Kite's color notation: Difference between revisions

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* 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.

'''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 (/a/, /ɛ/, /i/, /o/, and /u/) 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).

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).

{| class="right-1"

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|}

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 make a red-yellow-green-blue rainbow, with warm/cool colors indicating sharp/flat intervals. The rainbow of 3rds runs {{dash|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.

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: {{nowrap|z3 + g3 {{=}} zg5}} {{nowrap|{{=}} zogu 5th}}. Zogu, not guzo; higher primes always come first. Opposite colors cancel: {{nowrap|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.<imagemap>

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.

File:Lattice32.png | 694x694px

# yellow

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{| class="wikitable" style="text-align: center;"

|-

! ~~'''ratio'''~~

! Ratio

! ~~'''cents'''~~

! Cents

! colspan="2" | ~~'''color~~ & degree~~'''~~

! colspan="2" | Color & degree

|-

| 1/1

| 0¢

| 0{{c}}

| wa unison

| w1

|-

| 21/20

| ~~84¢~~

| 84{{c}}

| zogu 2nd

| zg2

|-

| 16/15

| ~~112¢~~

| 112{{c}}

| gu 2nd

| g2

|-

| 15/14

| ~~119¢~~

| 119{{c}}

| ruyo unison

| ry1

|-

| 10/9

| ~~182¢~~

| 182{{c}}

| yo 2nd

| y2

|-

| 9/8

| ~~204¢~~

| 204{{c}}

| wa 2nd

| w2

|-

| 8/7

| ~~231¢~~

| 231{{c}}

| ru 2nd

| r2

|-

| 7/6

| ~~267¢~~

| 267{{c}}

| zo 3rd

| z3

|-

| 6/5

| ~~316¢~~

| 316{{c}}

| gu 3rd

| g3

|-

| 5/4

| ~~386¢~~

| 386{{c}}

| yo 3rd

| y3

|-

| 9/7

| ~~435¢~~

| 435{{c}}

| ru 3rd

| r3

|-

| 21/16

| ~~471¢~~

| 471{{c}}

| zo 4th

| z4

|-

| 4/3

| ~~498¢~~

| 498{{c}}

| wa 4th

| w4

|-

| 7/5

| ~~583¢~~

| 583{{c}}

| zogu 5th

| zg5

|-

| 10/7

| ~~617¢~~

| 617{{c}}

| ruyo 4th

| ry4

|-

| 3/2

| ~~702¢~~

| 702{{c}}

| wa 5th

| w5

|-

| 32/21

| ~~729¢~~

| 729{{c}}

| ru 5th

| r5

|-

| 14/9

| ~~765¢~~

| 765{{c}}

| zo 6th

| z6

|-

| 8/5

| ~~814¢~~

| 814{{c}}

| gu 6th

| g6

|-

| 5/3

| ~~884¢~~

| 884{{c}}

| yo 6th

| y6

|-

| 12/7

| ~~933¢~~

| 933{{c}}

| ru 6th

| r6

|-

| 7/4

| ~~969¢~~

| 969{{c}}

| zo 7th

| z7

|-

| 16/9

| ~~996¢~~

| 996{{c}}

| wa 7th

| w7

|-

| 9/5

| ~~1018¢~~

| 1018{{c}}

| gu 7th

| g7

|-

| 28/15

| ~~1081¢~~

| 1081{{c}}

| zogu octave

| zg8

|-

| 15/8

| ~~1088¢~~

| 1088{{c}}

| yo 7th

| y7

|-

| 40/21

| ~~1116¢~~

| 1116{{c}}

| ruyo 7th

| ry7

|-

| 2/1

| ~~1200¢~~

| 1200{{c}}

| 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 notation#Color%20Names%20for%20Higher%20Primes | #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:

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 notation#Color%20Names%20for%20Higher%20Primes|#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:

[[File:Lattice41a.png|833x833px]]

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.

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. {{nowrap|0 {{=}} central|1 {{=}} large|2 = double large}}, etc. {{nowrap|81/64 {{=}} {{vector| -6 4 }}}}, and 4/7 rounds to 1, so {{nowrap|81/64 {{=}} Lw3}}. Similarily, {{nowrap|135/128 {{=}} {{map| -7 3 1 }}}} = Ly1. Unfortunately, magnitudes do not add up predictably like colors and degrees: {{nowrap|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 can be doubled or tripled, which are abbreviated '''bi-''' ("bee") and '''tri-''' ("tree"): {{nowrap|49/25 {{=}} bizogu 9th}} = zzgg9 and {{nowrap|128/125 {{=}} trigu 2nd}} = g32. Bi- is only used if it shortens the name: {{nowrap|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.

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 {{nowrap|121/64 {{=}} 1102{{c}}}}. Going from a {{nowrap|yo M7 {{=}} 1088{{c}}}} up to this m7 raises the pitch, and 121/120 is a dim unison.

'''Compound''', abbreviated '''co-''' or '''c''', is a [[wikipedia:Interval_(music)#Compound_intervals|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#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.

'''Compound''', abbreviated '''co-''' or '''c''', is a [[wikipedia:Interval_(music)#Compound_intervals|conventional music theory term]] that means widened by an octave. 15/4 is a compound {{nowrap|yo 7th {{=}} coyo 7th}} = cy7. 5/1 is a double-compound {{nowrap|yo 3rd {{=}} cocoyo 3rd}} = ccy3. 9/1 is a {{nowrap|tricowa 2nd {{=}} c3w2}}. 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 mentally 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, {{nowrap|D + y3 {{=}} yF#}}, and from yE to {{nowrap|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 = no-fives 7-limit. 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. {{nowrap|Ya {{=}} the 2.3.5 prime subgroup}} = 5-limit. {{nowrap|'''Za''' {{=}} 7-all}} = 2.3.7 {{nowrap|{{=}} no-fives 7-limit}}. {{nowrap|Yaza {{=}} 2.3.5.7}} {{nowrap|{{=}} the full 7-limit}}. '''Nowa''' means without wa, and {{nowrap|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.

Prime 2 (even more colorless than wa) is '''clear''', abbreviated '''ca''', and {{nowrap|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.

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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 [[243/242 | 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.

{{nowrap|'''Lo''' {{=}} 11-over|'''lu''' {{=}} 11-under}}, and {{nowrap|'''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 {{nowrap|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 [[243/242 | 7.1{{c}}]] 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.

(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.)

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|}

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:

Note that 23/16 = 628{{c}} is a 5th, not a 4th (but see po & qu below). Furthermore, 31/16 = 1145{{c}} is a 7th not an 8ve, and 37/32 = 251{{c}} 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:

{| class="wikitable"

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! 8ve

|-

| 0-~~50¢~~

| 0-50{{c}}

| 50-~~250¢~~

| 50-250{{c}}

| 250-~~450¢~~

| 250-450{{c}}

| 450-~~600¢~~

| 450-600{{c}}

| 600-~~750¢~~

| 600-750{{c}}

| 750-~~950¢~~

| 750-950{{c}}

| 950-~~1150¢~~

| 950-1150{{c}}

| 1150-~~1200¢~~

| 1150-1200{{c}}

|}

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 [[Val#Shorthand_notation | 7klmrs-edo]].

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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 (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]].

'''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{{c}}, 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

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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 [[Color notation/Temperament names | 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.

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{{c}} use the -bi- syllable (see the [[Color notation/Temperament names | 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. [[Meantone family#Septimal meantone | Septimal Meantone]] is Gu & Ruyoyo and [[Meantone family#Dominant | 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]].

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| colspan="2" | nowa

| remove 3 (wa) from the prime subgroup, i.e. no-threes

| 2.5.7 = yaza nowa, 2.5.7 & 50/49 = Biruyoti nowa

|-

| colspan="2" | noca

| remove 2 (clear) from the prime subgroup, i.e. non-8ve

| 3.5.7 = yaza noca, 3.5.7 & 245/243 = Zozoyoti noca

|-

| colspan="2" | nowaca

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|-

| and

| style="text-align: center;" | &

| joins commas that are tempered out

| 7-limit Porcupine = 2.3.5.7 with 250/243 & 64/63 = Triyo & Ru

|-

| -ward

@@ Line 7: / Line 7: @@
 * 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.
+'''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 (/a/, /ɛ/, /i/, /o/, and /u/) 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).
+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,&nbsp;9/8), under (4/3,&nbsp;16/9) and neither (1/1,&nbsp;2/1).
 {| class="right-1"
@@ Line 30: / Line 30: @@
 |}
-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 make a red-yellow-green-blue rainbow, with warm/cool colors indicating sharp/flat intervals. The rainbow of 3rds runs {{dash|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.
+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: {{nowrap|z3 + g3 {{=}} zg5}} {{nowrap|{{=}} zogu 5th}}. Zogu, not guzo; higher primes always come first. Opposite colors cancel: {{nowrap|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.<imagemap>
+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.
+<imagemap>
 File:Lattice32.png | 694x694px
 # yellow
@@ Line 79: / Line 80: @@
 {| class="wikitable" style="text-align: center;"
 |-
-! '''ratio'''
+! Ratio
-! '''cents'''
+! Cents
-! colspan="2" | '''color & degree'''
+! colspan="2" | Color &amp; degree
 |-
 | 1/1
-| 0¢
+| 0{{c}}
 | wa unison
 | w1
 |-
 | 21/20
-| 84¢
+| 84{{c}}
 | zogu 2nd
 | zg2
 |-
 | 16/15
-| 112¢
+| 112{{c}}
 | gu 2nd
 | g2
 |-
 | 15/14
-| 119¢
+| 119{{c}}
 | ruyo unison
 | ry1
 |-
 | 10/9
-| 182¢
+| 182{{c}}
 | yo 2nd
 | y2
 |-
 | 9/8
-| 204¢
+| 204{{c}}
 | wa 2nd
 | w2
 |-
 | 8/7
-| 231¢
+| 231{{c}}
 | ru 2nd
 | r2
 |-
 | 7/6
-| 267¢
+| 267{{c}}
 | zo 3rd
 | z3
 |-
 | 6/5
-| 316¢
+| 316{{c}}
 | gu 3rd
 | g3
 |-
 | 5/4
-| 386¢
+| 386{{c}}
 | yo 3rd
 | y3
 |-
 | 9/7
-| 435¢
+| 435{{c}}
 | ru 3rd
 | r3
 |-
 | 21/16
-| 471¢
+| 471{{c}}
 | zo 4th
 | z4
 |-
 | 4/3
-| 498¢
+| 498{{c}}
 | wa 4th
 | w4
 |-
 | 7/5
-| 583¢
+| 583{{c}}
 | zogu 5th
 | zg5
 |-
 | 10/7
-| 617¢
+| 617{{c}}
 | ruyo 4th
 | ry4
 |-
 | 3/2
-| 702¢
+| 702{{c}}
 | wa 5th
 | w5
 |-
 | 32/21
-| 729¢
+| 729{{c}}
 | ru 5th
 | r5
 |-
 | 14/9
-| 765¢
+| 765{{c}}
 | zo 6th
 | z6
 |-
 | 8/5
-| 814¢
+| 814{{c}}
 | gu 6th
 | g6
 |-
 | 5/3
-| 884¢
+| 884{{c}}
 | yo 6th
 | y6
 |-
 | 12/7
-| 933¢
+| 933{{c}}
 | ru 6th
 | r6
 |-
 | 7/4
-| 969¢
+| 969{{c}}
 | zo 7th
 | z7
 |-
 | 16/9
-| 996¢
+| 996{{c}}
 | wa 7th
 | w7
 |-
 | 9/5
-| 1018¢
+| 1018{{c}}
 | gu 7th
 | g7
 |-
 | 28/15
-| 1081¢
+| 1081{{c}}
 | zogu octave
 | zg8
 |-
 | 15/8
-| 1088¢
+| 1088{{c}}
 | yo 7th
 | y7
 |-
 | 40/21
-| 1116¢
+| 1116{{c}}
 | ruyo 7th
 | ry7
 |-
 | 2/1
-| 1200¢
+| 1200{{c}}
 | 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 notation#Color%20Names%20for%20Higher%20Primes | #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:
+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 notation#Color%20Names%20for%20Higher%20Primes|#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:
 [[File:Lattice41a.png|833x833px]]
-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.
+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. {{nowrap|0 {{=}} central|1 {{=}} large|2 = double large}}, etc. {{nowrap|81/64 {{=}} {{vector| -6 4 }}}}, and 4/7 rounds to 1, so {{nowrap|81/64 {{=}} Lw3}}. Similarily, {{nowrap|135/128 {{=}} {{map| -7 3 1 }}}} =&nbsp;Ly1. Unfortunately, magnitudes do not add up predictably like colors and degrees: {{nowrap|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 = g<sup>3</sup>2. 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 can be doubled or tripled, which are abbreviated '''bi-''' ("bee") and '''tri-''' ("tree"): {{nowrap|49/25 {{=}} bizogu 9th}} =&nbsp;zzgg9 and {{nowrap|128/125 {{=}} trigu 2nd}} =&nbsp;g<sup>3</sup>2. Bi- is only used if it shortens the name: {{nowrap|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.
+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 {{nowrap|121/64 {{=}} 1102{{c}}}}. Going from a {{nowrap|yo M7 {{=}} 1088{{c}}}} up to this m7 raises the pitch, and 121/120 is a dim unison.
-'''Compound''', abbreviated '''co-''' or '''c''', is a [[wikipedia:Interval_(music)#Compound_intervals|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 = c<sup>3</sup>w2. 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.
+'''Compound''', abbreviated '''co-''' or '''c''', is a [[wikipedia:Interval_(music)#Compound_intervals|conventional music theory term]] that means widened by an octave. 15/4 is a compound {{nowrap|yo 7th {{=}} coyo 7th}} =&nbsp;cy7. 5/1 is a double-compound {{nowrap|yo 3rd {{=}} cocoyo 3rd}} =&nbsp;ccy3. 9/1 is a {{nowrap|tricowa 2nd {{=}} c<sup>3</sup>w2}}. 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 mentally 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, {{nowrap|D + y3 {{=}} yF#}}, and from yE to {{nowrap|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 = no-fives 7-limit. 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. {{nowrap|Ya {{=}} the 2.3.5 prime subgroup}} =&nbsp;5-limit. {{nowrap|'''Za''' {{=}} 7-all}} =&nbsp;2.3.7 {{nowrap|{{=}} no-fives 7-limit}}. {{nowrap|Yaza {{=}} 2.3.5.7}} {{nowrap|{{=}} the full 7-limit}}. '''Nowa''' means without wa, and {{nowrap|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.
+Prime 2 (even more colorless than wa) is '''clear''', abbreviated '''ca''', and {{nowrap|yaza '''noca''' {{=}} 3.5.7}} =&nbsp;[[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.
@@ Line 253: / Line 254: @@
 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 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 [[243/242 | 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.
+{{nowrap|'''Lo''' {{=}} 11-over|'''lu''' {{=}} 11-under}}, and {{nowrap|'''la''' {{=}} 11-all}} =&nbsp;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 {{nowrap|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 [[243/242 | 7.1{{c}}]] 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.
 (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.)
@@ Line 402: / Line 403: @@
 |}
-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:
+Note that 23/16 = 628{{c}} is a 5th, not a 4th (but see po &amp; qu below). Furthermore, 31/16 = 1145{{c}} is a 7th not an 8ve, and 37/32 = 251{{c}} 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:
 {| class="wikitable"
@@ Line 415: / Line 416: @@
 ! 8ve
 |-
-| 0-50¢
+| 0-50{{c}}
-| 50-250¢
+| 50-250{{c}}
-| 250-450¢
+| 250-450{{c}}
-| 450-600¢
+| 450-600{{c}}
-| 600-750¢
+| 600-750{{c}}
-| 750-950¢
+| 750-950{{c}}
-| 950-1150¢
+| 950-1150{{c}}
-| 1150-1200¢
+| 1150-1200{{c}}
 |}
 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 [[Val#Shorthand_notation | 7klmrs-edo]].
@@ Line 448: / Line 449: @@
 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 (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]].
+'''<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{{c}}, 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
@@ Line 544: / Line 545: @@
 {{Main | 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 y<sup>3</sup>T. Certain commas over 90¢ use the -bi- syllable (see the [[Color notation/Temperament names | 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.
+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 y<sup>3</sup>T. Certain commas over 90{{c}} use the -bi- syllable (see the [[Color notation/Temperament names | 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. [[Meantone family#Septimal meantone | Septimal Meantone]] is Gu & Ruyoyo and [[Meantone family#Dominant | 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.
+Multi-comma temperaments have multiple commas in their name. [[Meantone family#Septimal meantone | Septimal Meantone]] is Gu &amp; Ruyoyo and [[Meantone family#Dominant | Dominant Meantone]] is Gu &amp; 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]].
@@ Line 742: / Line 743: @@
 | colspan="2" | nowa
 | remove 3 (wa) from the prime subgroup, i.e. no-threes
-| 2.5.7 = yaza nowa, 2.5.7 & 50/49 = Biruyoti nowa
+| 2.5.7 = yaza nowa, 2.5.7 &amp; 50/49 = Biruyoti nowa
 |-
 | colspan="2" | noca
 | remove 2 (clear) from the prime subgroup, i.e. non-8ve
-| 3.5.7 = yaza noca, 3.5.7 & 245/243 = Zozoyoti noca
+| 3.5.7 = yaza noca, 3.5.7 &amp; 245/243 = Zozoyoti noca
 |-
 | colspan="2" | nowaca
@@ Line 758: / Line 759: @@
 |-
 | and
-| style="text-align: center;" | &
+| style="text-align: center;" | &amp;
 | joins commas that are tempered out
-| 7-limit Porcupine = 2.3.5.7 with 250/243 & 64/63 = Triyo & Ru
+| 7-limit Porcupine = 2.3.5.7 with 250/243 &amp; 64/63 = Triyo &amp; Ru
 |-
 |  -ward