Kite's color notation: Difference between revisions

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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. (One might be tempted to write 11o instead of 1o. This would work on a score, but not in chord names. The triad C11o would look like a diminished 11th chord.) 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.

'''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. (One might be tempted to write 11o instead of 1o. This would work on a score, but not in chord names. The triad C11o would look like a diminished 11th chord.) 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.

'''Tho''' = 13-over, '''thu''' = 13-under, and '''tha''' = 13-all. Tho and thu are abbreviated as '''3o''' and '''3u''' on the score and in interval names, e.g. 13/8 = 3o6 = tho 6th, 14/13 = 3uz2 = thuzo 2nd. (See the preceding paragraph for why it's 3o and not 13o.)

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|sisisa

|}

Note that 23/16 = 628¢ is a 5th, not a 4th. 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¢ 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:

{| class="wikitable"

!unison

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== Exponents ==

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

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

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

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

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

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

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[[File:Lattice63.png|639x639px]]

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

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

[[File:Lattice64.png|660x660px]]

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Omissions are indicated by "no". The za [[Hendrix chord]] is Ch7z10no5. (To write it as a sharp-9 chord, use qu: Ch7zq9no5.) A no3 tetrad can also be written as a 5 chord with an added 6th or 7th: Cy6no3 = C5y6, and Cz7(zg5)no3 = C(zg5)z7.

The [[4:5:6:7|y,z7 chord]] is called the h7 chord ("har-seven"), because it's part of the harmonic series. [[4:5:6:7:9|Ch9]] = Cy,z7,9 and [[4:5:6:7:9:11|Ch11]] = Cy,z7,w9,1o11. The [[60:70:84:105|s7 ("sub-seven") chord]] is part of the subharmonic series. It's the first 7 subharmonics, with the 7th subharmonic becoming the root. [[140:180:210:252:315|Cs9]] = Cr,g7,9 and Cs11 = C1o11(1or5,1og9). Note that s9 is not s7 plus a 9th, but a completely different chord. Usually the 9th ascends from the root, but in a sub9 chord it descends from the top note, and becomes the new root. Thus the s7 chord is contained in the ~~~~upper~~~~ four notes of the s9 chord, not the lower four.

The [[4:5:6:7|y,z7 chord]] is called the h7 chord ("har-seven"), because it's part of the harmonic series. [[4:5:6:7:9|Ch9]] = Cy,z7,9 and [[4:5:6:7:9:11|Ch11]] = Cy,z7,w9,1o11. The [[60:70:84:105|s7 ("sub-seven") chord]] is part of the subharmonic series. It's the first 7 subharmonics, with the 7th subharmonic becoming the root. [[140:180:210:252:315|Cs9]] = Cr,g7,9 and Cs11 = C1o11(1or5,1og9). Note that s9 is not s7 plus a 9th, but a completely different chord. Usually the 9th ''ascends'' from the root, but in a sub9 chord it ''descends'' from the top note, and becomes the new root. Thus the s7 chord is contained in the ''upper'' four notes of the s9 chord, not the lower four.

Cs6 = Cg,r6 = [[70:84:105:120|12:10:8:7]]. Other than the s6 chord, all harmonic/subharmonic numbers must be odd, e.g. Ch6 and Ch8 are invalid. For any odd number N greater than 5, ChN is 1:3:5...N and CsN is 1/(N...5:3:1). Additions, alterations and omissions refer to degrees, not harmonics or subharmonics: Ch7,11 adds w11, not 1o11. Ch9no5 omits w5, not y3. However, all numbers > 13 refer to (sub)harmonics, e.g. Ch9,15 adds y7 and Ch19no15 omits it.

All wa chords can be named conventionally, since wa is the default color. Thus w1-w3-w5 is both Cw and Cm. And w1-Lw3-w5-w6 is both CLw6 and C6. For aesthetic reasons, the conventional name is preferred only when neither "M" nor "m" appears in the name. This is especially true ~~for~~ non-wa ~~chords~~: w1-w3-w5-y6 is Cw,y6 not Cm,y6.

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

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

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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, scales 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 of A minor is A gu and the scale is the gu scale. The Bbh7 - Ebh7 - Bbh7 - Fh9 example in the staff notation section is in Bb yozo. Like chords, scales can be classified as bicolored (A gu), tricolored (Bb yo zo), etc.

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.

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

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

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

Analogous to the relative and parallel major or minor, one can modulate to relative gu, parallel ru, etc. Modulating from a yo key to the relative gu means using gu chords on yo roots. Modulating from yo to the parallel gu means using gu chords on wa roots. Going from yo zo to the relative gu means using chords with gu and/or ru in them on yo roots. Going to the relative ru means using the same chords on zo roots. Going from yo zo to the parallel gu ru means using the same chords on wa roots. One can also modulate '''fourthward''' or '''fifthward''', abbreviated '''4thwd''' or '''5thwd'''. Modulating in either direction is modulating '''waward'''. Modulating from a yo key to the relative gu, and perhaps from there to the parallel yo is modulating '''yoward'''. A root movement by a yo interval (e.g. Iy - yVIg) is a yoward move. Likewise, there's '''guward''', and '''yaward''' includes both. Likewise, there's '''zoward''', '''ruward''', '''zaward''', '''iloward''', etc.

== Temperament names and comma names ==

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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.

Multi-comma temperaments have multiple commas in their name. [[Meantone family#Septimal meantone|Septimal Meantone]] is Gu & ~~Ruyoyoti~~ and [[Meantone family#Dominant|Dominant Meantone]] is Gu & ~~Ruguti~~. 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 & 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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|}

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

Higher exponents: 6 is tribi (triply-doubled), 8 is quadbi, 9 is tritri, 10 is quinbi, etc.

Pronunciation: exponent syllables like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Bi'''ru'''yo, Bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''Sa'''sa-'''gu'''gu, '''Zo'''zotri'''gu'''.

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!meaning

!example

|-

| colspan="2" |central

|refers to a ratio centrally located in the lattice

|every ratio of odd limit < 81 is central (but only some > 81 are not central)

|-

|la-

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|small, diminished by 2187/2048 from the central ratio

|27/16 = wa 6th = w6, 128/81 = sawa 6th = sw6

|-

| colspan="2" |magnitude

|refers to central, la, sa, lala, trisa, quadla, etc.

|the sum of all prime exponents except the 1st, divided by 7 and rounded off

|-

| colspan="2" |i-

| disambiguation prefix

|no 3rd = omit the 3rd, ino 3rd = 19/16

|no 3rd = omit the 3rd, but ino 3rd = 19/16

|-

| colspan="2" | -a-

| delimits an exponent such as bi-, tri-, etc.

|Trizogu = z3g3 = 1029/1000, Trizo-agu = z3g = 343/320

|Trizogu = z3g3 = 1029/1000, but Trizo-agu = z3g = 343/320

|-

|co-

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| style="text-align:center" |#2

|as a suffix, 2nd smallest comma in the row segment

|~~Meantone = 81/80 =~~ Guti = gT, ~~Father = 16/15 =~~ Gubiti = g#2T

|Guti = gT is Meantone, but Gubiti = g#2T is [[Father]] (16/15 vanishes)

|-

| Wa-

|w-

|alternate interval format, only used for 3-limit commas

|Mercator's comma = Wa-53 = w-53

|[[Mercator's comma]] = Wa-53 = w-53

|-

| colspan="2" |nowa

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

|2.5.7 = yaza nowa, 2.5.7 ~~with~~ 50/49 = ~~Biruyo~~ nowa

|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 ~~with~~ 245/243 = ~~Zozoyo~~ noca

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

|-

| colspan="2" |nowaca

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* "bi" means both "doubled" as in biruyo and "2nd smallest" as in Layobi. The meaning is always clear from context.

Temperaments use "virtual colors" represented with ^ v and / \:

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

{| class="wikitable"

|+

! colspan="2" |word

!meaning

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|v

|lowered by some comma

|-

| colspan="2" |arrow

|refers collectively to both ups and downs

|-

|lift

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|\

|lowered by some other comma

|-

| colspan="2" |slash

|refers collectively to both lifts and drops

|-

|plain

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|~

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

~~for 4ths,~~ halfway-augmented, and ~~for 5ths,~~ halfway-diminished

a mid 4th is halfway-augmented, and a mid 5th is halfway-diminished

|}

@@ Line 249: / Line 249: @@
 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. (One might be tempted to write 11o instead of 1o. This would work on a score, but not in chord names. The triad C11o would look like a diminished 11th chord.) 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.
+'''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. (One might be tempted to write 11o instead of 1o. This would work on a score, but not in chord names. The triad C11o would look like a diminished 11th chord.) 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.
 '''Tho''' = 13-over, '''thu''' = 13-under, and '''tha''' = 13-all. Tho and thu are abbreviated as '''3o''' and '''3u''' on the score and in interval names, e.g. 13/8 = 3o6 = tho 6th, 14/13 = 3uz2 = thuzo 2nd. (See the preceding paragraph for why it's 3o and not 13o.)
@@ Line 370: / Line 370: @@
 |sisisa
 |}
-Note that 23/16 = 628¢ is a 5th, not a 4th. 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¢ 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:
 {| class="wikitable"
 !unison
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 == Exponents ==
-Exponent syllables aka multiplier syllables provide a way to shorten names that have repeated syllables. For example, 250/243 = 2<sup>1</sup> * 3<sup>-5</sup> * 5<sup>3</sup> is a yoyoyo unison which shortens to triyo 1sn. Exponents can apply to magnitudes (Wa-22 = sasasawa 4th --> trisawa 4th) or octaves (13/1 = cococotho 6th --> tricotho 6th).
+Exponent syllables aka multiplier syllables provide a way to shorten names that have repeated syllables. For example, 250/243 = 2<sup>1</sup> * 3<sup>-5</sup> * 5<sup>3</sup> is a yoyoyo unison which shortens to triyo 1sn. Exponents can also apply to magnitudes (Wa-22 = sasasawa 4th --> trisawa 4th) and octaves (13/1 = cococotho 6th --> tricotho 6th).
 We've seen bi- for double and tri- for triple. Quadruple and quintuple are abbreviated '''quad-''' and '''quin-''', as in quadyo or quingu. Colorspeak syllables usually end in one of the five basic vowels. Quad and quin are both exceptions, so quad may optionally be spoken as "kwah", and quin as "kwee".
-Except for quad, all exponent syllables are prime numbers. Septuple is '''sep-'''. Above 7, all exponent syllables are the root color word plus -e. Eleven-fold is '''le-''' = "e'''<u>l</u>'''even '''<u>e</u>'''xponent", pronounced as in "<u>le</u>gitimate". Thirteen-fold is '''the-''' as in "<u>the</u>saurus". Note that sep- means seven-fold and '''se-''' means seven<u>teen</u>-fold.
+Except for quad, all exponent syllables are prime numbers. Septuple is '''sep-'''. Above 7, all exponent syllables are the root color word plus -e. Eleven-fold is '''le-''' = "e'''<u>l</u>'''even '''<u>e</u>'''xponent", pronounced as in "<u>le</u>ns". Thirteen-fold is '''the-''' as in "<u>the</u>saurus". Note that sep- means seven-fold and '''se-''' means seven<u>teen</u>-fold.
 Exponents can be combined: sextuple = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. The component syllables are simply the number's prime factors in descending order, except that quad replaces bibi and comes before tri.
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 [[File:Lattice63.png|639x639px]]
-A 9th chord contains a 3rd, 5th and 7th. An 11th chord contains all these plus a 9th. A 13th chord contains all these plus an 11th. The 5th, 9th and/or 13th default to wa. The 6th, 7th, and/or 11th default to the color of the 3rd. Thus Cy13 = w1 y3 w5 y7 w9 y11 w13, and Cy9 and Cy11 are subsets of this chord. However, an <u>added</u> 11th defaults to wa, as in z7,11:
+A 9th chord contains a 3rd, 5th and 7th. An 11th chord contains all these plus a 9th. A 13th chord contains all these plus an 11th. The 5th, 9th and/or 13th default to wa. The 6th, 7th, and/or 11th default to the color of the 3rd. Mnemonic: every other note of a stacked-thirds chord is non-wa: <u>6th</u>-root-<u>3rd</u>-5th-<u>7th</u>-9th-<u>11th</u>-13th. Thus Cy13 = w1 y3 w5 y7 w9 y11 w13, and Cy9 and Cy11 are subsets of this chord. However, an <u>added</u> 11th defaults to wa, as in z7,11:
 [[File:Lattice64.png|660x660px]]
@@ Line 482: / Line 482: @@
 Omissions are indicated by "no". The za [[Hendrix chord]] is Ch7z10no5. (To write it as a sharp-9 chord, use qu: Ch7zq9no5.) A no3 tetrad can also be written as a 5 chord with an added 6th or 7th: Cy6no3 = C5y6, and Cz7(zg5)no3 = C(zg5)z7.
-The [[4:5:6:7|y,z7 chord]] is called the h7 chord ("har-seven"), because it's part of the harmonic series. [[4:5:6:7:9|Ch9]] = Cy,z7,9 and [[4:5:6:7:9:11|Ch11]] = Cy,z7,w9,1o11. The [[60:70:84:105|s7 ("sub-seven") chord]] is part of the subharmonic series. It's the first 7 subharmonics, with the 7th subharmonic becoming the root. [[140:180:210:252:315|Cs9]] = Cr,g7,9 and Cs11 = C1o11(1or5,1og9). Note that s9 is not s7 plus a 9th, but a completely different chord. Usually the 9th ascends from the root, but in a sub9 chord it descends from the top note, and becomes the new root. Thus the s7 chord is contained in the <u>upper</u> four notes of the s9 chord, not the lower four.
+The [[4:5:6:7|y,z7 chord]] is called the h7 chord ("har-seven"), because it's part of the harmonic series. [[4:5:6:7:9|Ch9]] = Cy,z7,9 and [[4:5:6:7:9:11|Ch11]] = Cy,z7,w9,1o11. The [[60:70:84:105|s7 ("sub-seven") chord]] is part of the subharmonic series. It's the first 7 subharmonics, with the 7th subharmonic becoming the root. [[140:180:210:252:315|Cs9]] = Cr,g7,9 and Cs11 = C1o11(1or5,1og9). Note that s9 is not s7 plus a 9th, but a completely different chord. Usually the 9th ''ascends'' from the root, but in a sub9 chord it ''descends'' from the top note, and becomes the new root. Thus the s7 chord is contained in the ''upper'' four notes of the s9 chord, not the lower four.
 Cs6 = Cg,r6 = [[70:84:105:120|12:10:8:7]]. Other than the s6 chord, all harmonic/subharmonic numbers must be odd, e.g. Ch6 and Ch8 are invalid. For any odd number N greater than 5, ChN is 1:3:5...N and CsN is 1/(N...5:3:1).  <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.
-<u>All wa chords can be named conventionally</u>, since wa is the default color. Thus w1-w3-w5 is both Cw and Cm. And w1-Lw3-w5-w6 is both CLw6 and C6. For aesthetic reasons, the conventional name is preferred only when neither "M" nor "m" appears in the name. This is especially true for non-wa chords: w1-w3-w5-y6 is Cw,y6 not Cm,y6.
+<u>All wa chords can be named conventionally</u>, since wa is the default color. Thus w1-w3-w5 is both Cw and Cm. And w1-Lw3-w5-w6 is both CLw6 and C6. For aesthetic reasons, the conventional name is preferred only when neither "M" nor "m" appears in the name (since color notation doesn't use major/minor). This is especially true if the chord includes non-wa notes: w1-w3-w5-y6 is Cw,y6 not Cm,y6.
 Chords can be classified as '''bicolored''' (e.g. g7 or r6), '''tricolored''' (e.g. z7(zg5) or z,y6), '''quadricolored''' (e.g. s6(zg5) or h7,zg9), etc.
@@ Line 497: / Line 497: @@
 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, scales 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 of A minor is A gu and the scale is the gu scale. The Bbh7 - Ebh7 - Bbh7 - Fh9 example in the staff notation section is in Bb yozo. Like chords, scales can be classified as bicolored (A gu), tricolored (Bb yo zo), etc.
-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.
+Scales can be named more precisely analogous to how chords are named. The tonic, 2nd, 4th and 5th default to wa. Thus a yo scale is w1 w2 y3 w4 w5 y6 y7 w8. If the 2nd were instead yo, it would be a yo yo-2 scale, written y(y2). If the 2nd is sometimes yo, sometimes wa, the scale is yo plus yo-2, written y+y2. (A hexatonic scale might use "minus".) The 5-limit harmonic minor scale is gu yo-7. The Bbh7 - Ebh7 - Bbh7 - Fh9 scale is Bb yo plus zo-3-4-7, written Bb y+z347.
+(Occasionally, the 6th or the 7th may be La or sa. For example, the wa scale has a wa 3rd, because the 3rd of the scale always matches the scale name exactly. The 6th and 7th default to a perfect 4th/5th from the 3rd, so the 6th is sa, not central. Thus the wa scale is minor, and the Lawa scale is major.)
+Just as there is a har7 chord, there is a har15 scale: w1 w2 y3 1o4 w5 3o6 z7 y7 w8. A har-N scale (where N is odd) is harmonics (N+1)/2 to N+1. The tonic of the scale is always a power of 2. Thus the har9 scale is not 5:6:7:8:9:10 but 8:9:10:12:14:16 = w1 w2 y3 w5 z7 w8. Likewise there are sub scales.
+Analogous to the relative and parallel major or minor, one can modulate to relative gu, parallel ru, etc. Modulating from a yo key to the relative gu means using gu chords on yo roots. Modulating from yo to the parallel gu means using gu chords on <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, and perhaps from there to the parallel yo is modulating '''yoward'''. A root movement by a yo interval (e.g. Iy - yVIg) is a yoward move. Likewise, there's '''guward''', and '''y<u>a</u>ward''' includes both. Likewise, there's '''zoward''', '''ruward''', '''zaward''', '''iloward''', etc.
 == Temperament names and comma names ==
@@ Line 506: / Line 512: @@
 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.
-Multi-comma temperaments have multiple commas in their name. [[Meantone family#Septimal meantone|Septimal Meantone]] is Gu & Ruyoyoti and [[Meantone family#Dominant|Dominant Meantone]] is Gu & Ruguti. 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 & 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]].
@@ Line 622: / Line 628: @@
 |}
 Higher primes: 29o = tweno, 31o = thiwo, 37o = thiso, 41o = fowo, 43o = fotho, 47o = foso, 53o = fitho, 59o = fino, 61o = siwo, 67o = siso.
+Higher exponents: 6 is tribi (triply-doubled), 8 is quadbi, 9 is tritri, 10 is quinbi, etc.
 <u>Pronunciation</u>: exponent syllables like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Bi'''ru'''yo, Bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''Sa'''sa-'''gu'''gu, '''Zo'''zotri'''gu'''.
@@ Line 629: / Line 637: @@
 !meaning
 !example
+|-
+| colspan="2" |central
+|refers to a ratio centrally located in the lattice
+|every ratio of odd limit < 81 is central (but only some > 81 are not central)
 |-
 |la-
@@ Line 639: / Line 651: @@
 |small, diminished by 2187/2048 from the central ratio
 |27/16 = wa 6th = w6, 128/81 = sawa 6th = sw6
+|-
+| colspan="2" |magnitude
+|refers to central, la, sa, lala, trisa, quadla, etc.
+|the sum of all prime exponents except the 1st, divided by 7 and rounded off
 |-
 | colspan="2" |i-
 | disambiguation prefix
-|no 3rd = omit the 3rd, ino 3rd = 19/16
+|no 3rd = omit the 3rd, but ino 3rd = 19/16
 |-
 | colspan="2" |  -a-
 | delimits an exponent such as bi-, tri-, etc.
-|Trizogu = z<sup>3</sup>g<sup>3</sup> = 1029/1000, Trizo-agu = z<sup>3</sup>g = 343/320
+|Trizogu = z<sup>3</sup>g<sup>3</sup> = 1029/1000, but Trizo-agu = z<sup>3</sup>g = 343/320
 |-
 |co-
@@ Line 681: / Line 697: @@
 | style="text-align:center" |#2
 |as a suffix, 2nd smallest comma in the row segment
-|Meantone = 81/80 = Guti = gT, Father = 16/15 = Gubiti = g#2T
+|Guti = gT is Meantone, but Gubiti = g#2T is [[Father]] (16/15 vanishes)
 |-
 | Wa-
 |w-
 |alternate interval format, only used for 3-limit commas
-|Mercator's comma = Wa-53 = w-53
+|[[Mercator's comma]] = Wa-53 = w-53
 |-
 | colspan="2" |nowa
 |remove 3 (wa) from the prime subgroup, i.e. no-threes
-|2.5.7 = yaza nowa, 2.5.7 with 50/49 = Biruyo nowa
+|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 with 245/243 = Zozoyo noca
+| 3.5.7 = yaza noca, 3.5.7 & 245/243 = Zozoyoti noca
 |-
 | colspan="2" |nowaca
@@ Line 726: / Line 742: @@
 * "bi" means both "doubled" as in biruyo and "2nd smallest" as in Layobi. The meaning is always clear from context.
-Temperaments use "virtual colors" represented with ^ v and / \:
+Temperaments use "virtual colors" represented with arrows ^ v and perhaps slashes / \
 {| class="wikitable"
-|+
 ! colspan="2" |word
 !meaning
@@ Line 739: / Line 754: @@
 |v
 |lowered by some comma
+|-
+| colspan="2" |arrow
+|refers collectively to both ups and downs
 |-
 |lift
@@ Line 747: / Line 765: @@
 |\
 |lowered by some other comma
+|-
+| colspan="2" |slash
+|refers collectively to both lifts and drops
 |-
 |plain
@@ Line 755: / Line 776: @@
 |~
 |for 2nds, 3rd, 6ths and 7ths, exactly halfway between major and minor
-for 4ths, halfway-augmented, and for 5ths, halfway-diminished
+a mid 4th is halfway-augmented, and a mid 5th is halfway-diminished
 |}