Kite's color notation: Difference between revisions

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Color notation was mostly developed by [[Kite Giedraitis]]. This is a brief summary. For a full explanation, see [http://www.tallkite.com/AlternativeTunings.html "Alternative Tunings: Theory, Notation and Practice"]. For a great webapp that converts to/from ratios, monzos and color notation, see [https://www.yacavone.net/xen-calc/ yacavone.net/xen-calc/]

Color notation was mostly developed by [[Kite Giedraitis]]. This is a brief summary. For a full explanation, see [http://www.tallkite.com/AlternativeTunings.html "Alternative Tunings: Theory, Notation and Practice"]. For a great webapp that converts to/from ratios, monzos and color notation, see [https://www.yacavone.net/xen-calc/ yacavone.net/xen-calc/] It also does [[Ups and Downs Notation|ups and downs]].

Color notation has many advantages over other forms of microtonal notation:

* 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.

* No unnamed accidentals. Imagine rehearsal conversation if # and b were unnamed! "C pound-sign" / "C hashtag", "E little-b", etc.

* Every accidental name in colorspeak is one syllable ending with a vowel, making concise but flowing names.

* Accidentals can be combined and even exponentiatied, minimizing the number of new accidentals to learn.

* Most important of all, there are names for not only notes but also intervals.

* As a result, color notation can name scales, chords, chord progressions and even prime subgroups and temperaments.

== 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).

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). These three vowels (plus e and i which are discussed later) are pronounced as in Spanish or Italian.

3-all = '''Wa''' = white (strong but colorless) = often perfect

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7-under = '''Ru''' = red (alarming, inflamed) = often supermajor

~~The vowels~~

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.

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 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 ==

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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.

The ~~lalawa negative 2nd (LLw-2, the~~ [[~~Pythagorean comma~~|~~pythagorean or 12~~-edo comma]]) is ~~simply called~~ the wa comma~~. The~~ [[~~256~~/~~243~~|5-edo comma]] is the ~~sawa~~ comma, and the [[~~2187/2048~~|7-edo comma]] is the ~~lawa~~ comma. All other ~~3-limit~~ commas such as L3w-2 = (-30, 19) are named as w-19, the wa-19 comma.

The [[256/243|5-edo comma]] is the Sawa comma, the [[2187/2048|7-edo comma]] is the Lawa comma, and the [[Pythagorean comma|pythagorean or 12-edo comma]] is the Lalawa comma. All other wa commas such as L3w-2 = (-30 19) are named as W-19, the Wa-19 comma.

Compound, abbreviated '''co-''' or '''c''', is a conventional 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. More examples in the [[Gallery of just intervals#Intervals larger than an octave|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 superimposed on this absolute-notation lattice to name every note and interval. For example, D + y3 = yF#, and from yE to ryF# = r2.

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.

[[File:Lattice51.png|frameless|962x962px]]

== 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.

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. 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.

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.

== 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. 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=24_17&limit=2_3_11 Lulu aka Neutral] temperament. IIo notes could be called lovender, and lu notes could be called luvender. Both are "shades" of lavender.

'''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 [http://x31eq.com/cgi-bin/rt.cgi?ets=24_17&limit=2_3_11 Lulu aka Neutral] 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.

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'''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.

'''Ino''' = 19-over, '''nu''' = 19-under, and '''na''' = 19-all, abbreviated as '''19o''', '''19u''' and '''19a'''. 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.

'''Ino''' = 19-over, '''nu''' = 19-under, and '''na''' = 19-all, abbreviated as '''19o''', '''19u''' and '''19a'''. Ino 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". 12edo implies yasana = 2.3.5.17.19.

'''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 = bitwetho 2nd (not twethotho, because that means a ratio that's both 23-over and 13-over).

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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 ~~used~~:

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

'''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 "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 (i.e. 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]].

'''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 "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 = zg6. 63/20 would be zg13 = czg6.

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 [https://gist.github.com/m-yac/2236a03dd9fe89a992477fbcbc63746c here]) Convert the monzo to a ratio.

Example: interval = sgg2

Example: interval = sgg2 = sagugu 2nd

* S = 2 - 1 = 1 step, M = small = -1, C = 0. Monzo = [a b -2>

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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 [http://tallkite.com/misc_files/Sagittal-JI-Translated-To-Colors.png 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.[[File:Notation example 5a.png|992x992px]]

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. ~~This text~~ can be copied and pasted. The font used here is Arial Black.

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.

== Chord Names ==

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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 called the h7 chord ("har-seven"), because it's part of the harmonic series. Ch9 = Cy,z7,w9 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,w9 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.

The y,z7 chord is called the h7 chord ("har-seven"), because it's part of the harmonic series. Ch9 = Cy,z7,w9 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,w9 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. Other than the s6 chord, all harmonic/subharmonic numbers must be odd, Ch6 and Ch8 are invalid. For any odd number N >= 7, ChN is 1:3:5:7...N and CsN is N/(1:3:5:7...N). 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.

Cs6 = Cg,r6 = 12/(12:10:8:7). Other than the s6 chord, all harmonic/subharmonic numbers must be odd, Ch6 and Ch8 are invalid. For any odd number N >= 7, ChN is 1:3:5:7...N and CsN is N/(1:3:5:7...N). 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.

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 and Modulations ==

== 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 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". Never use lower-case roman numerals for minor chords: ii becomes IIg or IIz.

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. The pattern is [''old''=''new'']: the previous chord implies yDg and the following chord implies wDg. ~~For more examples, see~~ [[Comma pump examples]].

In adaptive JI, chords are just, but roots move by tempered intervals. Comma pumps are indicated with brackets roughly halfway through he 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/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.

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, scales 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 in either direction is modulating '''waward'''. Modulating from a yo key to the relative gu, then 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.

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, then 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 ==

Temperaments are named after the color of the comma(s) they temper out. The degree is omitted. Sextuple is tribi- and septuple is sep-. Eightfold is quadbi-, ninefold is tritri-, and tenfold is quinbi. Eleven-fold is le- = "e'''l'''even '''e'''xponent".

Temperaments are named after the color of the comma(s) they temper out. The degree is omitted. Remote commas have complex colors and/or magnitudes. We've seen bi-, tri-, quad- and quin-. Sextuple is tribi- and septuple is sep-. Eightfold is quadbi-, ninefold is tritri-, and tenfold is quinbi. Eleven-fold is le- = "e'''l'''even '''e'''xponent". See the glossary for more.

~~[[Meantone]] is~~ the ~~Gu temperament~~. [[Srutal]] is Sagugu. [[Porcupine]] is Triyo. Some multi-comma temperaments have multiple commas in their name, and some don't. [[Meantone family#Septimal meantone|Septimal Meantone]] is Gu, but [[Meantone family#Dominant|Dominant Meantone]] is Gu & Rugu. Untempered primes are included with a plus sign. The 2.3.5.7 prime subgroup with 81/80 tempered out is Gu + za.

~~Every wa comma implies an edo, and a~~ temperament ~~with such a~~ comma ~~uses that edo~~ in ~~its~~ name. ~~Thus Catler~~ is ~~12-edo~~ plus za. ~~An exception is made for~~ 5~~-edo and~~ 7~~-edo: Blackwood~~ is ~~Sawa~~ + ~~ya and Whitewood is Lawa + ya~~.

[[Meantone]] is the Gu temperament. [[Srutal]] is Sagugu. [[Porcupine]] is Triyo. Multi-comma temperaments have multiple commas in their name. [[Meantone family#Septimal meantone|Septimal Meantone]] is Gu & Ruyoyo and [[Meantone family#Dominant|Dominant Meantone]] is Gu & Rugu. Untempered primes are included with a plus sign. The 2.3.5.7 prime subgroup with 81/80 tempered out is Gu + za.

The color name of a rank-2 temperament can be used to name MOS and MODMOS scales, as in Triyo[8]. Individual modes can be named as 2nd Triyo[8], 3rd Triyo[7] b7, etc. See [[Naming Rank-2 Scales using Mode Numbers]].

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== 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|~~edo~~]] ~~notes~~, and colors can be used as well. A more precise notation uses [[Ups and Downs Notation|'''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.

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 [[EDO|edos]], and colors can be used as well. A more precise notation uses [[Ups and Downs Notation|'''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 (5, 7, 9, 12, 16, 19, 23, etc.) 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.

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-Color notation was mostly developed by [[Kite Giedraitis]]. This is a brief summary. For a full explanation, see [http://www.tallkite.com/AlternativeTunings.html "Alternative Tunings: Theory, Notation and Practice"]. For a great webapp that converts to/from ratios, monzos and color notation, see [https://www.yacavone.net/xen-calc/ yacavone.net/xen-calc/]
+Color notation was mostly developed by [[Kite Giedraitis]]. This is a brief summary. For a full explanation, see [http://www.tallkite.com/AlternativeTunings.html "Alternative Tunings: Theory, Notation and Practice"]. For a great webapp that converts to/from ratios, monzos and color notation, see [https://www.yacavone.net/xen-calc/ yacavone.net/xen-calc/] It also does [[Ups and Downs Notation|ups and downs]].
+Color notation has many advantages over other forms of microtonal notation:
+* No new symbols: all new accidentals are familiar characters, hence they are immediately <u>speed-readable</u>.
+* Furthermore, they are all on the QWERTY keyboard, making the notation easily <u>typeable</u>.
+* No unnamed accidentals. Imagine rehearsal conversation if # and b were unnamed! "C pound-sign" / "C hashtag", "E little-b", etc.
+* Every accidental name in colorspeak is one syllable ending with a vowel, making concise but flowing names.
+* Accidentals can be combined and even exponentiatied, minimizing the number of new accidentals to learn.
+* Most important of all, there are names for not only notes but also <u>intervals</u>.
+* As a result, color notation can name scales, chords, chord progressions and even prime subgroups and temperaments.
 == 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).
+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). These three vowels (plus e and i which are discussed later) are pronounced as in Spanish or Italian.
 &nbsp; &nbsp; &nbsp;3-all = '''Wa''' = white (strong but colorless) = often perfect<br />
@@ Line 11: / Line 21: @@
 -under = '''Ru''' = red (alarming, inflamed) = often supermajor
-The vowels
+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.
-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 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 ==
@@ Line 216: / Line 224: @@
 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.
-The lalawa negative 2nd (LLw-2, the [[Pythagorean comma|pythagorean or 12-edo comma]]) is simply called the wa comma. The [[256/243|5-edo comma]] is the sawa comma, and the [[2187/2048|7-edo comma]] is the lawa comma. All other 3-limit commas such as L<sup>3</sup>w-2 = (-30, 19) are named as w-19, the wa-19 comma.
+The [[256/243|5-edo comma]] is the Sawa comma, the [[2187/2048|7-edo comma]] is the Lawa comma, and the [[Pythagorean comma|pythagorean or 12-edo comma]] is the Lalawa comma. All other wa commas such as L<sup>3</sup>w-2 = (-30 19) are named as W-19, the Wa-19 comma.
 Compound, abbreviated '''co-''' or '''c''', is a conventional 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. More examples in the [[Gallery of just intervals#Intervals larger than an octave|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 superimposed on this absolute-notation lattice to name every note and interval. For example, D + y3 = yF#, and from yE to ryF# = r2.
+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.
 [[File:Lattice51.png|frameless|962x962px]]
 == 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.
+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. 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.
+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.
 == 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. 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 1u<sup>3</sup>, 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=24_17&limit=2_3_11 Lulu aka Neutral] temperament. IIo notes could be called lovender, and lu notes could be called luvender. Both are "shades" of lavender.
+'''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 1u<sup>3</sup>, 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=24_17&limit=2_3_11 Lulu aka Neutral] 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.
@@ Line 243: / Line 251: @@
 '''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.
-'''Ino''' = 19-over, '''nu''' = 19-under, and '''na''' = 19-all, abbreviated as '''19o''', '''19u''' and '''19a'''. 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.
+'''Ino''' = 19-over, '''nu''' = 19-under, and '''na''' = 19-all, abbreviated as '''19o''', '''19u''' and '''19a'''. Ino 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". 12edo implies yasana = 2.3.5.17.19.
 '''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 = bitwetho 2nd (not twethotho, because that means a ratio that's both 23-over and 13-over).
@@ Line 314: / Line 322: @@
 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 <u>not</u> 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 used:
+For more complex ratios, a more direct method is needed:
-'''<u>Converting a ratio</u>:''' 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 "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 (i.e. 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]].
+'''<u>Converting a ratio</u>:''' 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 "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 = zg6. 63/20 would be zg13 = czg6.
+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)
 <u>'''Converting a color name'''</u>: 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 [https://gist.github.com/m-yac/2236a03dd9fe89a992477fbcbc63746c here]) Convert the monzo to a ratio.
-Example: interval = sgg2
+Example: interval = sgg2 = sagugu 2nd
 * S = 2 - 1 = 1 step, M = small = -1, C = 0. Monzo = [a b -2>
@@ Line 342: / Line 357: @@
 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 [http://tallkite.com/misc_files/Sagittal-JI-Translated-To-Colors.png 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.[[File:Notation example 5a.png|992x992px]]
-L and s never appear on the staff. Tripled colors are written as y3 not y<sup>3</sup> or yyy. In MuseScore, color accidentals are made by adding fingerings to the notes, then editing the fingering text. This text can be copied and pasted. The font used here is Arial Black.
+L and s never appear on the staff. Tripled colors are written as y3 not y<sup>3</sup> 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.
 == Chord Names ==
@@ Line 361: / Line 376: @@
 Omissions are indicated by "no". The Hendrix chord might be Ch7z10no5. Unless using po or qu, <u>enharmonic substitutions aren't allowed</u>. 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 called the h7 chord ("har-seven"), because it's part of the harmonic series. Ch9 = Cy,z7,w9 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,w9 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.
+The y,z7 chord is called the h7 chord ("har-seven"), because it's part of the harmonic series. Ch9 = Cy,z7,w9 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,w9 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 <u>upper</u> four notes of the s9 chord, not the lower four.
-Cs6 = Cg,r6. Other than the s6 chord, all harmonic/subharmonic numbers must be odd, Ch6 and Ch8 are invalid. For any odd number N >= 7, ChN is 1:3:5:7...N and CsN is N/(1:3:5:7...N).  <u>Additions, a</u><u>lterations and omissions refer to degrees</u>, not harmonics or subharmonics: Ch7,11 adds w11, not 1o11. Ch9no5 omits w5, not y3. However, <u>all numbers > 13 refer to (sub)harmonics</u>, e.g. Ch9,15 adds y7 and Ch19no15 omits it.
+Cs6 = Cg,r6 = 12/(12:10:8:7). Other than the s6 chord, all harmonic/subharmonic numbers must be odd, Ch6 and Ch8 are invalid. For any odd number N >= 7, ChN is 1:3:5:7...N and CsN is N/(1:3:5:7...N).  <u>Additions, a</u><u>lterations and omissions refer to degrees</u>, not harmonics or subharmonics: Ch7,11 adds w11, not 1o11. Ch9no5 omits w5, not y3. However, <u>all numbers > 13 refer to (sub)harmonics</u>, e.g. Ch9,15 adds y7 and Ch19no15 omits it.
 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 and Modulations ==
+== 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 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 <u>must</u> 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". Never use lower-case roman numerals for minor chords: ii becomes IIg or IIz.
-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. The pattern is [''old''=''new'']: the previous chord implies yDg and the following chord implies wDg. For more examples, see [[Comma pump examples]].
+In adaptive JI, chords are just, but roots move by tempered intervals. Comma pumps are indicated with brackets roughly halfway through he 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/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.
+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, scales 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. 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, then 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.
+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. 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, then 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 '''y<u>a</u>ward''' includes both. Likewise, there's '''zoward''', '''ruward''', '''zaward''', '''iloward''', etc.
 == Temperament Names ==
-Temperaments are named after the color of the comma(s) they temper out. The degree is omitted. Sextuple is tribi- and septuple is sep-. Eightfold is quadbi-, ninefold is tritri-, and tenfold is quinbi. Eleven-fold is le- = "e'''<u>l</u>'''even '''<u>e</u>'''xponent".
+Temperaments are named after the color of the comma(s) they temper out. The degree is omitted. Remote commas have complex colors and/or magnitudes. We've seen bi-, tri-, quad- and quin-. Sextuple is tribi- and septuple is sep-. Eightfold is quadbi-, ninefold is tritri-, and tenfold is quinbi. Eleven-fold is le- = "e'''<u>l</u>'''even '''<u>e</u>'''xponent". See the glossary for more.
-[[Meantone]] is the Gu temperament. [[Srutal]] is Sagugu. [[Porcupine]] is Triyo. Some multi-comma temperaments have multiple commas in their name, and some don't. [[Meantone family#Septimal meantone|Septimal Meantone]] is Gu, but [[Meantone family#Dominant|Dominant Meantone]] is Gu & Rugu. Untempered primes are included with a plus sign. The 2.3.5.7 prime subgroup with 81/80 tempered out is Gu + za.
-Every wa comma implies an edo, and a temperament with such a comma uses that edo in its name. Thus Catler is 12-edo plus za. An exception is made for 5-edo and 7-edo: Blackwood is Sawa + ya and Whitewood is Lawa + ya.
+[[Meantone]] is the Gu temperament. [[Srutal]] is Sagugu. [[Porcupine]] is Triyo. Multi-comma temperaments have multiple commas in their name. [[Meantone family#Septimal meantone|Septimal Meantone]] is Gu & Ruyoyo and [[Meantone family#Dominant|Dominant Meantone]] is Gu & Rugu. Untempered primes are included with a plus sign. The 2.3.5.7 prime subgroup with 81/80 tempered out is Gu + za.
 The color name of a rank-2 temperament can be used to name MOS and MODMOS scales, as in Triyo[8]. Individual modes can be named as 2nd Triyo[8], 3rd Triyo[7] b7, etc. See [[Naming Rank-2 Scales using Mode Numbers]].
@@ Line 390: / Line 403: @@
 == 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|edo]] notes, and colors can be used as well. A more precise notation uses [[Ups and Downs Notation|'''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.
+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 [[EDO|edos]], and colors can be used as well. A more precise notation uses [[Ups and Downs Notation|'''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 (5, 7, 9, 12, 16, 19, 23, etc.) 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.