58¢ | ||
| Line 1,386: | Line 1,386: | ||
| | -100¢ + 5c = 180-190¢ | | | -100¢ + 5c = 180-190¢ | ||
|- | |- | ||
| | Superpyth | | | Superpyth aka Sasayo | ||
| | (12,-9,1) = m2 | | | (12,-9,1) = m2 | ||
| | c = 9¢ to 10¢ | | | c = 9¢ to 10¢ | ||
| Line 1,399: | Line 1,399: | ||
The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad. | The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad. | ||
For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-comma is the pythagorean comma (-19,12). | For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12). | ||
A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv. | A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Ru aka Archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv. | ||
==Notating rank-3 pergens== | ==Notating rank-3 pergens== | ||
| Line 1,449: | Line 1,449: | ||
| | E = d2, E' = vvA1 = vvm2 | | | E = d2, E' = vvA1 = vvm2 | ||
|- | |- | ||
| | pythagorean | | | 3-limit JI aka pythagorean | ||
| | (P8, P5) | | | (P8, P5) | ||
| | rank-2 | | | rank-2 | ||
| Line 1,457: | Line 1,457: | ||
| | --- | | | --- | ||
|- | |- | ||
| | Meantone | | | Meantone aka Gu | ||
| | (P8, P5) | | | (P8, P5) | ||
| | rank-2 | | | rank-2 | ||
| Line 1,465: | Line 1,465: | ||
| | --- | | | --- | ||
|- | |- | ||
| | | | | Diaschismic aka Sagugu | ||
| | (P8/2, P5) | | | (P8/2, P5) | ||
| | rank-2 | | | rank-2 | ||
| Line 1,473: | Line 1,473: | ||
| | E = ^^d2 | | | E = ^^d2 | ||
|- | |- | ||
| | Semaphore | | | Semaphore aka Zozo | ||
| | (P8, P4/2) | | | (P8, P4/2) | ||
| | rank-2 | | | rank-2 | ||
| Line 1,481: | Line 1,481: | ||
| | E = vvm2 | | | E = vvm2 | ||
|- | |- | ||
| | Decimal | | | Decimal aka Yoyo & Zozo | ||
| | (P8/2, P4/2) | | | (P8/2, P4/2) | ||
| | rank-2 | | | rank-2 | ||
| Line 1,497: | Line 1,497: | ||
| | --- | | | --- | ||
|- | |- | ||
| | Marvel | | | Marvel aka Ruyoyo | ||
| | (P8, P5, ^1) | | | (P8, P5, ^1) | ||
| | rank-3 | | | rank-3 | ||
| Line 1,505: | Line 1,505: | ||
| | --- | | | --- | ||
|- | |- | ||
| | Breedsmic | | | Breedsmic aka Bizozogu | ||
| | (P8, P5/2, ^1) | | | (P8, P5/2, ^1) | ||
| | rank-3 | | | rank-3 | ||
| Line 1,543: | Line 1,543: | ||
! | enharmonic | ! | enharmonic | ||
|- | |- | ||
| | Marvel | | | Marvel aka Ruyoyo | ||
| | 225/224 | | | 225/224 | ||
| | (P8, P5, ^1) | | | (P8, P5, ^1) | ||
| Line 1,583: | Line 1,583: | ||
| | ^^^\\\dd3 | | | ^^^\\\dd3 | ||
|- | |- | ||
| | Breedsmic | | | Breedsmic aka Bizozogu | ||
| | 2401/2400 | | | 2401/2400 | ||
| | (P8, P5/2, ^1) | | | (P8, P5/2, ^1) | ||
| Line 1,593: | Line 1,593: | ||
| | ^^\<span style="vertical-align: super;">4</span>dd3 | | | ^^\<span style="vertical-align: super;">4</span>dd3 | ||
|- | |- | ||
| | Demeter | | | Demeter aka Trizo-agugu | ||
| | 686/675 | | | 686/675 | ||
| | (P8, P5, \m3/2) | | | (P8, P5, \m3/2) | ||
| Line 1,611: | Line 1,611: | ||
This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyo doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip. | This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyo doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip. | ||
Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals. | Unlike the previous examples, Demeter aka Trizo-agugu's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals. | ||
There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. | There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. | ||
| Line 1,772: | Line 1,772: | ||
A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section. | A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section. | ||
A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples: | A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawa plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples: | ||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
| Line 1,785: | Line 1,785: | ||
! | /1 ratio | ! | /1 ratio | ||
|- | |- | ||
| | Blackwood | | | Blackwood aka Sawa+ya | ||
| | (P8/5, ^1) | | | (P8/5, ^1) | ||
| | rank-2 5-edo | | | rank-2 5-edo | ||
| Line 1,794: | Line 1,794: | ||
| | --- | | | --- | ||
|- | |- | ||
| | Whitewood | | | Whitewood aka Lawa+ya | ||
| | (P8/7, ^1) | | | (P8/7, ^1) | ||
| | rank-2 7-edo | | | rank-2 7-edo | ||
| Line 1,887: | Line 1,887: | ||
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. | In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. | ||
But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyo | But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals. | ||
Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians. | Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians. | ||
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Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the | Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the first 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous. | ||
Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)<span style="vertical-align: super;">3</span> * (64/63)<span style="vertical-align: super;">2</span>] = about 60¢. | Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)<span style="vertical-align: super;">3</span> * (64/63)<span style="vertical-align: super;">2</span>] = about 60¢. | ||
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| | (P8, P11/3) [7] | | | (P8, P11/3) [7] | ||
| | third-11th heptatonic | | | third-11th heptatonic | ||
| | fifth-double- | | | fifth-double-compound-4th, sixth-double-compound-5th | ||
|- | |- | ||
| | 3L 4s | | | 3L 4s | ||
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| | (P8, P5) [7] | | | (P8, P5) [7] | ||
| | unsplit heptatonic | | | unsplit heptatonic | ||
| | sixth-double- | | | sixth-double-compound-4th | ||
|- | |- | ||
| | 6L 1s | | | 6L 1s | ||
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The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the | The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyo generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen. | ||
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquingu Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4. | ||
==Pergens and EDOs== | ==Pergens and EDOs== | ||
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For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2). | For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2). | ||
To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot). 11/9 also works, it yields 243/242 (Mohajira). | To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot aka Yoyo). 11/9 also works, it yields 243/242 (Mohajira aka Lulu). | ||
If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen. | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen. | ||
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genchain<br /> | genchain<br /> | ||
perchain<br /> | perchain<br /> | ||
compound (increased by an octave)<br /> | |||
single-split, double-split<br /> | single-split, double-split<br /> | ||
single-pair, double-pair (number of new accidentals in the notation)<br /> | single-pair, double-pair (number of new accidentals in the notation)<br /> | ||