Color notation

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Color notation was mostly developed by Kite Giedraitis. This is a brief summary. For a full explanation, see "Alternative Tunings: Theory, Notation and Practice".

Color Names for Primes 3, 5 and 7

Every prime above 3 has two colors, an over color (prime in the numerator) and an under color (prime in the denominator). Over colors end with -o, and under colors end with -u. The color for 3-limit ends in -a for all, which includes over (3/2, 9/8), under (4/3, 16/9) and neither (1/1, 2/1).

     3-all = Wa = white (strong but colorless) = often perfect
  5-over = Yo = yellow (warm and sunny) = often major
5-under = Gu ("goo") = green (not as bright as yellow) = often minor
  7-over = Zo = blue/azure (dark and bluesy) = often subminor
7-under = Ru = red (alarming, inflamed) = often supermajor

The colors come in a red-yellow-green-blue rainbow, with warm/cool colors indicating sharp/flat intervals. The rainbow of 3rds runs 9/7 - 5/4 - 6/5 - 7/6. Colors are abbreviated as w, y, g, z and r. Use z (azure) not b (blue), because b already means flat. Mnemonic: Z looks like 7 with an extra line on the bottom.

Interval Names

A color and a degree indicates a ratio, and vice versa. Every ratio has a spoken name and a written name. For 3/2, they are wa 5th and w5. Colors and degrees always add up predictably: z3 + g3 = zg5 = zogu 5th. Zogu not guzo, higher primes always come first. Opposite colors cancel: y3 + g3 = w5.

The JI lattice consists of many rows, each one a chain of 5ths. Each row has its own color, and each color has its own row.

Lattice32.png

The next table lists all the intervals in this lattice. See the Gallery of Just Intervals for many more examples.

ratio cents color & degree
1/1 wa unison w1
21/20 84¢ zogu 2nd zg2
16/15 112¢ gu 2nd g2
10/9 182¢ yo 2nd y2
9/8 204¢ wa 2nd w2
8/7 231¢ ru 2nd r2
7/6 267¢ zo 3rd z3
6/5 316¢ gu 3rd g3
5/4 386¢ yo 3rd y3
9/7 435¢ ru 3rd r3
21/16 471¢ zo 4th z4
4/3 498¢ wa 4th w4
7/5 583¢ zogu 5th zg5
10/7 617¢ ruyo 4th ry4
3/2 702¢ wa 5th w5
32/21 729¢ ru 5th r5
14/9 765¢ zo 6th z6
8/5 814¢ gu 6th g6
5/3 884¢ yo 6th y6
12/7 933¢ ru 6th r6
7/4 969¢ zo 7th z7
16/9 996¢ wa 7th w7
9/5 1018¢ gu 7th g7
15/8 1088¢ yo 7th y7
40/21 1116¢ ruyo 7th ry7
2/1 1200¢ wa octave w8

Yo and ru intervals tend to be major, and gu and zo ones tend to be minor. But interval quality is redundant (if a third is yo, it must be major), it's not unique (there are other major thirds available), and quality isn't used with color names (see "Higher Primes" below for why). Instead of augmented and diminished, remote intervals are large (fifthward) and small (fourthward), abbreviated L and s. Central, the default, means neither large nor small. The magnitude is the sum all the monzo exponents except the first one, divided by 7, and rounded off. 0 = central, 1 = large, 2 = double large, etc. 81/64 = Lw3, 135/128 = Ly1. Unfortunately, magnitudes do not add up predictably like colors and degrees do: w2 + w2 = Lw3.

Colors can be doubled or tripled: 25/16 = yoyo 5th = yy5 and 128/125 = triple gu 2nd = g32. Quadruple and quintuple are abbreviated quad and quint, as in quadyo or quintlarge. Higher multipliers use -fold as in sixfold, tenfold, 11-fold, etc.

Lattice41a.png

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. There are also diminished unisons, which raise the pitch but diminish the quality. For example, if 11/8 is a P4, two of them are a m7 of 121/64 = 1102¢. Going from a yo M7 = 1088¢ up to this m7 raises the pitch, and 121/120 is a d1.

Any ratio under 50¢ can be called a comma, allowing us to omit the magnitude and degree in the spoken name. Thus sgg2 is not the small gugu 2nd, but simply the gugu comma. The double-large wa negative 2nd (LLw-2, the pyth comma) is simply the wa comma. 81/80 = g1 is the gu comma. LLg-2 (the sum of g1 and LLw-2) is also gu and also a comma, but LLg-2 is not the gu comma, because its double odd limit is higher. Thus its name can't be shortened. The comma of a certain color is the ratio < 50¢ with the least double odd limit that isn't a multiple of another comma. 3-limit commas such as L3w-2 = (-30, 19) can be abbreviated as w-19, the wa-19 comma.

Wide, abbreviated W, means widened by an octave. 15/4 = Wy7 = wide yo 7th. 5/1 = WWy3 = double-wide yo 3rd.

Note Names

Notes are named zEb, yyG#, etc. spoken as "zo E flat", "yoyo G sharp". Notes are never large or small, only intervals are. Uncolored notes default to wa. The relative-notation lattice above can be superimposed on this absolute-notation lattice to name every note and interval. For example, D + y3 = yF#, and from yE to ryF# = r2.

Lattice51.png

Prime Subgroup Names

Just as wa means 3-all or 3-limit, ya means 5-all and includes wa, yo, gu, yoyo, gugu, etc. Ya = the 2.3.5 prime subgroup = 5-limit. Za = 7-all = 2.3.7. Yaza = 2.3.5.7 = the full 7-limit. Nowa means without wa, and yaza nowa = 2.5.7.

Prime 2 (even more colorless than wa) is clear, abbreviated ca, and yaza noca = 3.5.7. 2-limit intervals like 2/1 are called wa not clear, for simplicity. Nowaca means without 2 or 3, thus 5.7.11 is yazala nowaca.

Color Names for Higher Primes

Colors for primes greater than 7 are named after the number itself, using the prefix i- for disambiguation as needed:

Lo = 11-over, lu = 11-under, and la = 11-all = 2.3.11 Because "lo C" sounds like "low C", lo when by itself becomes ilo ("ee-LOW"). But with other words it doesn't use i-, as in 11/7 = loru 5th. Lo and lu are abbreviated to 1o and 1u on the score and in interval names and chord names, e.g. ilo A = 1oA, ilo 4th = 1o4 = 11/8, and C ilo seven = C1o7 = 1/1 - 11/9 - 3/2 - 11/6 on C. Lolo is 1oo, triple-lu is 1u3, etc. The associated color is lavender (mnemonic: "e-leven-der"), which refers to both ilo and lu, since they are only 7.1¢ apart. Lavender is a pseudocolor that implies the Neuter temperament. IIo notes could be called lovender, and lu notes could be called luvender. Both are "shades" of lavender.

Tho = 13-over, thu = 13-under, and tha = 13-all. Tho and thu are abbreviated as 3o and 3u on the score and in interval names, e.g. 13/8 = 3o6 = tho 6th, 14/13 = 3uz2 = thuzo 2nd.

Prime subgroups: yala = 2.3.5.11, zalatha nowa = 2.7.11.13, and yazalatha = 2.3.5.7.11.13 = the full 13-limit. Noya is a general term, not used in actual subgroup names, that indicates the absence of 5 and the presence of higher primes, e.g. zala, latha and zalatha are all noya. Likewise, there's noza and noyaza.

On the score and in note names, the 1o accidental either raises by 33/32 or lowers by 729/704. The meaning will usually be clear from context, however it's safer to write at the top of the page either "1o4 = P4" or "1o4 = A4". Likewise, 3o6 should be noted as either m6 or M6. While the note 11/8 above C can be written two ways, either as 1oF or as 1oF#, the interval 11/8 can only be written one way, as 1o4. Likewise, 13/8 above C is either 3oA or 3oAb, but 13/8 is only 3o6. This is the rationale for using large/small/central rather than major/minor. 11/9 is ambiguously major or minor, but unambiguously central. Intervals names and chord names become unambiguous for la and tha intervals. Another rationale: commonly used intervals and chords are all central, and get concise names: gu 3rd not gu minor 3rd, A gu not A gu minor, etc. (see chord names below).

So = 17-over, su = 17-under, and sa = 17-all, abbreviated as 17o and 17u. Iso is an alternate form of so, to distinguish it from the solfege syllable So. 17/12 = 17o5 = iso So.

Ino = 19-over, nu = 19-under, and na = 19-all, abbreviated as 19o and 19u. Ino because "no 3rd" could mean either 19/16 or thirdless. Inu is an alternate form of nu, to distinguish "the nu key" from "the new key". 12edo implies yasana = 2.3.5.17.19.

Twenty-tho = 23-over, twenty-thu = 23-under, twenty-tha =23-all, abbreviated as 23o, 23u and 23a. 2.3.5.7.23 = yaza23a = yaza-twenty-tha. 23/16 = 23o5 = twenty-tho 5th, and 23/22 = 23o1u2 = twenty-tholu 2nd.

Similarly, twenty-no/-nu/-na = 29o/29u/29a, thirty-wo/-wu/-wa = 31o/31u/31a, thirty-so/-su/-sa = 37o/37u/37a, etc.

The prefix i- is only needed when confusion is possible. Thus 19/15 = nogu 4th, not inogu 4th, and 29o = twenty-no, not twenty-ino.

For any prime P, the degree of the ratio P/1 is determined by its 8ve-reduced cents, and how it relates to 12edo: 0-50¢ = 1sn, 50-250¢ = 2nd, 250-450¢ = 3rd, 450-600¢ = 4th, 600-750¢ = 5th, 750-950¢ = 6th, 950-1150¢ = 7th, and 1150-1200¢ = 8ve. Thus 23/16 = 628¢ is a 5th, 31/16 = 1145¢ is a 7th, and 37/32 = 251¢ is a 3rd. This makes the "pseudo-edomapping" <7 11 16 20 24 26 29 30 32 34 34 37...|. (An alternate method is to use the 7edo edomapping, but that requires using every other 14edostep as boundaries, less convenient than the 24edo boundaries used here.)

Converting a ratio to/from a color name

Often a ratio can be converted by breaking it down into simpler, familiar ratios. For example, 45/32 is 5/4 times 9/8, which is y3 plus w2. The colors and degrees are summed, making y4. The magnitude is not summed, and must be found either visually from the lattices above, or from the monzo directly. 45/32 = |-5 2 1>, and (2+1)/7 rounds to 0, so y4 is central, and 45/32 = y4.

For more complex ratios, a more direct method is used:

Converting a ratio: Find the 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 "pseudo-edomapping" discussed above <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 = 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.

Example: interval = sgg2, S = 2 - 1 = 1 step, 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 = 69 mod 7 - 7 - 3 = -4, a = (1 - (-32) - 11·(-4)) / 7 = 77/7 = 11, monzo = |11 -4 -2>, ratio = 2048/2025.

Staff notation

Notes on the staff default to wa. Non-wa notes have a color accidental like g, ry, etc. Like conventional sharp/flat accidentals, they apply to every such note in the measure and in the same octave. Unlike conventional accidentals which apply to a note (e.g. A), color accidentals only apply to one specific "version" of that note (e.g. A flat or A natural). For example, the yo accidental in the first chord applies to all the D naturals in that measure, but not to the D flats.

Notation example 1.png

Staff notation can optionally include a color signature written above the staff. This makes color notation more similar to Johnston notation.

Notation example 2.png

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. (See Sagittal-JI-Translated-To-Colors.png.) For example, yF# = ypGb, and zEb = zqD#. This allows trills to always be written as a 2nd, less cluttered.Notation example 5a.png 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. The font used here is Arial Black.

Chord names

Triads are named after their 3rd, e.g. a yo chord has a yo 3rd. A yo chord rooted on C is a Cy chord = "C yo" = C yE G. Qualities such as major and minor aren't used, because a chord with an 11/9 3rd is hard to classify. Thirdless dyads are written C5 = w1 w5 or C(zg5) = w1 zg5. The four main yaza triads:

lattice62.png

Tetrads are named e.g. "C yo six" = Cy6 = C yE G yA. The 11 main yaza tetrads, with chord homonyms (same shape, different root) equated:

Lattice63.png

A 9th chord contains a 3rd, 5th and 7th. An 11th chord contains all these plus a 9th. A 13th chord contains all these plus an 11th. The 5th, 9th and/or 13th default to wa. The 6th, 7th, and/or 11th default to the color of the 3rd. Thus Cy13 = w1 y3 w5 y7 w9 y11 w13, and Cy9 and Cy11 are subsets of this chord. However, an add-11 chord defaults to a wa 11, see z7,11:

Lattice64.png

Alterations are always in parentheses, additions never are, e.g. z7(zg5) and z,y6. An alteration's degree must match a note in the chord, e.g. Cz7(y6) is invalid. But an exception is made for sus chords, where degree 2 or 4 alter the 3rd: C(z4) = w1 z4 w5. The sus note defaults to wa: Cy9(4) = w1 w4 w5 y7 w9.

Omissions are indicated by "no", the Hendrix chord might be Ch7z10no5. Unless using po or qu, enharmonic substitutions aren't allowed. 7/3 is a 10th, never a 9th unless it's a qu 9th (e.g. Ch7zq9no5). A no3 tetrad can also be written as a 5 chord with an added 6th or 7th: Cy6no3 = C5y6, and Cz7(zg5)no3 = C(zg5)z7.

The y,z7 chord is also called the h7 chord ("har-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 = Cy,z7,w9 and Ch11 = Cy,z7,w9,1o11. Cs9 = Cg,r6,w11 and Cs11 = Cg,r6,w11,1u9. All harmonic numbers must be odd, Ch8 is invalid. For any odd number N > 7, ChN would be 1:3:5:7...N and CsN would be 3/(1:3:5:7...N). Additions refer to harmonics or subharmonics, not degrees: Cs7,11 adds 1u9, not w11. To add w11, use colors: Cs7,w11. Alterations and omissions refer to degrees, not (sub)harmonics: Cs7(zg5) alters the w5, not the 5th subharmonic g3. Ch9no5 omits w5, not y3. However, all numbers > 13 refer to (sub)harmonics (e.g. Ch19no15).

Chords can be classified as bicolored (e.g. g7 or r6), tricolored (e.g. z7(zg5) or z,y6), quadricolored (e.g. s7(zg5) or h7,zg9), etc.

Chord Progressions, Keys and Modulations

The tonic is always wa. The root of each chord has a color, which defaults to wa. C - Am - F - G7 might be Cy - yAg - Fy - Gy,w7, spoken as "C yo, yo A gu, F yo, G yo wa-seven". If the root isn't wa, the root color is added to each interval's color. Thus yAg = yA + (w1 g3 w5) = yA + wC + yE.

In relative notation, the I, IV and V chords default to a wa root. But II, III, VI and VII must have an explicit root-color. The previous example becomes Iy - yVIg - IVy - Vy,w7, spoken as "one yo, yo-six gu, four yo, five yo wa-seven".

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 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 wa 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. One can also modulate fourthward or fifthward, abbreviated 4thwd or 5thwd. Modulating from a yo key to the relative gu, then from there to the parallel yo is modulating yoward. Likewise, there's guward, zoward, iloward, etc.

Temperament Names

Temperaments are named after the color of the comma(s) they temper out. Meantone = the gu temperament = gT. Porcupine = triple yo temperament = y3T. 7-limit Porcupine = triple yo and ru = y3&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: Schismatic is LyT and Diaschismic 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¢.

If the comma is wa, an edo is implied. The temperament is named after the edo, not the comma. 2.3.5.7 and the pyth comma and 225/224 is 12edo&ryyT. If the comma(s) don't include every prime in the subgroup, some primes are untempered. These are added with plus: Blackwood is 5edo+yT = 5-edo plus ya. 2.3.5.7.11 and 81/80 = g+z1aT = gu plus zala.

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. y3&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The prime name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.

Ups and Downs, Lifts and Drops, Plain and Mid

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#. Some edos like 9, 12, 16, 19, 23 and 26 don't require ups and downs.

Unlike actual colors, virtual colors generally add up to something simpler, e.g. three of 22edo's ups adds up to an A1. Unlike actual colors, virtual colors combine with major, minor, etc. Intervals are named upmajor 3rd = ^M3, up 4th = ^4, downaug 5th = vA5, etc. Chords are named C upminor 7th = C^m7 = C ^Eb G ^Bb, etc.

Plain means neither up nor down, analogous to natural meaning neither sharp nor flat. Mid, abbreviated ~, means exactly midway between major and minor. Mid simplifies 72edo notation: m2, ^m2, v~2, ~2, ^~2, vM2, M2. Upmid (^~) means one edostep above mid in 72edo, but one half edostep above mid in 53edo. Mid is only used in relative notation, it never applies to notes and never appears on the staff.

Rank-2 temperaments can be notated with ups and downs as well. Plain and mid are also used in this context. Some temperaments require an additional pair of virtual colors, lifts and drops (/ and \). Notes are named lift C = /C, downdrop F sharp = v\F#, etc. Intervals are named drop 4th = \4, uplift major 3rd = ^/M3, etc. Plain means neither up nor down nor lifted nor dropped. There may be upmid or liftmid intervals. Chords are named C-up lift-seven = C^,/7 = C ^E G /Bb, C uplift-seven = C^/7 = C ^/E G ^/Bb, etc. See pergens.

Translations

For translations of color notation terms into other languages, see Color notation/Translations.