Kite's ups and downs notation: Difference between revisions

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Sharp-1, flat-2, etc. refer to the number of EDOsteps made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.

A ring is a circle of 5ths. In multi-ring edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation ~~(no ups and downs),~~ if notes are permitted to be out of order (e.g. 22edo could have C Db B# C#). But multi-ring edos absolutely require ups and downs.

A ring is a circle of 5ths. In multi-ring edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C#). But multi-ring edos absolutely require ups and downs.

12-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2.

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| P8

|}

18b-edo contains 2 rings of 9-edo: an up/down-ring and a plain-ring. There are two ways to notate it.

18b-edo contains 2 rings of 9-edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic intervals: either ^^A1 and vvM2, or vvA1 and vvm2.

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

| rowspan="4" |'''[[18-edo|18b-edo]]'''

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| m2

| ~2

| ~~mM2~~/m3

| M2/m3

| ~3

| M3

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| etc.

|}

23-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it.

23-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A32 or d32.

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

| rowspan="4" |'''[[23-edo]]'''

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The major chord and various alterations of it:

* C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")

C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")

* C ^E G = C^ = "C up" or "C upmajor"

* C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")

C ^E G = C^ = "C up" or "C upmajor"

* C vvE G = Cvv = "C double-down" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 25, 32, 39, 46, etc. C^~ = "C upmid")

C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")

C vvE G = Cvv = "C double-down" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 25, 32, 39, 46, etc. C^~ = "C upmid")

This table shows how altering the 3rd or the 5th affects the name of the triad (+ and o are conventional abbreviations for aug and dim).

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

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|}

Many EDOs have notes between the major 3rd and the perfect 4th, creating triads impossible in 12-edo, such as:

* C Fb G = C(d4) or C(b4) = "C dim-four" or "C flat-four"

C Fb G = C(d4) or C(b4) = "C dim-four" or "C flat-four"

* C E# G = C(A3) or C(#3) = "C aug-three" or "C sharp-three"

* C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C double-flat-three"

C E# G = C(A3) or C(#3) = "C aug-three" or "C sharp-three"

* C D# G = C(A2) or C(#2) = "C aug-two" or "C sharp-two"

C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C double-flat-three"

C D# G = C(A2) or C(#2) = "C aug-two" or "C sharp-two"

For each chord, care must be taken in pronouncing the latter of the two names, that C(#2) doesn't sound like C#2, which is C# D# G#.

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If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th.

* C E G Bb = C7 = "C seven"

C E G Bb = C7 = "C seven"

* C vE G Bb = Cv,7 = "C down add-seven"

* C E G vBb = C,v7 = "C add down-seven"

C vE G Bb = Cv,7 = "C down add-seven"

* C vE G vBb = Cv7 = "C down seven"

C E G vBb = C,v7 = "C add down-seven"

C vE G vBb = Cv7 = "C down seven"

All 7th chords follow this same pattern. Likewise, if a 6th is not a P4 or A4 above the 3rd, it's an "add-6" chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can be replaced with "dim add-7", written Co,7.

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|}

Various unusual tetrads:

* C vE G ^Bb = Cv,^7 = "C down up-seven" (in EDOs 17, 24, 31, etc. C~7 = "C mid-seven")

C vE G ^Bb = Cv,^7 = "C down up-seven" (in EDOs 17, 24, 31, etc. C~7 = "C mid-seven")

* C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"

* C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"

C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"

* C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)

* C E G B# is C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"

C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"

* C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"

C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)

C E G B# is C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"

C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"

'''Ninth chords:'''

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In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yo. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is downmajor, it doesn't work in that progression.

Another example: I7 - bVII7 - IV7 - I7. To ~~make~~ this ~~work~~, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under)~~. If you want to play this progression in 19-edo, using the downminor (zo) 7th will cause shifts or drifts in the progression~~.

Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under).

==Ups and downs solfege==

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This suffices for many but not all EDOs, as some require triple sharps or quadruple ups.

Fixed-do solfege:

Fixed-do solfege examples:

* Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb

Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb

* Da = C, Dae = ^C, Dai = ^^C, Dao = vC, Dau = vvC, Daa = ^^^C

* De = C#, Dee = ^C#, Dei = ^^C#, Deo = vC#, Deu = vvC#, Dea = vvvC#

Da = C, Dae = ^C, Dai = ^^C, Dao = vC, Dau = vvC, Daa = ^^^C

De = C#, Dee = ^C#, Dei = ^^C#, Deo = vC#, Deu = vvC#, Dea = vvvC#

~~etc.~~

Moveable-do solfege:

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Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim

* Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1

Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1

* Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1

Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1

~~etc.~~

Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim

* Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2

Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2

* Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2

Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2

~~etc.~~

[[Category:notation]]

@@ Line 77: / Line 77: @@
 Sharp-1, flat-2, etc. refer to the number of EDOsteps made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.
-A ring is a circle of 5ths. In multi-ring edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation (no ups and downs), if notes are permitted to be out of order (e.g. 22edo could have C Db B# C#). But multi-ring edos absolutely require ups and downs.
+A ring is a circle of 5ths. In multi-ring edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C#). But multi-ring edos absolutely require ups and downs.
 -edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2.
@@ Line 376: / Line 376: @@
 |  P8
 |}
-b-edo contains 2 rings of 9-edo: an up/down-ring and a plain-ring. There are two ways to notate it.
+b-edo contains 2 rings of 9-edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic intervals: either ^^A1 and vvM2, or vvA1 and vvm2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="4" |'''[[18-edo|18b-edo]]'''
@@ Line 448: / Line 448: @@
 |  m2
 | ~2
-| mM2/m3
+| M2/m3
 |  ~3
 | M3
@@ Line 644: / Line 644: @@
 | etc.
 |}
--edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it.
+-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A<sup>3</sup>2 or d<sup>3</sup>2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="4" |'''[[23-edo]]'''
@@ Line 806: / Line 806: @@
 <span style="display: block; text-align: left;">The major chord and various alterations of it:</span>
+* C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")
-C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")
+* C ^E G = C^ = "C up" or "C upmajor"
+* C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")
-C ^E G = C^ = "C up" or "C upmajor"
+* C vvE G = Cvv = "C double-down" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 25, 32, 39, 46, etc. C^~ = "C upmid")
-C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")
-C vvE G = Cvv = "C double-down" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 25, 32, 39, 46, etc. C^~ = "C upmid")
 This table shows how altering the 3rd or the 5th affects the name of the triad (+ and o are conventional abbreviations for aug and dim).
 {| class="wikitable" style="text-align:center;"
@@ Line 875: / Line 870: @@
 |}
 Many EDOs have notes between the major 3rd and the perfect 4th, creating triads impossible in 12-edo, such as:
+* C Fb G = C(d4) or C(b4) = "C dim-four" or "C flat-four"
-C Fb G = C(d4) or C(b4) = "C dim-four" or "C flat-four"
+* C E# G = C(A3) or C(#3) = "C aug-three" or "C sharp-three"
+* C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C double-flat-three"
-C E# G = C(A3) or C(#3) = "C aug-three" or "C sharp-three"
+* C D# G = C(A2) or C(#2) =  "C aug-two" or "C sharp-two"
-C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C double-flat-three"
-C D# G = C(A2) or C(#2) =  "C aug-two" or "C sharp-two"
 For each chord, care must be taken in pronouncing the latter of the two names, that C(#2) doesn't sound like C#2, which is C# D# G#.
@@ Line 889: / Line 879: @@
 If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th.
+* C E G Bb = C7 = "C seven"
-C E G Bb = C7 = "C seven"
+* C vE G Bb = Cv,7 = "C down add-seven"
+* C E G vBb = C,v7 = "C add down-seven"
-C vE G Bb = Cv,7 = "C down add-seven"
+* C vE G vBb = Cv7 = "C down seven"
-C E G vBb = C,v7 = "C add down-seven"
-C vE G vBb = Cv7 = "C down seven"
 All 7th chords follow this same pattern. Likewise, if a 6th is not a P4 or A4 above the 3rd, it's an "add-6" chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can be replaced with "dim add-7", written Co,7.
@@ Line 1,018: / Line 1,003: @@
 |}
 Various unusual tetrads:
+* C vE G ^Bb = Cv,^7 = "C down up-seven" (in EDOs 17, 24, 31, etc. C~7 = "C mid-seven")
-C vE G ^Bb = Cv,^7 = "C down up-seven" (in EDOs 17, 24, 31, etc. C~7 = "C mid-seven")
+* C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"
+* C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"
-C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"
+* C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)
+* C E G B# is C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"
-C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"
+* C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"
-C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)
-C E G B# is C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"
-C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"
 <u>'''Ninth chords:'''</u>
@@ Line 1,219: / Line 1,197: @@
 In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yo. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is <u>down</u>major, it doesn't work in that progression.
-Another example: I7 - bVII7 - IV7 - I7. To make this work, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under). If you want to play this progression in 19-edo, using the downminor (zo) 7th will cause shifts or drifts in the progression.
+Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under).
 ==Ups and downs solfege==
@@ Line 1,239: / Line 1,217: @@
 This suffices for many but not all EDOs, as some require triple sharps or quadruple ups.
-Fixed-do solfege:
+Fixed-do solfege examples:
+* Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
-Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
+* Da = C, Dae = ^C, Dai = ^^C, Dao = vC, Dau = vvC, Daa = ^^^C
+* De = C#, Dee = ^C#, Dei = ^^C#, Deo = vC#, Deu = vvC#, Dea = vvvC#
-Da = C, Dae = ^C, Dai = ^^C, Dao = vC, Dau = vvC, Daa = ^^^C
-De = C#, Dee = ^C#, Dei = ^^C#, Deo = vC#, Deu = vvC#, Dea = vvvC#
-etc.
 Moveable-do solfege:
@@ Line 1,254: / Line 1,226: @@
 Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
+* Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
-Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
+* Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1
-Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1
-etc.
 Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim
+* Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2
-Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2
+* Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2
-Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2
-etc.
 [[Category:notation]]