Kite's color notation: Difference between revisions
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[[File:Lattice32.png|694x694px]] | [[File:Lattice32.png|694x694px]] | ||
Colors can be doubled or tripled: 25/16 = yoyo 5th = yy5 and 128/125 = triple gu 2nd = g<sup>3</sup>2. Quadruple and quintuple are abbreviated '''quad''' and '''quint''', as in the quadgu comma. Degrees can be negative: 50/49 = double ruyo negative 2nd = rryy-2. It's a negative 2nd because it goes up in pitch but down the scale: zg5 + rryy-2 = ry4. Negative is different than descending, from ry4 to zg5 is a descending negative 2nd. | Colors can be doubled or tripled: 25/16 = yoyo 5th = yy5 and 128/125 = triple gu 2nd = g<sup>3</sup>2. Quadruple and quintuple are abbreviated '''quad''' and '''quint''', as in the quadgu comma 648/625. Degrees can be negative: 50/49 = double ruyo negative 2nd = rryy-2. It's a negative 2nd because it goes up in pitch but down the scale: zg5 + rryy-2 = ry4. Negative is different than descending, from ry4 to zg5 is a descending negative 2nd. | ||
The next table lists all the intervals in the lattice above. See the [[Gallery of Just Intervals]] for many more examples. | The next table lists all the intervals in the lattice above. See the [[Gallery of Just Intervals]] for many more examples. | ||
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Converting a ratio: Find the [[Monzos|monzo]] by prime factorization. To find the color, combine all the appropriate colors for each prime > 3, higher primes first. To find the degree, first find the stepspan, which is the dot product of the monzo with the 7edo [[Patent val|edomapping]] <7 11 16 20 24 26 29 30...|. Then add 1, or subtract 1 if the stepspan is negative. To find the magnitude, add up all the monzo exponents except the first one, divide by 7, and round off. Combine the magnitude, color and degree to make the color name. | Converting a ratio: Find the [[Monzos|monzo]] by prime factorization. To find the color, combine all the appropriate colors for each prime > 3, higher primes first. To find the degree, first find the stepspan, which is the dot product of the monzo with the 7edo [[Patent val|edomapping]] <7 11 16 20 24 26 29 30...|. Then add 1, or subtract 1 if the stepspan is negative. To find the magnitude, add up all the monzo exponents except the first one, divide by 7, and round off. Combine the magnitude, color and degree to make the color name. | ||
Example: ratio = 63/40, monzo = |-3 2 -1 1>, color = | Example: ratio = 63/40, monzo = |-3 2 -1 1>, color = zogu, stepspan = <7 11 16 20| dot |-3 2 -1 1> = -21 + 22 - 16 + 20 = 5 steps, degree = 5 + 1 = a 6th, magnitude = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central, interval = zg6. | ||
Converting a color name: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let the monzo be |a b c d e...>. The colors directly give you all the monzo entries except a and b. Let X = the dot product of |0 0 c d e...> with the 7edo edomapping. Then b = (2S - 2X + 3) mod 7 + 7M - 3, and a = (S - X - 11b) / 7. Convert the monzo to a ratio. | Converting a color name: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let the monzo be |a b c d e...>. The colors directly give you all the monzo entries except a and b. Let X = the dot product of |0 0 c d e...> with the 7edo edomapping. Then b = (2S - 2X + 3) mod 7 + 7M - 3, and a = (S - X - 11b) / 7. Convert the monzo to a ratio. | ||
Example: interval = sgg2, S = 2-1 = 1, M = -1, monzo = |a b -2>, X = <7 11 16| dot |0 0 -2> = -32, b = ( | Example: interval = sgg2, S = 2-1 = 1, M = small = -1, monzo = |a b -2>, X = <7 11 16| dot |0 0 -2> = -32, b = (2·1 - 2·(-32) + 3) mod 7 + 7·(-1) - 3 = 6 - 7 - 3 = -4, a = (1 - (-32) - 11·(-4)) / 7 = 77/7 = 11, monzo = |11 -4 -1>, ratio = 2048/2025. | ||
== Staff notation == | == Staff notation == | ||
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[[File:Notation example 2.png|786x786px]] | [[File:Notation example 2.png|786x786px]] | ||
Color notation can optionally be made more similar to Sagittal notation by including two more accidentals, p and q (long forms po and qu = "ku"), to indicate raising/lowering by a pythagorean comma. For example, yF# = ypGb, and zEb = zqD#. This allows trills to always be written as a 2nd, less cluttered.[[File:Notation example 5a.png|992x992px]] | Color notation can optionally be made more similar to Sagittal notation by including two more accidentals, p and q (long forms po and qu = "ku"), to indicate raising/lowering by a pythagorean comma. For example, yF# = ypGb, and zEb = zqD#. This allows trills to always be written as a 2nd, less cluttered.[[File:Notation example 5a.png|992x992px]]L and s never appear on the staff. Tripled colors are written as y3 not yyy. In MuseScore, color accidentals are made by adding fingerings to the notes, then editing the fingering text. | ||
== Chord names == | == Chord names == | ||
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The y,z7 chord is also called the h7 chord ("aitch-seven"), because it's part of the harmonic series. The s7 ("sub-seven") chord is part of the subharmonic series. It's the first 7 subharmonics, with the 3rd subharmonic becoming the root. Note that s7 has no 7th. Ch9 = y,z7,w9 and Ch11 = y,z7,w9,1o11. Cs9 = g,r6,w11 and Cs11 = g,r6,w11,1u9. All harmonic numbers must be odd, Ch8 is invalid. <u>Additions refer to harmonics or subharmonics</u>, not degrees: Cs7,11 adds 1u9, not w11. To add w11, use colors: Cs7,w11. <u>Alterations and omissions refer to degrees</u>, not (sub)harmonics; Cs7(zg5) alters the w5, not the 5th subharmonic g3. Ch9no5 omits w5, not y3. However, Ch19no15 refers to the 15th harmonic, since degrees above 13 aren't conventionally used. | The y,z7 chord is also called the h7 chord ("aitch-seven"), because it's part of the harmonic series. The s7 ("sub-seven") chord is part of the subharmonic series. It's the first 7 subharmonics, with the 3rd subharmonic becoming the root. Note that s7 has no 7th. Ch9 = y,z7,w9 and Ch11 = y,z7,w9,1o11. Cs9 = g,r6,w11 and Cs11 = g,r6,w11,1u9. All harmonic numbers must be odd, Ch8 is invalid. <u>Additions refer to harmonics or subharmonics</u>, not degrees: Cs7,11 adds 1u9, not w11. To add w11, use colors: Cs7,w11. <u>Alterations and omissions refer to degrees</u>, not (sub)harmonics; Cs7(zg5) alters the w5, not the 5th subharmonic g3. Ch9no5 omits w5, not y3. However, Ch19no15 refers to the 15th harmonic, since degrees above 13 aren't conventionally used. | ||
Chords can be classified as '''bicolored''' (e.g. g7 or r6), '''tricolored''' (e.g. z7(zg5) or z,y6), '''quadricolored''', etc. | |||
== Chord Progressions, Keys and Modulations == | == Chord Progressions, Keys and Modulations == | ||
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In adaptive JI, chords are just, but roots move by tempered intervals. Comma pumps are indicated with brackets: Cy - yAg - [y=w]Dg - Gy - Cy. | In adaptive JI, chords are just, but roots move by tempered intervals. Comma pumps are indicated with brackets: Cy - yAg - [y=w]Dg - Gy - Cy. | ||
Keys are named after the colors used. In | Keys and scales are loosely named after the colors used. Wa is assumed present. In 5-limit JI, the key/scale of A minor is A gu. The Bbh7 - Ebh7 - Bbh7 - Fh9 example in the staff notation section is in Bb yo zo. Like chords, keys can be classified as bicolored (A gu), tricolored (Bb yo zo), etc. | ||
Analogous to the relative and parallel major or minor, one can modulate to relative gu, parallel ru, etc. Modulating from a yo key to the relative gu means using gu chords on yo roots. Modulating from yo to the parallel gu means using gu chords on <u>wa</u> roots. Going from yo zo to the relative gu means using chords with gu and/or ru in them on yo roots. Going to the relative ru means using the same chords on zo roots. Going from yo zo to the parallel gu ru means using the same chords on wa roots. | |||
== Temperament Names == | == Temperament Names == | ||
Temperaments are named after the color of the comma(s) they temper out. Meantone = the gu temperament = gT. Porcupine = triple yo temperament = y<sup>3</sup>T. 7-limit Porcupine = triple yo and ru = y<sup>3</sup>&rT. Each porcupine has a different name, thus color names provide more information than standard temperament names. Both porcupines have the same [[pergen]], third-4th, thus pergens group together similar temperaments. | Temperaments are named after the color of the comma(s) they temper out. Meantone = the gu temperament = gT. Porcupine = triple yo temperament = y<sup>3</sup>T. 7-limit Porcupine = triple yo and ru = y<sup>3</sup>&rT. Each variety of porcupine has a different name, thus color names provide more information than standard temperament names. Both porcupines have the same [[pergen]], third-4th, thus pergens group together similar temperaments. | ||
The magnitude is part of the name: Schismic is LyT and shrutal is sggT. The degree is too, but only if the comma is not the smallest of the 7 ratios of that magnitude and color: Mavila is Ly1T and [[Father]] is g2T. The degree is never needed if the comma is ≤ 90¢. | The magnitude is part of the name: Schismic is LyT and shrutal is sggT. The degree is too, but only if the comma is not the smallest of the 7 ratios of that magnitude and color: Mavila is Ly1T and [[Father]] is g2T. The degree is never needed if the comma is ≤ 90¢. | ||
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The temperament name indicates the prime subgroup and the rank of the temperament. For example, ryyT ([[Marvel]]) is rank-3 because it has 2 explicit colors ru and yo and 2 implicit colors wa and clear, and 4 colors minus 1 comma = rank-3. Edos count as commas, but plusses don't. Both 12edo&ryyT and 5edo+yT are rank-2. | The temperament name indicates the prime subgroup and the rank of the temperament. For example, ryyT ([[Marvel]]) is rank-3 because it has 2 explicit colors ru and yo and 2 implicit colors wa and clear, and 4 colors minus 1 comma = rank-3. Edos count as commas, but plusses don't. Both 12edo&ryyT and 5edo+yT are rank-2. | ||
There are two obvious ways to name multi-comma temperaments. The odd name minimizes the double odd limit of the comma set, and the prime name minimizes the number and size of the primes used by each comma. The odd name for 7-limit [[Pajara]] is rryy&rT, and the prime name is sgg&rT. Often the two names are identical, e.g. y<sup>3</sup>&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The prime name shows relationships between single-comma temperaments better. The question of which name to use is not yet fully resolved. | There are two obvious ways to name multi-comma temperaments. The odd name minimizes the [[Odd limit|double odd limit]] of the comma set, and the prime name minimizes the number and size of the primes used by each comma. The odd name for 7-limit [[Pajara]] is rryy&rT, and the prime name is sgg&rT. Often the two names are identical, e.g. y<sup>3</sup>&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The prime name shows relationships between single-comma temperaments better. The question of which name to use is not yet fully resolved. | ||
== Ups and Downs, Lifts and Drops == | == Ups and Downs, Lifts and Drops == | ||
Color notation merely renames ratios more conveniently, and strictly speaking, it only applies to just intonation. However, ratios are often used to loosely describe edo notes, and colors can be used as well. A more precise application is to use ups and downs (^ and v) as "virtual colors", accidentals that always map to exactly one edostep. Ups and downs are used on the score just like color accidentals are. Notes are named e.g. up C sharp = ^C#. Intervals are named upmajor 3rd = ^M3, down 4th = v4, etc. Chords are named upminor 7th = ^m7, etc. | Color notation merely renames ratios more conveniently, and strictly speaking, it only applies to just intonation. However, ratios are often used to loosely describe edo notes, and colors can be used as well. A more precise application is to use ups and downs (^ and v) as "virtual colors", accidentals that always map to exactly one edostep. Ups and downs are used on the score just like color accidentals are. Notes are named e.g. up C sharp = ^C#. Intervals are named upmajor 3rd = ^M3, down 4th = v4, etc. Chords are named upminor 7th = ^m7, etc. | ||
Rank-2 temperaments can be notated with ups and downs as well. Some require an additional accidental pair, lifts and drops (/ and \). See [[Pergen|pergens]]. | Rank-2 temperaments can be notated with ups and downs as well. Some require an additional accidental pair, lifts and drops (/ and \). See [[Pergen|pergens]]. | ||
'''(OBSOLETE LINKS, IGNORE THEM):''' | '''(OBSOLETE LINKS, IGNORE THEM):''' | ||
[[Paradoxical_intervals|Paradoxical intervals]] | [[Paradoxical_intervals|Paradoxical intervals]] | ||