Kite's thoughts on pergens: Difference between revisions
updated the temperament names, e.g. small gugu to sagugu, and triple yo to triyo. |
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! | comma(s) | ! | comma(s) | ||
! | name | ! | name | ||
! colspan="2" | color name | ! colspan="2" | [[Color notation|color name]] | ||
|- | |- | ||
| style="text-align:center;" | (P8, P5) | | style="text-align:center;" | (P8, P5) | ||
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| style="text-align:center;" | (-14,8,1) | | style="text-align:center;" | (-14,8,1) | ||
| style="text-align:center;" | schismic | | style="text-align:center;" | schismic | ||
| style="text-align:center;" | | | style="text-align:center;" | layo | ||
| style="text-align:center;" | LyT | | style="text-align:center;" | LyT | ||
|- | |- | ||
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| style="text-align:center;" | (11, -4, -2) | | style="text-align:center;" | (11, -4, -2) | ||
| style="text-align:center;" | srutal | | style="text-align:center;" | srutal | ||
| style="text-align:center;" | | | style="text-align:center;" | sagugu | ||
| style="text-align:center;" | sggT | | style="text-align:center;" | sggT | ||
|- | |- | ||
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| style="text-align:center;" | 81/80, 50/49 | | style="text-align:center;" | 81/80, 50/49 | ||
| style="text-align:center;" | injera | | style="text-align:center;" | injera | ||
| style="text-align:center;" | | | style="text-align:center;" | biruyo and gu | ||
| style="text-align:center;" | rryy&gT | | style="text-align:center;" | rryy&gT | ||
|- | |- | ||
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| style="text-align:center;" | 250/243 | | style="text-align:center;" | 250/243 | ||
| style="text-align:center;" | porcupine | | style="text-align:center;" | porcupine | ||
| style="text-align:center;" | | | style="text-align:center;" | triyo | ||
| style="text-align:center;" | y<span style="vertical-align: super;">3</span>T | | style="text-align:center;" | y<span style="vertical-align: super;">3</span>T | ||
|- | |- | ||
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| style="text-align:center;" | third-11th | | style="text-align:center;" | third-11th | ||
| style="text-align:center;" | (12,-1,0,0,-3) | | style="text-align:center;" | (12,-1,0,0,-3) | ||
| style="text-align:center;" | | | style="text-align:center;" | satrilu | ||
| style="text-align:center;" | | | style="text-align:center;" | satrilu | ||
| style="text-align:center;" | s1u<span style="vertical-align: super;">3</span>T | | style="text-align:center;" | s1u<span style="vertical-align: super;">3</span>T | ||
|- | |- | ||
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| style="text-align:center;" | (3,4,-4) | | style="text-align:center;" | (3,4,-4) | ||
| style="text-align:center;" | diminished | | style="text-align:center;" | diminished | ||
| style="text-align:center;" | | | style="text-align:center;" | quadgu | ||
| style="text-align:center;" | g<span style="vertical-align: super;">4</span>T | | style="text-align:center;" | g<span style="vertical-align: super;">4</span>T | ||
|- | |- | ||
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| style="text-align:center;" | half-8ve quarter-tone | | style="text-align:center;" | half-8ve quarter-tone | ||
| style="text-align:center;" | (-17,2,0,0,4) | | style="text-align:center;" | (-17,2,0,0,4) | ||
| style="text-align:center;" | | | style="text-align:center;" | laquadlo | ||
| style="text-align:center;" | | | style="text-align:center;" | laquadlo | ||
| style="text-align:center;" | L1o<span style="vertical-align: super;">4</span>T | | style="text-align:center;" | L1o<span style="vertical-align: super;">4</span>T | ||
|- | |- | ||
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| style="text-align:center;" | (-10,-1,5) | | style="text-align:center;" | (-10,-1,5) | ||
| style="text-align:center;" | magic | | style="text-align:center;" | magic | ||
| style="text-align:center;" | | | style="text-align:center;" | laquinyo | ||
| style="text-align:center;" | Ly<span style="vertical-align: super;">5</span>T | | style="text-align:center;" | Ly<span style="vertical-align: super;">5</span>T | ||
|} | |} | ||
(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half. | (P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half. | ||
The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: | The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...). | ||
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. | ||
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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 (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: | 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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Another obvious application is to name 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. | Another obvious application is to name 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 | 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 and 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 can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below. | Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below. | ||
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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, | 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. | ||
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. | ||
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| style="text-align:center;" | P4/2 = vA2 = ^d3 | | style="text-align:center;" | P4/2 = vA2 = ^d3 | ||
| style="text-align:center;" | C - D#v=Ebb^ - F | | style="text-align:center;" | C - D#v=Ebb^ - F | ||
| style="text-align:center;" | | | style="text-align:center;" | lalayoyo | ||
^1 = 81/80 | ^1 = 81/80 | ||
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C - F^/=Gv\ - C | C - F^/=Gv\ - C | ||
| style="text-align:center;" | semaphore & | | style="text-align:center;" | semaphore & mohajira | ||
^1 = 33/32 | ^1 = 33/32 | ||
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C - Eb^/=Ev\ - G | C - Eb^/=Ev\ - G | ||
| style="text-align:center;" | | | style="text-align:center;" | diaschismic & semaphore | ||
^1 = 81/80 | ^1 = 81/80 | ||
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C - Dv/=Eb^\ - F | C - Dv/=Eb^\ - F | ||
| style="text-align:center;" | | | style="text-align:center;" | diaschismic and mohajira | ||
^1 = 81/80 | ^1 = 81/80 | ||
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| style="text-align:center;" | P11/3 = vA4 = ^^dd5 | | style="text-align:center;" | P11/3 = vA4 = ^^dd5 | ||
| style="text-align:center;" | C - F#v - Cb^ - F | | style="text-align:center;" | C - F#v - Cb^ - F | ||
| style="text-align:center;" | | | style="text-align:center;" | satrilu, if 11/8 = A4 | ||
^1 = 729/704 | ^1 = 729/704 | ||
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| style="text-align:center;" | P11/3 = ^4 = vv5 | | style="text-align:center;" | P11/3 = ^4 = vv5 | ||
| style="text-align:center;" | C - F^ - Cv - F | | style="text-align:center;" | C - F^ - Cv - F | ||
| style="text-align:center;" | | | style="text-align:center;" | satrilu, if 11/8 = P4 | ||
^1 = 33/32 | ^1 = 33/32 | ||
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C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F | C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F | ||
| style="text-align:center;" | | | style="text-align:center;" | tribilo, if 11/8 = P4 | ||
^1 = 33/32 | ^1 = 33/32 | ||
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C - Eb/=E\ - G | C - Eb/=E\ - G | ||
| style="text-align:center;" | | | style="text-align:center;" | satribizo | ||
^1 = 49/48, /1 = 343/324 | ^1 = 49/48, /1 = 343/324 | ||
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C - D\ - Eb/ - F | C - D\ - Eb/ - F | ||
| style="text-align:center;" | | | style="text-align:center;" | latribiru | ||
^1 = 1029/1024, /1 = 49/48 | ^1 = 1029/1024, /1 = 49/48 | ||
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C - D#vv - Fb^^ - G | C - D#vv - Fb^^ - G | ||
| style="text-align:center;" | | | style="text-align:center;" | lartribiyo | ||
^1 = 81/80 | ^1 = 81/80 | ||
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C - F^^ - Cvv - F | C - F^^ - Cvv - F | ||
| style="text-align:center;" | | | style="text-align:center;" | latribilo, if 11/8 = P4 | ||
^1 = 33/32 | ^1 = 33/32 | ||
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C - Dv/ - F^\ - G | C - Dv/ - F^\ - G | ||
| style="text-align:center;" | | | style="text-align:center;" | porcupine & triru | ||
^1 = 64/63 | ^1 = 64/63 | ||
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C - Dv\ - Eb^/ - F | C - Dv\ - Eb^/ - F | ||
| style="text-align:center;" | | | style="text-align:center;" | augmented & latrizo | ||
^1 = 81/80 | ^1 = 81/80 | ||
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C - Ev/ - Ab^\ - C | C - Ev/ - Ab^\ - C | ||
| style="text-align:center;" | | | style="text-align:center;" | triyo & latrizo | ||
^1 = 81/80 | ^1 = 81/80 | ||
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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). | ||
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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 | 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 lalayoyo, 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. | ||
For example, | 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 (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. | ||
| Line 1,286: | Line 1,286: | ||
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, 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 | 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. | ||
==Notating unsplit pergens== | ==Notating unsplit pergens== | ||
| Line 1,336: | Line 1,336: | ||
| style="text-align:center;" | -100¢ - 7c = 47¢-54¢ | | style="text-align:center;" | -100¢ - 7c = 47¢-54¢ | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | lagu | ||
| style="text-align:center;" | (-15,11,-1) = A1 | | style="text-align:center;" | (-15,11,-1) = A1 | ||
| style="text-align:center;" | c = -10¢ to -12¢ | | style="text-align:center;" | c = -10¢ to -12¢ | ||
| Line 1,358: | Line 1,358: | ||
| style="text-align:center;" | 12c = 20¢-24¢ | | style="text-align:center;" | 12c = 20¢-24¢ | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | lalagu | ||
| style="text-align:center;" | (-23,16,-1) = -d2 | | style="text-align:center;" | (-23,16,-1) = -d2 | ||
| style="text-align:center;" | c = -0.9¢ to -1.2¢ | | style="text-align:center;" | c = -0.9¢ to -1.2¢ | ||
| Line 1,557: | Line 1,557: | ||
| style="text-align:center;" | ^^\d2 | | style="text-align:center;" | ^^\d2 | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | biruyo | ||
| style="text-align:center;" | 50/49 | | style="text-align:center;" | 50/49 | ||
| style="text-align:center;" | (P8/2, P5, ^1) | | style="text-align:center;" | (P8/2, P5, ^1) | ||
| Line 1,567: | Line 1,567: | ||
| style="text-align:center;" | ^^\\d2 | | style="text-align:center;" | ^^\\d2 | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | trizogu | ||
| style="text-align:center;" | 1029/1000 | | style="text-align:center;" | 1029/1000 | ||
| style="text-align:center;" | (P8, P11/3, ^1) | | style="text-align:center;" | (P8, P11/3, ^1) | ||
| Line 1,607: | Line 1,607: | ||
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'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 vm3, 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''' (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 vm3, 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. | ||
If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's. | If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's. | ||
| Line 1,853: | Line 1,853: | ||
! | ^1 ratio | ! | ^1 ratio | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | laquinzo | ||
| style="text-align:center;" | 2.3.7 | | style="text-align:center;" | 2.3.7 | ||
| style="text-align:center;" | (-14,0,0,5) | | style="text-align:center;" | (-14,0,0,5) | ||
| Line 1,863: | Line 1,863: | ||
| style="text-align:center;" | 49/48 | | style="text-align:center;" | 49/48 | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | saquinru | ||
| style="text-align:center;" | 2.3.7 | | style="text-align:center;" | 2.3.7 | ||
| style="text-align:center;" | (22,-5,0,-5) | | style="text-align:center;" | (22,-5,0,-5) | ||
| Line 1,881: | Line 1,881: | ||
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, | 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. | ||