Kite's color notation: Difference between revisions

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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.

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. Clear/ca is only ever used in the terms noca and nowaca.

== Color Names for Higher Primes ==

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'''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'''.

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 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 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. '''Isa''' is an alternate form of sa, to distinguish it from small, and from the Indian saregam syllable Sa.

'''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 small, and from the Indian saregam syllable Sa.

'''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.

'''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.

'''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. 529/512 = bi-twenty-tho 2nd.

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The prefix i- is only used 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.)

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, harder to remember and much less convenient than the 24edo boundaries used here.)

== Converting a Ratio to/from a Color Name ==

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Example: ratio = 63/40, monzo = |-3 2 -1 1>, color = zogu, stepspan = <7 11 16 20| dot |-3 2 -1 1> = -21 + 22 - 16 + 20 = 5 steps, degree = 5 + 1 = a 6th, magnitude = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central, interval = zg6.

Converting a color name: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let the monzo be |a b c d e...>. The colors directly give you all the monzo entries except a and b. Let X = the dot product of |0 0 c d e...> with the ~~7edo~~ edomapping. Then b = (2S - 2X + 3) mod 7 + 7M - 3, and a = (S - X - 11b) / 7. Convert the monzo to a ratio.

Converting a color name: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let the monzo be |a b c d e...>. The colors directly give you all the monzo entries except a and b. Let X = the dot product of |0 0 c d e...> with the pseudo-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 = 6 - 10 = -4, a = (1 - (-32) - 11·(-4)) / 7 = 77/7 = 11, monzo = |11 -4 -2>, ratio = 2048/2025.

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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 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.

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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. 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.

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.

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.

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.

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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 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]].

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[[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. Blackwood is 5-edo + ya. The 2.3.5.7 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. This notation is discussed here: [[Naming Rank-2 Scales using Mode Numbers]].

Temperament names are further explained here: [[Color notation/Temperament Names]].

== 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 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 [[EDO|edo]] notes, and colors can be used as well. A more precise application is to use [[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.

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.~~

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. 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.

'''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. half-augmented, and the mid 5th is half-diminished. There is no mid unison or octave. 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. In 24-edo or 31-edo, the 3rd of C~ is vE or ^Eb, but in 41-edo, it's vvE or ^^Eb.

~~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 [~~[Pergen|pergens]~~].

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 [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf notation guide for edos 5-72]

~~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. ~~This notation is discussed here:~~ [[~~Naming Rank-2 Scales using Mode Numbers~~]].

[[Tour of Regular Temperaments|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 [[Pergen|pergens]].

== Glossary / Crash Course ==

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|-

| -ward

| style="text-align:center" | wd

| style="text-align:center" | -wd

|refers to the direction of chord root movement

|I - IV = 4thwd, I - V = 5thwd, I - III = yoward, i - iii = guward

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|mid

|~

|exactly halfway between major and minor

|for 2nds, 3rd, 6ths and 7ths, exactly halfway between major and minor

for 4ths, half-aug, and for 5ths, half-dim

|}

==Translations==

:''For translations of color notation terms into other languages, see [[Color notation/Translations]].'' [[Category:color_notation]] [[Category:ji]] [[Category:notation]]

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 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.
+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. Clear/ca is only ever used in the terms noca and nowaca.
 == Color Names for Higher Primes ==
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 '''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'''.
+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 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. <u>This is the rationale for using large/small/central rather than major/minor</u>. 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. '''Isa''' is an alternate form of sa, to distinguish it from small, and from the Indian saregam syllable Sa.
+'''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 small, and from the Indian saregam syllable Sa.
-'''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.
+'''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.
 '''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. 529/512 = bi-twenty-tho 2nd.
@@ Line 199: / Line 199: @@
 The prefix i- is only used 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.)
+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, harder to remember and much less convenient than the 24edo boundaries used here.)
 == Converting a Ratio to/from a Color Name ==
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 Example: ratio = 63/40, monzo = |-3 2 -1 1>, color = zogu, stepspan = <7 11 16 20| dot |-3 2 -1 1> = -21 + 22 - 16 + 20 = 5 steps, degree = 5 + 1 = a 6th, magnitude = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central, interval = zg6.
-Converting a color name: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let the monzo be |a b c d e...>. The colors directly give you all the monzo entries except a and b. Let X = the dot product of |0 0 c d e...> with the 7edo edomapping. Then b = (2S - 2X + 3) mod 7 + 7M - 3, and a = (S - X - 11b) / 7. Convert the monzo to a ratio.
+Converting a color name: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let the monzo be |a b c d e...>. The colors directly give you all the monzo entries except a and b. Let X = the dot product of |0 0 c d e...> with the pseudo-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 = 6 - 10 = -4, a = (1 - (-32) - 11·(-4)) / 7 = 77/7 = 11, monzo = |11 -4 -2>, ratio = 2048/2025.
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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 y<sup>3</sup> 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.
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 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. 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.
+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.
+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.
 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.
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 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".
+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]].
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 [[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. Blackwood is 5-edo + ya. The 2.3.5.7 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. This notation is discussed here: [[Naming Rank-2 Scales using Mode Numbers]].
 Temperament names are further explained here: [[Color notation/Temperament Names]].
 == 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 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 [[EDO|edo]] notes, and colors can be used as well. A more precise application is to use [[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.
-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.
+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. 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.
+'''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. half-augmented, and the mid 5th is half-diminished. There is no mid unison or octave. 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. In 24-edo or 31-edo, the 3rd of C~ is vE or ^Eb, but in 41-edo, it's vvE or ^^Eb.
-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 [[Pergen|pergens]].
+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 [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf notation guide for edos 5-72]
-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. This notation is discussed here: [[Naming Rank-2 Scales using Mode Numbers]].
+[[Tour of Regular Temperaments|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 [[Pergen|pergens]].
 == Glossary / Crash Course ==
@@ Line 437: / Line 442: @@
 |-
 | -ward
-| style="text-align:center" | wd
+| style="text-align:center" | -wd
 |refers to the direction of chord root movement
 |I - IV = 4thwd, I - V = 5thwd, I - III = yoward, i - iii = guward
@@ Line 488: / Line 493: @@
 |mid
 |~
-|exactly halfway between major and minor
+|for 2nds, 3rd, 6ths and 7ths, exactly halfway between major and minor
+for 4ths, half-aug, and for 5ths, half-dim
 |}
 ==Translations==
-:''For translations of color notation terms into other languages, see [[Color notation/Translations]].''                         [[Category:color_notation]]                         [[Category:ji]]                         [[Category:notation]]
+:''For translations of color notation terms into other languages, see [[Color notation/Translations]].''                          [[Category:color_notation]]                          [[Category:ji]]                          [[Category:notation]]