Kite's color notation: Difference between revisions

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Example: ratio = 63/40

*~~monzo~~ = [-3 2 -1 1>

* Monzo = {{vector| -3 2 -1 1 }}

*~~color~~ = zogu

* Color = zogu

*~~stepspan~~ = <7 11 16 20~~] dot [~~-3 2 -1 1> = -21 + 22 - 16 + 20 = 5 steps

* Stepspan = {{vmprod| 7 11 16 20 | -3 2 -1 1 }} = -21 + 22 - 16 + 20 = 5 steps

*~~degree~~ = 5 + 1 = a 6th

* Degree = 5 + 1 = a 6th

*~~magnitude~~ = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central

* Magnitude = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central

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

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

'''Converting a color name''': Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let C be the number of "co-" prefixes. Let the monzo be [a b c d e~~...>~~. The colors directly give you all the monzo entries except a and b. Let S' = the dot product of [0 0 c d e~~...>~~ with the pseudo-edomapping. Let M' = round ((2 (S - S') + c + d + e + ...) / 7). Then a = -3 (S - S') - 11 (M - M') + C and b = 2 (S - S') + 7 (M - M'). (Derivation [https://gist.github.com/m-yac/2236a03dd9fe89a992477fbcbc63746c here]) Convert the monzo to a ratio.

'''Converting a color name''': Let S be the stepspan of the interval, {{nowrap|S {{=}} degree − sign (degree)}}. Let M be the magnitude of the color name, with {{nowrap|L {{=}} 1|LL {{=}} 2}}, etc. Small is negative and central is zero. Let C be the number of "co-" prefixes. Let the monzo be {{vector| a b c d e … }}. The colors directly give you all the monzo entries except a and b. Let S' be the dot product of {{vector| 0 0 c d e … }} with the pseudo-edomapping. Let {{nowrap|M' {{=}} round((2(S − S') + c + d + e + ...) / 7)}}. Then {{nowrap|a {{=}} −3 (S – S') – 11 (M – M') + C}} and {{nowrap|b {{=}} 2 (S − S') + 7 (M − M')}}. (Derivation [https://gist.github.com/m-yac/2236a03dd9fe89a992477fbcbc63746c here]) Convert the monzo to a ratio.

Example: interval = sgg2 = sagugu 2nd

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

* S = 2 - 1 = 1 step, M = small = -1, C = 0. Monzo = {{vector| a b -2 }}

*S' = <7 11 16~~] dot [~~0 0 -2> = -32. S - S' = 1 - (-32) = 33.

* S' = {{vmprod| 7 11 16 | 0 0 -2 }} = -32. S - S' = 1 - (-32) = 33.

*M' = round ((2·33 + (-2)) / 7) = round (64 / 7) = 9. M - M' = -1 - 9 = -10.

* M' = round ((2·33 + (-2)) / 7) = round (64 / 7) = 9. M - M' = -1 - 9 = -10.

*a = -3 (S - S') - 11 (M - M') + C = -3·33 - 11·(-10) + 0 = -99 + 110 = 11.

* a = -3 (S - S') - 11 (M - M') + C = -3·33 - 11·(-10) + 0 = -99 + 110 = 11.

*b = 2 (S - S') + 7 (M - M') = 2·33 + 7·(-10) = 66 - 70 = -4

* b = 2 (S - S') + 7 (M - M') = 2·33 + 7·(-10) = 66 - 70 = -4

*Monzo = [11 -4 -2>, ratio = 2048/2025.

* Monzo = {{vector| 11 -4 -2 }}, ratio = 2048/2025.

== Staff notation ==

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

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

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

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

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== Chord names ==

Triads are named after their 3rd, e.g. a [[4:5:6|yo chord]] has a yo 3rd. A yo chord rooted on C is a Cy chord = "C yo" = C yE G. Qualities such as major and minor aren't used, because a chord with an 11/9 3rd is hard to classify. Thirdless dyads are written C5 = w1 w5 or C(zg5) = w1 zg5. The four main yaza triads:

Triads are named after their 3rd, e.g. a [[4:5:6|yo chord]] has a yo 3rd. A yo chord rooted on C is a Cy chord {{nowrap|{{=}} "C yo"}} {{nowrap|{{=}} {{dash|C yE G}}}}. Qualities such as major and minor aren't used, because a chord with an 11/9 3rd is hard to classify. Thirdless dyads are written {{nowrapC5 {{=}} w1 w5}} or {{nowrap|C(zg5) {{=}} w1 zg5}}. The four main yaza triads:

[[File:lattice62.png|alt=lattice62.png|640x138px|lattice62.png]]

Tetrads are named e.g. "C yo-six" = [[12:15:18:20|Cy6]] = C yE G yA. The 11 main yaza tetrads, with chord homonyms (same shape, different root) equated:

Tetrads are named e.g. {{nowrap|"C yo-six" {{=}} [[12:15:18:20|Cy6]]}} {{nowrap|{{=}} C yE G yA}}. The 11 main yaza tetrads, with chord homonyms (same shape, different root) equated:

[[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. 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:

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 {{nowrap|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]]

Alterations are always in parentheses, additions never are, e.g. z7(zg5) and z,y6. An alteration's degree must match a note in the chord, e.g. Cz7(y6) is invalid. But an exception is made for sus chords, where degree 2 or 4 alter the 3rd. The sus note defaults to wa. A [[6:8:9|6:8:9 chord]] could be written C(4), but the parentheses rule is relaxed to allow the conventional C4. Likewise [[8:9:12]] is C2. But if the sus note isn't wa, parentheses must be used. Thus w1 z4 w5 = C(z4) = "C zo-four". More examples:

Alterations are always in parentheses, additions never are, e.g. z7(zg5) and z,y6. An alteration's degree must match a note in the chord, e.g. Cz7(y6) is invalid. But an exception is made for sus chords, where degree 2 or 4 alter the 3rd. The sus note defaults to wa. A [[6:8:9|6:8:9 chord]] could be written C(4), but the parentheses rule is relaxed to allow the conventional C4. Likewise [[8:9:12]] is C2. But if the sus note isn't wa, parentheses must be used. Thus {{nowrap|w1 z4 w5 {{=}} C(z4)}} {{nowrap|{{=}} "C zo-four"}}. More examples:

* [[6:7:8:9]] = Cz,4 = "C zo add-four"

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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. {{nowrap|[[4:5:6:7:9|Ch9]] {{=}} Cy,z7,9}} and {{nowrap|[[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. {{nowrap|[[140:180:210:252:315|Cs9]] {{=}} Cr,g7,9}} and {{nowrap|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 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.

{{nowrap|Cs6 {{=}} Cg,r6}} {{nowrap|{{=}} [[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 N...5:3:1. Additions, alterations and omissions refer to degrees, not harmonics or subharmonics: Ch7,11 adds w11, not 1o11. Ch9no5 omits w5, not y3. However, all numbers > 13 refer to (sub)harmonics, e.g. Ch9,15 adds y7 and Ch19no15 omits it.

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

All wa chords can be named conventionally, since wa is the default color. Thus {{dash|w1, w3, w5}} is both Cw and Cm. And {{dash|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: {{dash|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 453: / Line 453: @@
 Example: ratio = 63/40
-*monzo = [-3 2 -1 1>
+* Monzo = {{vector| -3 2 -1 1 }}
-*color = zogu
+* Color = zogu
-*stepspan = <7 11 16 20] dot [-3 2 -1 1> = -21 + 22 - 16 + 20 = 5 steps
+* Stepspan = {{vmprod| 7 11 16 20 | -3 2 -1 1 }} = -21 + 22 - 16 + 20 = 5 steps
-*degree = 5 + 1 = a 6th
+* Degree = 5 + 1 = a 6th
-*magnitude = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central
+* Magnitude = round [(2 + (-1) + 1) / 7] = round (2/7) = 0 = central
-*interval = zogu 6th or zg6 (63/20 would be zg13 = czg6)
+* Interval = zogu 6th or zg6 (63/20 would be zg13 = czg6)
-<u>'''Converting a color name'''</u>: Let S be the stepspan of the interval, S = degree - sign (degree). Let M be the magnitude of the color name, with L = 1, LL = 2, etc. Small is negative and central is zero. Let C be the number of "co-" prefixes. Let the monzo be [a b c d e...>. The colors directly give you all the monzo entries except a and b. Let S' = the dot product of [0 0 c d e...> with the pseudo-edomapping. Let M' = round ((2 (S - S') + c + d + e + ...) / 7). Then a = -3 (S - S') - 11 (M - M') + C and b = 2 (S - S') + 7 (M - M'). (Derivation [https://gist.github.com/m-yac/2236a03dd9fe89a992477fbcbc63746c here]) Convert the monzo to a ratio.
+<u>'''Converting a color name'''</u>: Let S be the stepspan of the interval, {{nowrap|S {{=}} degree − sign (degree)}}. Let M be the magnitude of the color name, with {{nowrap|L {{=}} 1|LL {{=}} 2}}, etc. Small is negative and central is zero. Let C be the number of "co-" prefixes. Let the monzo be {{vector| a b c d e … }}. The colors directly give you all the monzo entries except a and b. Let S' be the dot product of {{vector| 0 0 c d e … }} with the pseudo-edomapping. Let {{nowrap|M' {{=}} round((2(S − S') + c + d + e + ...) / 7)}}. Then {{nowrap|a {{=}} −3 (S – S') – 11 (M – M') + C}} and {{nowrap|b {{=}} 2 (S − S') + 7 (M − M')}}. (Derivation [https://gist.github.com/m-yac/2236a03dd9fe89a992477fbcbc63746c here]) Convert the monzo to a ratio.
 Example: interval = sgg2 = sagugu 2nd
-*S = 2 - 1 = 1 step, M = small = -1, C = 0. Monzo = [a b -2>
+* S = 2 - 1 = 1 step, M = small = -1, C = 0. Monzo = {{vector| a b -2 }}
-*S' = <7 11 16] dot [0 0 -2> = -32. S - S' = 1 - (-32) = 33.
+* S' = {{vmprod| 7 11 16 | 0 0 -2 }} = -32. S - S' = 1 - (-32) = 33.
-*M' = round ((2·33 + (-2)) / 7) = round (64 / 7) = 9. M - M' = -1 - 9 = -10.
+* M' = round ((2·33 + (-2)) / 7) = round (64 / 7) = 9. M - M' = -1 - 9 = -10.
-*a = -3 (S - S') - 11 (M - M') + C = -3·33 - 11·(-10) + 0 = -99 + 110 = 11.
+* a = -3 (S - S') - 11 (M - M') + C = -3·33 - 11·(-10) + 0 = -99 + 110 = 11.
-*b = 2 (S - S') + 7 (M - M') = 2·33 + 7·(-10) = 66 - 70 = -4
+* b = 2 (S - S') + 7 (M - M') = 2·33 + 7·(-10) = 66 - 70 = -4
-*Monzo = [11 -4 -2>, ratio = 2048/2025.
+* Monzo = {{vector| 11 -4 -2 }}, ratio = 2048/2025.
 == Staff notation ==
@@ Line 488: / Line 488: @@
 '''Po''' and '''qu''' ("coo") (short forms '''p''' and '''q''') are two optional accidentals that indicate raising/lowering by a pythagorean comma. (Mnemonics: p stands for pythagorean, and q is the mirror image of p.) Why would one want to do that? Because by first subtracting that comma and then adding it on again, one can rename a note as another note. This is similar to [[Sagittal notation | Sagittal]] notation (see [http://tallkite.com/misc_files/Sagittal-JI-Translated-To-Colors.png Sagittal-JI-Translated-To-Colors.png]).
-For example, F# minus a pythagorean comma is Gb. And Gb plus a pythagorean comma is po Gb. Thus an alternate name for F# is po Gb. <u>Adding po raises the degree by one</u>. The new note name is always a 12edo equivalent of the old note name. Adding qu lowers the degree: Gb = qu F#. If one is resolving from Gb to G, one can rename Gb as qF#.
+For example, F# minus a pythagorean comma is Gb. And Gb plus a pythagorean comma is po Gb. Thus an alternate name for F# is po Gb. <u>Adding po raises the degree by one</u>. The new note name is always a 12edo equivalent of the old note name. Adding qu lowers the degree: {{nowrap|Gb {{=}} qu F#}}. If one is resolving from Gb to G, one can rename Gb as qF#.
 <u>Subtracting po lowers the degree</u>. Thus ruyopo Db = ruyo C#.
@@ Line 497: / Line 497: @@
 == Chord names ==
-Triads are named after their 3rd, e.g. a [[4:5:6|yo chord]] has a yo 3rd. A yo chord rooted on C is a Cy chord = "C yo" = C yE G. Qualities such as major and minor aren't used, because a chord with an 11/9 3rd is hard to classify. Thirdless dyads are written C5 = w1 w5 or C(zg5) = w1 zg5. The four main yaza triads:
+Triads are named after their 3rd, e.g. a [[4:5:6|yo chord]] has a yo 3rd. A yo chord rooted on C is a Cy chord {{nowrap|{{=}} "C yo"}} {{nowrap|{{=}} {{dash|C yE G}}}}. Qualities such as major and minor aren't used, because a chord with an 11/9 3rd is hard to classify. Thirdless dyads are written {{nowrapC5 {{=}} w1 w5}} or {{nowrap|C(zg5) {{=}} w1 zg5}}. The four main yaza triads:
 [[File:lattice62.png|alt=lattice62.png|640x138px|lattice62.png]]
-Tetrads are named e.g. "C yo-six" = [[12:15:18:20|Cy6]] = C yE G yA. The 11 main yaza tetrads, with chord homonyms (same shape, different root) equated:
+Tetrads are named e.g. {{nowrap|"C yo-six" {{=}} [[12:15:18:20|Cy6]]}} {{nowrap|{{=}} C yE G yA}}. The 11 main yaza tetrads, with chord homonyms (same shape, different root) equated:
 [[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. 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:
+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 {{nowrap|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]]
-<u>Alterations are always in parentheses</u>, additions never are, e.g. z7(zg5) and z,y6. An alteration's degree must match a note in the chord, e.g. Cz7(y6) is invalid. But an exception is made for sus chords, where degree 2 or 4 alter the 3rd. The sus note defaults to wa. A [[6:8:9|6:8:9 chord]] could be written C(4), but the parentheses rule is relaxed to allow the conventional C4. Likewise [[8:9:12]] is C2. But if the sus note isn't wa, parentheses must be used. Thus w1 z4 w5 = C(z4) = "C zo-four". More examples:
+<u>Alterations are always in parentheses</u>, additions never are, e.g. z7(zg5) and z,y6. An alteration's degree must match a note in the chord, e.g. Cz7(y6) is invalid. But an exception is made for sus chords, where degree 2 or 4 alter the 3rd. The sus note defaults to wa. A [[6:8:9|6:8:9 chord]] could be written C(4), but the parentheses rule is relaxed to allow the conventional C4. Likewise [[8:9:12]] is C2. But if the sus note isn't wa, parentheses must be used. Thus {{nowrap|w1 z4 w5 {{=}} C(z4)}} {{nowrap|{{=}} "C zo-four"}}. More examples:
 * [[6:7:8:9]] = Cz,4 = "C zo add-four"
@@ Line 517: / Line 517: @@
 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. {{nowrap|[[4:5:6:7:9|Ch9]] {{=}} Cy,z7,9}} and {{nowrap|[[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. {{nowrap|[[140:180:210:252:315|Cs9]] {{=}} Cr,g7,9}} and {{nowrap|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 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.
+{{nowrap|Cs6 {{=}} Cg,r6}} {{nowrap|{{=}} [[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 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 >&nbsp;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 (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.
+<u>All wa chords can be named conventionally</u>, since wa is the default color. Thus {{dash|w1, w3, w5}} is both Cw and Cm. And {{dash|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: {{dash|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.