Kite's thoughts on pergens: Difference between revisions
m Removed protection from "Pergen" |
updated color names to yogu zoru lolu thothu |
||
| Line 32: | Line 32: | ||
| style="text-align:center;" | 81/80 | | style="text-align:center;" | 81/80 | ||
| style="text-align:center;" | meantone | | style="text-align:center;" | meantone | ||
| style="text-align:center;" | | | style="text-align:center;" | gu | ||
| style="text-align:center;" | gT | | style="text-align:center;" | gT | ||
|- | |- | ||
| Line 39: | Line 39: | ||
| style="text-align:center;" | 64/63 | | style="text-align:center;" | 64/63 | ||
| style="text-align:center;" | archy | | style="text-align:center;" | archy | ||
| style="text-align:center;" | | | style="text-align:center;" | ru | ||
| style="text-align:center;" | rT | | style="text-align:center;" | rT | ||
|- | |- | ||
| Line 46: | Line 46: | ||
| 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;" | large | | style="text-align:center;" | large yo | ||
| style="text-align:center;" | LyT | | style="text-align:center;" | LyT | ||
|- | |- | ||
| Line 53: | Line 53: | ||
| 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;" | small | | style="text-align:center;" | small gugu | ||
| style="text-align:center;" | sggT | | style="text-align:center;" | sggT | ||
|- | |- | ||
| Line 60: | Line 60: | ||
| 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;" | double ruyo and gu | ||
| style="text-align:center;" | rryy&gT | | style="text-align:center;" | rryy&gT | ||
|- | |- | ||
| Line 67: | Line 67: | ||
| style="text-align:center;" | 25/24 | | style="text-align:center;" | 25/24 | ||
| style="text-align:center;" | dicot | | style="text-align:center;" | dicot | ||
| style="text-align:center;" | | | style="text-align:center;" | yoyo | ||
| style="text-align:center;" | yyT | | style="text-align:center;" | yyT | ||
|- | |- | ||
| Line 74: | Line 74: | ||
| style="text-align:center;" | (-1,5,0,0,-2) | | style="text-align:center;" | (-1,5,0,0,-2) | ||
| style="text-align:center;" | mohajira | | style="text-align:center;" | mohajira | ||
| style="text-align:center;" | | | style="text-align:center;" | lulu | ||
| style="text-align:center;" | | | style="text-align:center;" | 1uuT | ||
|- | |- | ||
| style="text-align:center;" | (P8, P4/2) | | style="text-align:center;" | (P8, P4/2) | ||
| Line 81: | Line 81: | ||
| style="text-align:center;" | 49/48 | | style="text-align:center;" | 49/48 | ||
| style="text-align:center;" | semaphore | | style="text-align:center;" | semaphore | ||
| style="text-align:center;" | | | style="text-align:center;" | zozo | ||
| style="text-align:center;" | | | style="text-align:center;" | zzT | ||
|- | |- | ||
| style="text-align:center;" | (P8/2, P4/2) | | style="text-align:center;" | (P8/2, P4/2) | ||
| Line 88: | Line 88: | ||
| style="text-align:center;" | 25/24, 49/48 | | style="text-align:center;" | 25/24, 49/48 | ||
| style="text-align:center;" | decimal | | style="text-align:center;" | decimal | ||
| style="text-align:center;" | | | style="text-align:center;" | yoyo and zozo | ||
| style="text-align:center;" | yy& | | style="text-align:center;" | yy&zzT | ||
|- | |- | ||
| style="text-align:center;" | (P8, P4/3) | | style="text-align:center;" | (P8, P4/3) | ||
| Line 95: | Line 95: | ||
| 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;" | triple | | style="text-align:center;" | triple yo | ||
| 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 | ||
|- | |- | ||
| Line 101: | Line 101: | ||
| 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;" | small triple | | style="text-align:center;" | small triple lu | ||
| style="text-align:center;" | small triple | | style="text-align:center;" | small triple lu | ||
| style="text-align:center;" | | | style="text-align:center;" | s1u<span style="vertical-align: super;">3</span>T | ||
|- | |- | ||
| style="text-align:center;" | (P8/4, P5) | | style="text-align:center;" | (P8/4, P5) | ||
| Line 109: | Line 109: | ||
| 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;" | quadruple | | style="text-align:center;" | quadruple gu | ||
| 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 | ||
|- | |- | ||
| Line 115: | Line 115: | ||
| 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;" | large quadruple | | style="text-align:center;" | large quadruple lo | ||
| style="text-align:center;" | large quadruple | | style="text-align:center;" | large quadruple lo | ||
| style="text-align:center;" | | | style="text-align:center;" | L1o<span style="vertical-align: super;">4</span>T | ||
|- | |- | ||
| style="text-align:center;" | (P8, P12/5) | | style="text-align:center;" | (P8, P12/5) | ||
| Line 123: | Line 123: | ||
| 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;" | large quintuple | | style="text-align:center;" | large quintuple yo | ||
| 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: double splits something into two parts, triple 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) = | 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/lova, (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 (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: Triple | More examples: Triple zogu (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Double ruyo (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, double ruyo minus wa (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. | ||
| Line 164: | Line 164: | ||
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, | 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). | ||
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 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: | ||
| Line 314: | Line 314: | ||
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 triple | 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 triple yo, and the second one is triple yo 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. | ||
| Line 320: | Line 320: | ||
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, '''highs and lows''', written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, | 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, '''highs and lows''', written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, 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. | ||
| Line 406: | Line 406: | ||
| style="text-align:center;" | P8/2 = ^4 = vP5 | | style="text-align:center;" | P8/2 = ^4 = vP5 | ||
| style="text-align:center;" | C - F^=Gv - C | | style="text-align:center;" | C - F^=Gv - C | ||
| style="text-align:center;" | | | style="text-align:center;" | thotho, if 13/8 = M6 | ||
^1 = 27/26 | ^1 = 27/26 | ||
| Line 428: | Line 428: | ||
| 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;" | double large | | style="text-align:center;" | double large yoyo | ||
^1 = 81/80 | ^1 = 81/80 | ||
| Line 476: | Line 476: | ||
C - F^/=Gv\ - C | C - F^/=Gv\ - C | ||
| style="text-align:center;" | | | style="text-align:center;" | semaphore & lulu | ||
^1 = 33/32 | ^1 = 33/32 | ||
| Line 504: | Line 504: | ||
C - Eb^/=Ev\ - G | C - Eb^/=Ev\ - G | ||
| style="text-align:center;" | small | | style="text-align:center;" | small gugu & zozo | ||
^1 = 81/80 | ^1 = 81/80 | ||
| Line 532: | Line 532: | ||
C - Dv/=Eb^\ - F | C - Dv/=Eb^\ - F | ||
| style="text-align:center;" | small | | style="text-align:center;" | small gugu and lulu | ||
^1 = 81/80 | ^1 = 81/80 | ||
| Line 590: | Line 590: | ||
| 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;" | small triple | | style="text-align:center;" | small triple lu, if 11/8 = A4 | ||
^1 = 729/704 | ^1 = 729/704 | ||
| Line 600: | Line 600: | ||
| 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;" | small triple | | style="text-align:center;" | small triple lu, if 11/8 = P4 | ||
^1 = 33/32 | ^1 = 33/32 | ||
| Line 616: | Line 616: | ||
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;" | sixfold | | style="text-align:center;" | sixfold lo, if 11/8 = P4 | ||
^1 = 33/32 | ^1 = 33/32 | ||
| Line 654: | Line 654: | ||
C - Eb/=E\ - G | C - Eb/=E\ - G | ||
| style="text-align:center;" | small sixfold | | style="text-align:center;" | small sixfold zo | ||
^1 = 49/48, /1 = 343/324 | ^1 = 49/48, /1 = 343/324 | ||
| Line 674: | Line 674: | ||
C - D\ - Eb/ - F | C - D\ - Eb/ - F | ||
| style="text-align:center;" | large sixfold | | style="text-align:center;" | large sixfold ru | ||
^1 = 1029/1024, /1 = 49/48 | ^1 = 1029/1024, /1 = 49/48 | ||
| Line 690: | Line 690: | ||
C - D#vv - Fb^^ - G | C - D#vv - Fb^^ - G | ||
| style="text-align:center;" | large sixfold | | style="text-align:center;" | large sixfold yo | ||
^1 = 81/80 | ^1 = 81/80 | ||
| Line 724: | Line 724: | ||
C - F^^ - Cvv - F | C - F^^ - Cvv - F | ||
| style="text-align:center;" | large sixfold | | style="text-align:center;" | large sixfold lo, if 11/8 = P4 | ||
^1 = 33/32 | ^1 = 33/32 | ||
| Line 774: | Line 774: | ||
C - Dv\ - Eb^/ - F | C - Dv\ - Eb^/ - F | ||
| style="text-align:center;" | triple | | style="text-align:center;" | triple gu & large triple zo | ||
^1 = 81/80 | ^1 = 81/80 | ||
| Line 798: | Line 798: | ||
C - Ev/ - Ab^\ - C | C - Ev/ - Ab^\ - C | ||
| style="text-align:center;" | triple | | style="text-align:center;" | triple yo & large triple zo | ||
^1 = 81/80 | ^1 = 81/80 | ||
| Line 860: | Line 860: | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|} | |} | ||
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't | The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex. | ||
Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢). | Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢). | ||
| Line 1,116: | Line 1,116: | ||
==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 quadruple | 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 quadruple ru 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). | ||
| Line 1,130: | Line 1,130: | ||
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize: | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize: | ||
<ul><li>'<nowiki/>'' | <ul><li>'''A'''<nowiki/> '''double-split pergen is <u>explicitly false</u> if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a <u>true double</u> if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>'.'''</li><li>'''A double-split pergen is a <u>true double</u> if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul> | ||
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | ||
| Line 1,148: | Line 1,148: | ||
Sometimes the mapping comma needs to be inverted. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. 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. See also blackwood-like pergens below. | Sometimes the mapping comma needs to be inverted. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. 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. See also blackwood-like pergens below. | ||
Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + | Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. | ||
In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1. | In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1. | ||
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¢ (^ = 1/3 #) | |||
15-edo: # = 240¢, ^ = 80¢ (^ = 1/3 #) | * 16-edo: # = -75¢ | ||
* 17-edo: # = 141¢, ^ = 71¢ (^ = 1/2 #) | |||
16-edo: # = -75¢ | * 18b-edo: # = -133¢, ^ = 67¢ (^ = 1/2 #) | ||
* 19-edo: # = 63¢ | |||
17-edo: # = 141¢, ^ = 71¢ (^ = 1/2 #) | * 21-edo: ^ = 57¢ (if used, # = 0¢) | ||
* 22-edo: # = 164¢, ^ = 55¢ (^ = 1/3 #) | |||
18b-edo: # = -133¢, ^ = 67¢ (^ = 1/2 #) | * quarter-comma meantone: # = 76¢ | ||
* fifth-comma meantone: # = 84¢ | |||
19-edo: # = 63¢ | * third-comma archy: # = 177¢ | ||
* eighth-comma porcupine: # = 157¢, ^ = 52¢ (^ = 1/3 #) | |||
21-edo: ^ = 57¢ (if used, # = 0¢) | * sixth-comma srutal: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #) | ||
* third-comma injera: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80) | |||
22-edo: # = 164¢, ^ = 55¢ (^ = 1/3 #) | * eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243) | ||
quarter-comma meantone: # = 76¢ | |||
fifth-comma meantone: # = 84¢ | |||
third-comma archy: # = 177¢ | |||
eighth-comma porcupine: # = 157¢, ^ = 52¢ (^ = 1/3 #) | |||
sixth-comma srutal: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #) | |||
third-comma injera: # = 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) | |||
Hedgehog is half-8ve third-4th. 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. | Hedgehog is half-8ve third-4th. 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. | ||
| Line 1,268: | Line 1,253: | ||
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 double large | 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 double large 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. | ||
For example, small triple | For example, small triple lu 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 1,292: | Line 1,277: | ||
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 (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^. | ||
Highs and lows, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments | Highs and lows, 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. | ||
<u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3)</span></u> | <u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3)</span></u> | ||
| Line 1,310: | Line 1,295: | ||
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 large triple | Another "tippy" temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yo 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 1,360: | Line 1,345: | ||
| style="text-align:center;" | -100¢ - 7c = 47¢-54¢ | | style="text-align:center;" | -100¢ - 7c = 47¢-54¢ | ||
|- | |- | ||
| style="text-align:center;" | large | | style="text-align:center;" | large gu | ||
| 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 1,382: | Line 1,367: | ||
| style="text-align:center;" | 12c = 20¢-24¢ | | style="text-align:center;" | 12c = 20¢-24¢ | ||
|- | |- | ||
| style="text-align:center;" | double large | | style="text-align:center;" | double large gu | ||
| 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 1,581: | Line 1,566: | ||
| style="text-align:center;" | ^^\d2 | | style="text-align:center;" | ^^\d2 | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | double ruyo | ||
| 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 1,591: | Line 1,576: | ||
| style="text-align:center;" | ^^\\d2 | | style="text-align:center;" | ^^\\d2 | ||
|- | |- | ||
| style="text-align:center;" | triple | | style="text-align:center;" | triple zogu | ||
| 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 1,623: | Line 1,608: | ||
If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\. | If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\. | ||
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, | There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, double ruyo is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. | ||
With this standardization, the enharmonic can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler. | With this standardization, the enharmonic can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler. | ||
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 | 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 double ruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore double ruyo 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/highs/lows 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/highs/lows only for the other kind of accidentals. | ||
| Line 1,821: | Line 1,806: | ||
| style="text-align:center;" | --- | | style="text-align:center;" | --- | ||
|- | |- | ||
| style="text-align:center;" | 10edo+ | | style="text-align:center;" | 10edo+ya | ||
| style="text-align:center;" | (P8/10, /1) | | style="text-align:center;" | (P8/10, /1) | ||
| style="text-align:center;" | rank-2 10-edo | | style="text-align:center;" | rank-2 10-edo | ||
| Line 1,830: | Line 1,815: | ||
| style="text-align:center;" | 81/80 | | style="text-align:center;" | 81/80 | ||
|- | |- | ||
| style="text-align:center;" | 12edo+ | | style="text-align:center;" | 12edo+la | ||
| style="text-align:center;" | (P8/12, ^1) | | style="text-align:center;" | (P8/12, ^1) | ||
| style="text-align:center;" | rank-2 12-edo | | style="text-align:center;" | rank-2 12-edo | ||
| Line 1,848: | Line 1,833: | ||
| style="text-align:center;" | --- | | style="text-align:center;" | --- | ||
|- | |- | ||
| style="text-align:center;" | 17edo+ | | style="text-align:center;" | 17edo+ya | ||
| style="text-align:center;" | (P8/17, /1) | | style="text-align:center;" | (P8/17, /1) | ||
| style="text-align:center;" | rank-2 17-edo | | style="text-align:center;" | rank-2 17-edo | ||
| Line 1,863: | Line 1,848: | ||
All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Blackwood-like pergens are a small minority of rank-2 pergens. | All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Blackwood-like pergens are a small minority of rank-2 pergens. | ||
It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such: | It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such: | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 1,877: | Line 1,862: | ||
! | ^1 ratio | ! | ^1 ratio | ||
|- | |- | ||
| style="text-align:center;" | large quintuple | | style="text-align:center;" | large quintuple zo | ||
| 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 1,887: | Line 1,872: | ||
| style="text-align:center;" | 49/48 | | style="text-align:center;" | 49/48 | ||
|- | |- | ||
| style="text-align:center;" | small quintuple | | style="text-align:center;" | small quintuple ru | ||
| 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 1,905: | Line 1,890: | ||
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, double ruyo minus wa (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. | ||
| Line 2,094: | Line 2,079: | ||
C2 -- G2 | C2 -- G2 | ||
| . . . . | | | . . . . | | ||
| Line 2,117: | Line 2,103: | ||
C2 Ev2 G2 | C2 Ev2 G2 | ||
| . . . . . . | | | . . . . . . | | ||
| Line 2,124: | Line 2,111: | ||
C3 Ev3 G3 | C3 Ev3 G3 | ||
| . . . . . . | | | . . . . . . | | ||
C2 Ev2 G2 | C2 Ev2 G2 | ||
| . . . . . . | | | . . . . . . | | ||
| Line 2,134: | Line 2,123: | ||
C2 ---- G2 | C2 ---- G2 | ||
| . A^1 . | | | . A^1 . | | ||