Color notation

Color notation is a musical notation system for just intonation that was primarily developed by Kite Giedraitis.

Color notation has many features that other microtonal notations lack:

• No new symbols: all new accidentals are familiar characters; hence they are immediately speed-readable.
• Furthermore, they are all on the QWERTY keyboard, making the notation easily typeable.
• Every new accidental has a spoken name (colorspeak), making the notation speakable.
• Most importantly, one can name not only notes but also intervals. As a result, color notation can name scales, chords, chord progressions and even prime subgroups and temperaments.

Colorspeak is designed to be an international language, a sort of microtonal Esperanto quickly learned and spoken no matter what one's native language is. Almost every term in colorspeak is one syllable ending with a vowel. The five basic vowels are pronounced ah-eh-ee-oh-oo as in Spanish or Italian.

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 make 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 or Spanish/Portuguese azul), 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 indicate 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.

The following table lists all the intervals in this lattice. See the gallery of just intervals for many more examples.

ratio	cents	color & degree
1/1	0¢	wa unison	w1
21/20	84¢	zogu 2nd	zg2
16/15	112¢	gu 2nd	g2
15/14	119¢	ruyo unison	ry1
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
28/15	1081¢	zogu octave	zg8
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 #Color Names for Higher Primes below for why). Intervals on the lattice's far right or far left are called not augmented and diminished but large and small, written as L and s, and abbreviated as la and sa. La and sa can always be distinguished from solfege's La and saregam's Sa by context. Central, the default, means neither large nor small. This lattice shows the boundaries between the large, small and central zones:

The general term for large/small/central is magnitude. Only intervals have a magnitude, notes never do, and L and s never appear on the staff. A ratio's magnitude is the sum of all the monzo exponents except the first one, divided by 7, and rounded off. 0 = central, 1 = large, 2 = double large, etc. 81/64 = [-6 4>, and 4/7 rounds to 1, so 81/64 = Lw3. 135/128 = [-7 3 1> = Ly1. Unfortunately, magnitudes do not add up predictably like colors and degrees: w2 + w2 = Lw3.

Colors can be doubled or tripled, which are abbreviated bi- ("bee") and tri- ("tree"): 49/25= bizogu 9th = zzgg9 and 128/125 = trigu 2nd = g³2. Bi- is only used if it shortens the name: 25/16 = yoyo 5th not biyo 5th. Likewise with magnitudes: double-large is lala and triple-large is trila. For quadruple, etc., see #Exponents.

Colors using only one prime above 3 are called primary colors. Thus gu and yoyo are primary and ruyo is non-primary.

Degrees can be negative: 50/49 = biruyo 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 notated as 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 dim unison.

Compound, abbreviated co- or c, is a conventional music theory term that means widened by an octave. 15/4 is a compound yo 7th = coyo 7th = cy7. 5/1 is a double-compound yo 3rd = cocoyo 3rd = ccy3. 9/1 is a tricowa 2nd = c³w2. More examples in the Gallery of just intervals. Mnemonic: co- as in co-pilot means auxiliary, thus a 9th is a co-2nd. See #Prime Subgroup Names below for another mnemonic.

Note names

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

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 = no-fives 7-limit. 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 = Bohlen-Pierce. 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. Clear/ca is only ever used in the terms noca and nowaca, and in certain theoretical discussions. However, an additional mnemonic for "co-" (compound, widened by an octave) is "clear-over", in the sense that the ratio's numerator is multiplied by 2.

More on prime subgroups in the next section.

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 when with other words, it doesn't need i-, as in 11/7 = loru 5th. La when by itself becomes ila, to avoid confusion with the solfege note La, and also with La for large. 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, trilu 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 Lulu aka Neutral temperament. IIo notes could be called lovender, and lu notes could be called luvender. Both are "shades" of lavender.

(One might be tempted to write 11o instead of 1o. This would work on a score, but would be confusing in chord names. The triad C11o would look like a diminished 11th chord. In general, color notation avoids naming primes with the numbers found in chord names, which are 2 4 5 6 7 9 11 and 13.)

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. (See the preceding paragraph for why it's 3o and not 13o.)

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 descriptive adjective, not used in actual prime 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, noyaza, etc.

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 primary 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 is that commonly used intervals and chords are all central, and get concise names: gu 3rd not gu minor 3rd, E gu not E gu minor, etc. (see #Chord Names below).

So = 17-over, su = 17-under, and sa = 17-all, abbreviated as 17o, 17u and 17a. Iso is an alternate form of so, to distinguish it from the solfege syllable So. 17/12 = 17o5 = iso So. Isa is an alternate form of sa, to distinguish it from sa for small, and from the Indian saregam syllable Sa. (See the 3rd paragraph in this section for why it's 17o and not 7o.)

No = 19-over, nu = 19-under, and na = 19-all, abbreviated as 19o, 19u and 19a. Ino is an alternate form of no, because "no 3rd" could mean either 19/16 or thirdless. Inu is an alternate form of nu, to distinguish "the nu chord" from "the new chord". (See the 3rd paragraph in this section for why it's 19o and not 9o.)

The prefix i- is only used when confusion is possible. Thus 19/15 = nogu 4th not inogu 4th.

Twetho = 23-over, twethu = 23-under, and twetha = 23-all, abbreviated as 23o, 23u and 23a. 2.3.5.7.23 = yaza23a = yazatwetha. 23/16 = 23o5 = twetho 5th, and 23/22 = 23o1u2 = twetholu 2nd. 529/512 = 23oo2 = bitwetho 2nd (not twethotho, because that means 23-over 13-over).

Similarly, tweno/-nu/-na = 29o/29u/29a, thiwo/-wu/-wa = 31o/31u/31a, etc. The abbreviations are twe-, thi-, fo-, fi- and si-. Note that wa by itself means 3-limit, but -wa as a suffix means "-one-all".

prime	5/4	7/4	11/8	13/8	17/16	19/16	23/16	29/16	31/16	37/32	41/32	43/32	47/32	53/32	59/32	61/32	67/64
	y3	z7	1o4	3o6	17o2	19o3	23o5	29o7	31o7	37o3	41o3	43o4	47o5	53o6	59o7	61o7	67o2
word	ya	za	(i)la	tha	(i)sa	(i)na	twetha	twena	thiwa	thisa	fowa	fotha	fosa	fitha	fina	siwa	sisa
on the score	M3	m7	P4 or A4	m6 or M6	m2	m3	d5	m7	M7	m3	M3	P4	P5	M6	m7 or M7	M7	m2

Mnemonic (sung to the tune of "Supercalifragilisticexpialidocious"):

Yaza latha sana twetha twena thiwa thisa / Fowa fotha fosa fitha fina siwa sisa

Unfortunately seventy can't become se- because that already means 17-exponent (see #Exponents below). Setho means 13¹⁷-over, so it can't mean 73-over. So starting at 71, one might use the longer form: 71o is seventy-wo, 73o is seventy-tho, etc. 103o is hundred-tho and 113o is one-ten-tho. Or one might use these terms:

prime	71/64	73/64	79/64	83/64	89/64	97/64	101/64	103/64	107/64	109/64	113/64	127/64
	71o2	73o2	79o3	83o4	89o4	97o5	101o6	103o6	107o6	109o6	113o7	127o8
word	fitwewa	fitwetha	fitwena	fithitha	fithina	fifosa	fifiwa	fifitha	fifisa	fifina	fisitha	sisisa

Note that 23/16 = 628¢ is a 5th, not a 4th (but see po & qu below). Furthermore, 31/16 = 1145¢ is a 7th not an 8ve, and 37/32 = 251¢ is a 3rd not a 2nd. For any prime P, the degree of the ratio P/1 is chosen to minimize negative intervals. It is determined by its 8ve-reduced cents, and how it relates to 12edo:

unison	2nd	3rd	4th	5th	6th	7th	8ve
0-50¢	50-250¢	250-450¢	450-600¢	600-750¢	750-950¢	950-1150¢	1150-1200¢

This makes the "pseudo-edomapping" <7 11 16 20 24 26 29 30 32 34 37...]. An alternative method would use the actual 7edo edomapping, but that requires using every other 14edostep as boundaries, harder to remember and much less convenient than the 24edo boundaries used here. Since negative intervals will arise no matter what, convenience is prioritized. For the first 26 primes, the 24edo-based degrees correspond to 7klmrs-edo.

Exponents

Exponent syllables aka multiplier syllables provide a way to shorten names that have repeated syllables. For example, 250/243 = 2¹ * 3^-5 * 5³ is a yoyoyo unison which shortens to triyo 1sn. Exponents can also apply to magnitudes (Wa-22 = sasasawa 4th --> trisawa 4th) and octaves (13/1 = cococotho 6th --> tricotho 6th).

We've seen bi- for double and tri- for triple. Quadruple and quintuple are abbreviated quad- and quin-, as in quadyo or quingu. Colorspeak syllables usually end in one of the five basic vowels. Quad and quin are both exceptions, so quad may optionally be spoken as "kwah", and quin as "kwee".

Except for quad, all exponent syllables are prime numbers. Septuple is sep-. Above 7, all exponent syllables are the root color word plus -e. Eleven-fold is le- = "eleven exponent", pronounced as in "lens". Thirteen-fold is the- as in "thesaurus". Note that sep- means seven-fold and se- means seventeen-fold.

Exponents can be combined: sextuple = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. The component syllables are simply the number's prime factors in descending order, except that quad replaces bibi and comes before tri.

Exponents affect all subsequent syllables until the -a- delimiter occurs: trizogu = z³g³, but trizo-agu = z³g. The "a" in la- and sa- also acts as a delimiter: trilayo = L³y, not L³y³, which would be trila-triyo.

Long color names use hyphens to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity.

• Put a hyphen before every -a- delimiter
• Put a hyphen after the magnitude (after the final la- or sa-)
• Put a hyphen after coco-, trico-, etc.
• Put a hyphen before and after "seventy", "eighty", etc.

The hyphen is generally omitted if it would create a subunit of 1 syllable. Thus despite the 2nd rule, layo, lalagu and sagugu are all unhyphenated. And despite the 3rd rule, coyo, cozogu and cocowa are unhyphenated. However, the last rule always holds, e.g. 284/243 = 2² * 3^-5 * 71 is a sa-seventy-wo 3rd.

Converting a ratio to/from a color name

Often a ratio can be converted by breaking it down into simpler ratios with familiar color names, then adding. 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 it's central, and 45/32 = y4.

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

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. If the interval is > 1200¢, octave-reduce as desired (e.g. a 9th may or may not become a compound 2nd). Add one co- prefix for every octave removed. Combine repeated syllables so that three yo's becomes triyo, etc. For the exact combination "grammar", see Color notation/Temperament Names.

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 = zogu 6th or zg6 (63/20 would be zg13 = czg6)

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 C be the number of "co-" prefixes. Let the monzo be [a b c d e...>. The colors directly give you all the monzo entries except a and b. Let S' = the dot product of [0 0 c d e...> with the pseudo-edomapping. Let M' = round ((2 (S - S') + c + d + e + ...) / 7). Then a = -3 (S - S') - 11 (M - M') + C and b = 2 (S - S') + 7 (M - M'). (Derivation here) Convert the monzo to a ratio.

Example: interval = sgg2 = sagugu 2nd

• S = 2 - 1 = 1 step, M = small = -1, C = 0. Monzo = [a b -2>
• S' = <7 11 16] dot [0 0 -2> = -32. S - S' = 1 - (-32) = 33.
• M' = round ((2·33 + (-2)) / 7) = round (64 / 7) = 9. M - M' = -1 - 9 = -10.
• a = -3 (S - S') - 11 (M - M') + C = -3·33 - 11·(-10) + 0 = -99 + 110 = 11.
• b = 2 (S - S') + 7 (M - M') = 2·33 + 7·(-10) = 66 - 70 = -4
• 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.

L and s never appear on the staff. Tripled colors are written as y3 not y³ or yyy. In MuseScore, color accidentals are made by adding fingerings to the notes, then editing the fingering text. A fingering can be copied from one note and pasted to another note. The font used here is Arial Black.

This 10-page score uses the free open-source font Petaluma Script. The letters are 9pt, except that a "z" between two staff lines is 8pt. File:Evening Rondo colors.pdf

Color signatures

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

Po and qu

Po and qu ("coo") (short forms p and q) are two optional accidentals that indicate raising/lowering by a pythagorean comma. (Mnemonics: p stands for pythagorean, and q is the mirror image of p.) Why would one want to do that? Because by first subtracting that comma and then adding it on again, one can rename a note as another note. This is similar to Sagittal notation (see Sagittal-JI-Translated-To-Colors.png).

For example, F# minus a pythagorean comma is Gb. And Gb plus a pythagorean comma is po Gb. Thus an alternate name for F# is po Gb. Adding po raises the degree by one. The new note name is always a 12edo equivalent of the old note name. Adding qu lowers the degree: Gb = qu F#. If one is resolving from Gb to G, one can rename Gb as qF#.

Subtracting po lowers the degree. Thus ruyopo Db = ruyo C#.

Po and qu can be used with intervals as well. A ruyo 1sn becomes a ruyopo 2nd. Neither the color nor the magnitude changes.

One reason to change the degree is for ease of naming chords. See the Hendrix chord in the next section. Another reason is to avoid an awkward unison trill.

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:

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:

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. Mnemonic: every other note of a stacked-thirds chord is non-wa: 6th-root-3rd-5th-7th-9th-11th-13th. Thus Cy13 = w1 y3 w5 y7 w9 y11 w13, and Cy9 and Cy11 are subsets of this chord. However, an added 11th defaults to wa, as in z7,11:

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. The sus note defaults to wa. A 6:8:9 chord could be written C(4), but the parentheses rule is relaxed to allow the conventional C4. Likewise 8:9:12 is C2. But if the sus note isn't wa, parentheses must be used. Thus w1 z4 w5 = C(z4) = "C zo-four". More examples:

• 6:7:8:9 = Cz,4 = "C zo add-four"
• w1 w4 w5 y7 w9 = Cy9(4) = "C yo-nine sus-four"
• w1 z4 w5 z7 = Cz7(z4) or C(z4)z7 = "C zo-seven zo-four" or "C zo-four zo-seven"

Omissions are indicated by "no". The za Hendrix chord is Ch7z10no5. (To write it as a sharp-9 chord, use qu: 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 called the h7 chord ("har-seven"), because it's part of the harmonic series. Ch9 = Cy,z7,9 and Ch11 = Cy,z7,w9,1o11. The s7 ("sub-seven") chord is part of the subharmonic series. It's the first 7 subharmonics, with the 7th subharmonic becoming the root. Cs9 = Cr,g7,9 and Cs11 = C1o11(1or5,1og9). Note that s9 is not s7 plus a 9th, but a completely different chord. Usually the 9th ascends from the root, but in a sub9 chord it descends from the top note, and becomes the new root. Thus the s7 chord is contained in the upper four notes of the s9 chord, not the lower four.

Cs6 = Cg,r6 = 12:10:8:7. Other than the s6 chord, all harmonic/subharmonic numbers must be odd, e.g. Ch6 and Ch8 are invalid. For any odd number N greater than 5, ChN is 1:3:5...N and CsN is 1/(N...5:3:1). Additions, alterations and omissions refer to degrees, not harmonics or subharmonics: Ch7,11 adds w11, not 1o11. Ch9no5 omits w5, not y3. However, all numbers > 13 refer to (sub)harmonics, e.g. Ch9,15 adds y7 and Ch19no15 omits it.

All wa chords can be named conventionally, since wa is the default color. Thus w1-w3-w5 is both Cw and Cm. And w1-Lw3-w5-w6 is both CLw6 and C6. For aesthetic reasons, the conventional name is preferred only when neither "M" nor "m" appears in the name (since color notation doesn't use major/minor). This is especially true if the chord includes non-wa notes: w1-w3-w5-y6 is Cw,y6 not Cm,y6.

Chords can be classified as bicolored (e.g. g7 or r6), tricolored (e.g. z7(zg5) or z,y6), quadricolored (e.g. s6(zg5) or h7,zg9), etc.

Chord progressions, keys, scales and modulations

The tonic is always wa. The root of each chord has a color, which defaults to wa. C - Am - F - G7 might become 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 previous example becomes Iy - yVIg - IVy - Vy,w7, spoken as "one yo, yo-six gu, four yo, five yo wa-seven". Never use lower-case roman numerals for minor chords: ii becomes IIg or IIz. A IIIy chord has a w3 root, which is 32/27 not 81/64. The latter would be a LwIIIy chord (use L and s, not # and b; #IIIy is invalid).

In adaptive JI, chords are just, but roots move by tempered intervals. Comma pumps are indicated with brackets roughly halfway through the pump: Cy - yAg - [y=w]Dg - Gy - Cy. The pattern is [old=new]: the previous chord implies yDg and the following chord implies wDg. See Comma pump examples.

Keys and scales are loosely named after the colors used. Wa is assumed present. In 5-limit JI, the key of A minor is A gu and the scale is the gu scale. The Bbh7 - Ebh7 - Bbh7 - Fh9 example in the staff notation section is in Bb yozo. Like chords, scales can be classified as bicolored (A gu), tricolored (Bb yo zo), etc.

Scales can be named more precisely analogous to how chords are named. The tonic, 2nd, 4th and 5th default to wa. Thus a yo scale is w1 w2 y3 w4 w5 y6 y7 w8. If the 2nd were instead yo, it would be a yo yo-2 scale, written y(y2). If the 2nd is sometimes yo, sometimes wa, the scale is yo plus yo-2, written y+y2. (A hexatonic scale might use "minus".) The 5-limit harmonic minor scale is gu yo-7. The Bbh7 - Ebh7 - Bbh7 - Fh9 scale is Bb yo plus zo-3-4-7, written Bb y+z347.

(Occasionally, the 6th or the 7th may be La or sa. For example, the wa scale has a wa 3rd, because the 3rd of the scale always matches the scale name exactly. The 6th and 7th default to a perfect 4th/5th from the 3rd, so the 6th is sa, not central. Thus the wa scale is minor, and the Lawa scale is major.)

Just as there is a har7 chord, there is a har15 scale: w1 w2 y3 1o4 w5 3o6 z7 y7 w8. A har-N scale (where N is odd) is harmonics (N+1)/2 to N+1. The tonic of the scale is always a power of 2. Thus the har9 scale is not 5:6:7:8:9:10 but 8:9:10:12:14:16 = w1 w2 y3 w5 z7 w8. Likewise there are sub scales.

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 in either direction is modulating waward. Modulating from a yo key to the relative gu, and perhaps from there to the parallel yo is modulating yoward. A root movement by a yo interval (e.g. Iy - yVIg) is a yoward move. Likewise, there's guward, and yaward includes both. Likewise, there's zoward, ruward, zaward, iloward, etc.

Temperament names and comma names

Main article: Color notation/Temperament names

Temperaments are named after the comma(s) they temper out. Commas are named using an alternate format that omits the degree. 81/80 is the Gu comma, with the "G" capitalized to distinguish it from the gu color, which includes many ratios. Tempering out Gu creates Meantone or Guti or gT, where "-ti" and "T" stand for temperament. 2048/2025 is the Sagugu comma sgg2, and Srutal is Saguguti or sggT. Porcupine is Triyoti or y³T. Certain commas over 90¢ use the -bi- syllable (see the main article for details). For example, Schismic is Layoti or LyT, but Mavila is Layobiti or Ly#2T. Certain wa commas use yet another alternate format, e.g. Mercator's comma is Wa-53 or w-53.

Multi-comma temperaments have multiple commas in their name. Septimal Meantone is Gu & Ruyoyo and Dominant Meantone is Gu & Rugu (-ti can be omitted when the ampersand is used). Untempered primes are included with a plus sign. The 2.3.5.7 prime subgroup with 81/80 tempered out is Guti + za.

MOS and MODMOS scales can be named as e.g. Triyoti[8]. Individual modes can be named as 2nd Triyoti[8], 3rd Triyoti[7] b7, etc. See Naming Rank-2 Scales using Mode Numbers.

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 intervals in edos, and colors can be used as well. A more precise notation uses 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#. Sharp-1 and flat-1 edos 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.

Plain means neither up nor down, analogous to natural meaning neither sharp nor flat. Mid, abbreviated ~, means exactly midway between major and minor. The mid 4th is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th is halfway-diminished. There is no mid unison or octave. Mid simplifies 72edo notation: m2, ^m2, v~2, ~2, ^~2, vM2, M2. Mid is only used in relative notation, it never applies to notes and never appears on the staff. In 24-edo or 31-edo, the 3rd of C~ is vE or ^Eb, but in 41-edo, it's vvE or ^^Eb.

Chords are named similarly to color notation, with the various qualities downmajor, upminor, mid, etc. replacing colors. Major is the default quality, thus C = C major and Cv = C downmajor. The 6th, 7th and 11th inherit their quality from the 3rd, thus C upminor 9th = C ^Eb G ^Bb D. Chord roots can have ups and downs, as in Cv - Gv - vA^m - Fv or Iv - Vv - vVI^m - IVv. In roman numeral notation, chord roots can be downflat, mid, etc., as in Iv7 - vbIII^m6 - IVv7 or I~7 - ~III - V7. Lower-case roman numerals are never used for minor chords, because vii could mean either seven-minor or down-two-minor. Instead vii is written either VIIm or vIIm. See the notation guide for edos 5-72

Rank-2 temperaments can be notated with ups and downs as well. Plain and mid are also used in this context. Certain 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 add lift-seven = C^,/7 = C ^E G /Bb, C uplift-seven = C^/7 = C ^/E G ^/Bb, etc. See pergens.

Glossary / crash course

Over = prime in the numerator. Under = prime in the denominator. All = over, under or neither: wa = 3-limit, ya = 2.3.5, yaza = 2.3.5.7. Exponent = repeated syllable: triyo = yoyoyo = 125-over.

prime	-o ("oh") for over		-u ("oo") for under		-a ("ah") for all		-e ("eh") for exponent
2	—		—		(clear)	—	bi ("bee")	double
3	—		—		wa (white)	—	tri ("tree")	triple
							quad	quadruple
5	yo (yellow)	y	gu (green)	g	ya	—	quin	quintuple
7	zo (azul)	z	ru (red)	r	za	—	sep	septuple
11	(i)lo	1o	lu	1u	(i)la	1a	le	11-fold
13	tho	3o	thu	3u	tha	3a	the	13-fold
17	(i)so	17o	su	17u	(i)sa	17a	se	17-fold
19	(i)no	19o	(i)nu	19u	na	19a	ne	19-fold
23	twetho	23o	twethu	23u	twetha	23a	twethe	23-fold

Higher primes: 29o = tweno, 31o = thiwo, 37o = thiso, 41o = fowo, 43o = fotho, 47o = foso, 53o = fitho, 59o = fino, 61o = siwo, 67o = siso.

Exponents: sextuple is tribi (triply-doubled), octuple is quadbi, 9-fold is tritri, etc.

Pronunciation: exponent syllables like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Biruyo, Bizozogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: Sasa-gugu, Zozotrigu.

word		meaning	example
central		refers to a ratio centrally located in the lattice	every ratio of odd limit < 81 is central (but only some > 81 are not central)
la-	L	large, augmented by 2187/2048 from the central ratio	32/27 = wa 3rd = w3, 81/64 = lawa 3rd = Lw3
sa-	s	small, diminished by 2187/2048 from the central ratio	27/16 = wa 6th = w6, 128/81 = sawa 6th = sw6
magnitude		refers to central, la, sa, lala, trisa, quadla, etc.	the sum of all prime exponents except the 1st, divided by 7 and rounded off
i-		disambiguation prefix	no 3rd = omit the 3rd, but ino 3rd = 19/16
-a-		delimits an exponent such as bi-, tri-, etc.	Trizogu = z3g3 = 1029/1000, but Trizo-agu = z3g = 343/320
co-	c	compound (conventional term for widened by an 8ve)	7/4 = zo 7th = z7, 7/2 = compound zo 7th = cozo 7th = cz7
har	h	refers to a harmonic series (otonal) chord	4:5:6:7 = C har-seven = Ch7
sub	s	refers to a subharmonic series (utonal) chord	7:6:5:4 = C sub-seven = Cs7
po	p	adds a pythagorean comma, to change the degree	15/14 = ruyo unison = ry1 = ruyopo 2nd = ryp2
qu	q	subtracts a pythagorean comma	49/48 = zozo 2nd = zz2 = zozoqu unison = zzq1
-ti	T	changes a comma name to a temperament name	Gu = 81/80, Guti = meantone
-bi	#2	as a suffix, 2nd smallest comma in the row segment	Guti = gT is Meantone, but Gubiti = g#2T is Father (16/15 vanishes)
Wa-	w-	alternate interval format, only used for 3-limit commas	Mercator's comma = Wa-53 = w-53
nowa		remove 3 (wa) from the prime subgroup, i.e. no-threes	2.5.7 = yaza nowa, 2.5.7 & 50/49 = Biruyoti nowa
noca		remove 2 (clear) from the prime subgroup, i.e. non-8ve	3.5.7 = yaza noca, 3.5.7 & 245/243 = Zozoyoti noca
nowaca		remove both 2 and 3 from the prime subgroup	5.7.11 = yazala nowaca
plus	+	add an untempered prime to the temperament	Blackwood = 2.3.5 with 256/243 tempered out = Sawa + ya
and	&	joins commas that are tempered out	7-limit Porcupine = 2.3.5.7 with 250/243 & 64/63 = Triyo & Ru
-ward	-wd	refers to the direction of chord root movement	Iy - IVy = 4thwd, Iy - Vy = 5thwd, Iy - yIIIy = yoward, Ig - gIIIg = guward

Homonyms:

• "wa" means both "3-all" and "-one-all" (e.g. thiwa means 31-all). The meaning is always clear from context.
• "lo" means both "11-over" and "low", as in "low C". Thus 1o by itself becomes "ilo".
• "la" means both "11-all" and "large", and also the solfege note La. Thus 1a by itself becomes "ila".
• "so" means both "17-over" and the solfege note So. Thus 17o by itself becomes "iso".
• "sa" means both "17-all" and "small", and also the Saregam note Sa. Thus 17a by itself becomes "isa".
• "no" means both "19-over" and "omit", as in no3. Thus 19o by itself becomes "ino".
• "nu" means both "19-under" and "new", as in "the new key". Thus 19u by itself becomes "inu".
• "bi" means both "doubled" as in biruyo and "2nd smallest" as in Layobi. The meaning is always clear from context.

Temperaments use "virtual colors" represented with arrows ^ v and perhaps slashes / \

word		meaning
up	^	raised by some comma
down	v	lowered by some comma
arrow		refers collectively to both ups and downs
lift	/	raised by some other comma
drop	\	lowered by some other comma
slash		refers collectively to both lifts and drops
plain	♢	neither up nor down nor lifted nor dropped
mid	~	for 2nds, 3rd, 6ths and 7ths, exactly halfway between major and minor a mid 4th is halfway-augmented, and a mid 5th is halfway-diminished

Translations

For translations of color notation terms into other languages, see Color notation/Translations. Translating avoids using sounds not in one's native language. For example, in many European languages, "th-" for prime 13 becomes "tr-".

See also

• xen-calc - A web app that converts to/from ratios, monzos and color notation, and also supports ups and downs notation.
• ledzo notation - A similar competing notation system.

Further reading

• Giedraitis, Kite. Alternative Tunings: Theory, Notation and Practice.