Kite's ups and downs notation: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 16:38:54 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 17:44:54 UTC</tt>.

: The original revision id was <tt>~~594507750~~</tt>.

: The original revision id was <tt>594511442</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 144:

||= A minor chord ||= A major ||= A C# E ||= A Cb E ||

||= Eb major chord ||= E# minor ||= E# G# B# ||= Eb Gb Db ||

||= G7 = G + M3 + P5 + m7 ||= G + m3 + P5 + M7 ||= G Bb D F# ||= G B# D Fb ||

||= Gm7 = G + m3 + P5 + m7 ||= G + M3 + P5 + M7 ||= G B D F# ||= G B D Fb ||

||= ~~Ab aug~~ = Ab + M3 + A5 ||= A# + m3 + d5 ||= A# C# E ||= Ab Cb E ||

||= Ab7aug = Ab + M3 + A5 + m7 ||= A# + m3 + d5 + M7 ||= A# C# E G## ||= Ab Cb E Gbb ||

||= C major scale = C + M2 + M3

+ P4 + P5 + M6 + M7 + P8 ||= C + m2 + m3 + P4

Line 151:

G Ab Bb C ||= C D# E# F

G A# B# C ||

||= C# minor scale = C# + M2 + m3

||= C minor scale = C + M2 + m3

+ P4 + P5 + m6 + m7 + P8 ||= Cb + m2 + M3 + P4

+ P4 + P5 + m6 + m7 + P8 ||= C + m2 + M3 + P4

+ P5 + M6 + M7 + P8 ||= ~~Cb Dbb Eb Fb~~

+ P5 + M6 + M7 + P8 ||= C Db E F

~~Gb Ab Bb Cb~~ ||= C# D## E# F#

G A B C ||= C D# E F

G# A# B# C# ||

G A B C ||

Both approaches have their merit, but the first one will be used here.

Line 981:

Natural generators can be used for other EDOs as well. For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.

__**Chroma-2 ~~and chroma-5~~ edos:**__

__**Chroma-2 edos:**__

genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - d6 etc.

genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.

Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C

Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#

"Every Good Boy Deserves Fudge And Candy"

Line 990:

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

__**17-edo**__: (generator = 3\17 = perfect 3rd)

__**17-edo**__: (generator = 5\17 = perfect 3rd)

D * E * * F * G * * A * B * * C * D

P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8

__**24-edo**__: (generator = 7\24 = perfect 3rd)

D * * E * * * F * * G * * * A * * B * * * C * * D

P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4/d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8

__**31-edo**__: (generator = 9\31 = perfect 3rd)

D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D

P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...

etc.

__**Chroma-5 edos:**__

genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.

E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb

"Every Good Boy Deserves Fudge And Candy"

__**18-edo**__: see above

__**17-edo**__: (generator = 3\17 = perfect 3rd)

__**25-edo**__: (generator = 7\25 = perfect 3rd)

D * E * * F * G * * A * B * * C * D

D * * * E * * F * * * G * * A * * * B * * C * * * D

P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8

P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8

etc.

__**Chroma-3 edos**__**:**

genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.

D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.

**__15-edo__:** (generator = 2\15 = perfect 2nd)

D * E * F * G * * A * B * C * D

P1 - A1/d2 - P2 - A2/d3 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - A6/d7 - P7 - A7/d8 - P8

**__22-edo__:** (generator = 3\22 = perfect 2nd)

D * * E * * F * * G * * * A * * B * * C * * D

P1 - A1 - d2 - P2 - A2 - d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6 - d7 - P7 - A7 - d8 - P8

etc.

~~For 22~~-~~edo, it's a 3\22 2nd~~

__**Chroma-4 edos**__**:**

genchain: .~~..D# E# F# G#~~ A B C D E F G ~~Ab Bb Cb Db...~~

genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.

scale: D * * E * * F * * G * * * A ~~* *~~ B ~~* *~~ C ~~* *~~ D

Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.

**__20-edo__:** (generator = 3\20 = perfect 2nd)

D * * E * * F * * G * A * * B * * C * * D

P1 - A1 - d2 - P2 - A2/d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6/d7 - P7 - A7 - d8 - P8

~~For 17~~-~~edo, the natural~~ generator ~~is the 5~~\~~17 3rd~~.

**__27-edo__:** (generator = 4\27 = perfect 2nd)

D * * * E * * * F * * * G * * A * * * B * * * C * * * D

P1 - A1 - AA1/dd2 - d2 - P2 - A2 - d3 - m3 - M3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 etc.

~~For 15-edo, it's the 2\15 2nd.~~

etc.

~~For 12-edo, it's the 7\12 5th, which generates standard notation.~~

~~For 16-edo, it's the 9\16 5th.~~

~~For 21-edo, it's the 3\21 2nd, the 6\21 3rd, the 9\21 4th, the 12\21 5th,~~ etc~~. All these intervals generate the same scale, and the same notation~~.

~~You can easily find the natural generator using the scale tree. It's also related to the chroma. But it doesn't intrinsically depend on any ratios or commas, or on the prime limit.~~

=__Ups and downs solfege__=

Line 1,596:

Line 1,626:

</tr>

<tr>

<td style="text-align: center;">G7 = G + M3 + P5 + m7

<td style="text-align: center;">Gm7 = G + m3 + P5 + m7

</td>

<td style="text-align: center;">G + m3 + P5 + M7

<td style="text-align: center;">G + M3 + P5 + M7

</td>

<td style="text-align: center;">G Bb D F#

<td style="text-align: center;">G B D F#

</td>

<td style="text-align: center;">G B# D Fb

<td style="text-align: center;">G B D Fb

</td>

</tr>

<tr>

<td style="text-align: center;">~~Ab aug~~ = Ab + M3 + A5

<td style="text-align: center;">Ab7aug = Ab + M3 + A5 + m7

</td>

<td style="text-align: center;">A# + m3 + d5

<td style="text-align: center;">A# + m3 + d5 + M7

</td>

<td style="text-align: center;">A# C# E

<td style="text-align: center;">A# C# E G##

</td>

<td style="text-align: center;">Ab Cb E

<td style="text-align: center;">Ab Cb E Gbb

</td>

</tr>

Line 1,630:

Line 1,660:

</tr>

<tr>

<td style="text-align: center;">C# minor scale = C# + M2 + m3

<td style="text-align: center;">C minor scale = C + M2 + m3

+ P4 + P5 + m6 + m7 + P8

</td>

<td style="text-align: center;">Cb + m2 + M3 + P4

<td style="text-align: center;">C + m2 + M3 + P4

+ P5 + M6 + M7 + P8

</td>

<td style="text-align: center;">~~Cb Dbb Eb Fb~~

<td style="text-align: center;">C Db E F

~~Gb Ab Bb Cb~~

G A B C

</td>

<td style="text-align: center;">C# D## E# F#

<td style="text-align: center;">C D# E F

G# A# B# C#

G A B C

</td>

</tr>

Line 3,798:

Line 3,828:

Natural generators can be used for other EDOs as well. For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.

Chroma-2 ~~and chroma-5~~ edos:

Chroma-2 edos:

genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - d6 etc.

genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.

Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C

Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#

&quot;Every Good Boy Deserves Fudge And Candy&quot;

Line 3,807:

Line 3,837:

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

17-edo: (generator = 3\17 = perfect 3rd)

17-edo: (generator = 5\17 = perfect 3rd)

D * E * * F * G * * A * B * * C * D

P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8

24-edo: (generator = 7\24 = perfect 3rd)

D * * E * * * F * * G * * * A * * B * * * C * * D

P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4/d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8

31-edo: (generator = 9\31 = perfect 3rd)

D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D

P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...

etc.

Chroma-5 edos:

genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.

E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb

&quot;Every Good Boy Deserves Fudge And Candy&quot;

18-edo: see above

17-edo: (generator = 3\17 = perfect 3rd)

25-edo: (generator = 7\25 = perfect 3rd)

D * E * * F * G * * A * B * * C * D

D * * * E * * F * * * G * * A * * * B * * C * * * D

P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8

P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8

etc.

Chroma-3 edos:

genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.

D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.

15-edo: (generator = 2\15 = perfect 2nd)

D * E * F * G * * A * B * C * D

P1 - A1/d2 - P2 - A2/d3 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - A6/d7 - P7 - A7/d8 - P8

22-edo: (generator = 3\22 = perfect 2nd)

D * * E * * F * * G * * * A * * B * * C * * D

P1 - A1 - d2 - P2 - A2 - d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6 - d7 - P7 - A7 - d8 - P8

etc.

~~For 22~~-~~edo, it's a 3\22 2nd~~

Chroma-4 edos:

genchain: ~~...D# E# F# G# A B C D E F G Ab Bb Cb Db..~~.

genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.

~~scale:~~ D ~~* *~~ E ~~* *~~ F ~~* *~~ G ~~* * *~~ A ~~* *~~ B ~~* *~~ C ~~* *~~ D

Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.

20-edo: (generator = 3\20 = perfect 2nd)

D * * E * * F * * G * A * * B * * C * * D

P1 - A1 - d2 - P2 - A2/d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6/d7 - P7 - A7 - d8 - P8

~~For 17~~-edo~~, the natural~~ generator ~~is the 5~~\~~17 3rd~~.

27-edo: (generator = 4\27 = perfect 2nd)

D * * * E * * * F * * * G * * A * * * B * * * C * * * D

P1 - A1 - AA1/dd2 - d2 - P2 - A2 - d3 - m3 - M3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 etc.

~~For 15-edo, it's the 2\15 2nd. ~~

etc.

~~For 12-edo, it's the 7\12 5th, which generates standard notation. ~~

~~For 16-edo, it's the 9\16 5th. ~~

~~For 21-edo, it's the 3\21 2nd, the 6\21 3rd, the 9\21 4th, the 12\21 5th,~~ etc~~. All these intervals generate the same scale, and the same notation~~.

~~You can easily find the natural generator using the scale tree. It's also related to the chroma. But it doesn't intrinsically depend on any ratios or commas, or on the prime limit. ~~

<h1 id="toc14"><a name="Ups and downs solfege"></a>Ups and downs solfege</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 16:38:54 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 17:44:54 UTC</tt>.<br>
-: The original revision id was <tt>594507750</tt>.<br>
+: The original revision id was <tt>594511442</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 144: / Line 144: @@
 ||= A minor chord ||= A major ||= A C# E ||= A Cb E ||
 ||= Eb major chord ||= E# minor ||= E# G# B# ||= Eb Gb Db ||
-||= G7 = G + M3 + P5 + m7 ||= G + m3 + P5 + M7 ||= G Bb D F# ||= G B# D Fb ||
+||= Gm7 = G + m3 + P5 + m7 ||= G + M3 + P5 + M7 ||= G B D F# ||= G B D Fb ||
-||= Ab aug = Ab + M3 + A5 ||= A# + m3 + d5 ||= A# C# E ||= Ab Cb E ||
+||= Ab7aug = Ab + M3 + A5 + m7 ||= A# + m3 + d5 + M7 ||= A# C# E G## ||= Ab Cb E Gbb ||
 ||= C major scale = C + M2 + M3
 + P4 + P5 + M6 + M7 + P8 ||= C + m2 + m3 + P4
@@ Line 151: / Line 151: @@
 G Ab Bb C ||= C D# E# F
 G A# B# C ||
-||= C# minor scale = C# + M2 + m3
+||= C minor scale = C + M2 + m3
-+ P4 + P5 + m6 + m7 + P8 ||= Cb + m2 + M3 + P4
++ P4 + P5 + m6 + m7 + P8 ||= C + m2 + M3 + P4
-+ P5 + M6 + M7 + P8 ||= Cb Dbb Eb Fb
++ P5 + M6 + M7 + P8 ||= C Db E F
-Gb Ab Bb Cb ||= C# D## E# F#
+G A B C ||= C D# E F
-G# A# B# C# ||
+G A B C ||
 Both approaches have their merit, but the first one will be used here.
@@ Line 981: / Line 981: @@
 Natural generators can be used for other EDOs as well. For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.
-__**Chroma-2 and chroma-5 edos:**__
+__**Chroma-2 edos:**__
-genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - d6 etc.
+genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.
-Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C
+Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#
 "Every Good Boy Deserves Fudge And Candy"
@@ Line 990: / Line 990: @@
 P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8
-__**17-edo**__: (generator = 3\17 = perfect 3rd)
+__**17-edo**__: (generator = 5\17 = perfect 3rd)
 D * E * * F * G * * A * B * * C * D
 P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
+__**24-edo**__: (generator = 7\24 = perfect 3rd)
+D * * E * * * F * * G * * * A * * B * * * C * * D
+P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4/d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8
+__**31-edo**__: (generator = 9\31 = perfect 3rd)
+D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D
+P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...
+etc.
+__**Chroma-5 edos:**__
+genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.
+E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb
+"Every Good Boy Deserves Fudge And Candy"
 __**18-edo**__: see above
-__**17-edo**__: (generator = 3\17 = perfect 3rd)
+__**25-edo**__: (generator = 7\25 = perfect 3rd)
-D * E * * F * G * * A * B * * C * D
+D * * * E * * F * * * G * * A * * * B * * C * * * D
-P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
+P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8
+etc.
+__**Chroma-3 edos**__**:**
+genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.
+D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.
+**__15-edo__:** (generator = 2\15 = perfect 2nd)
+D * E * F * G * * A * B * C * D
+P1 - A1/d2 - P2 - A2/d3 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - A6/d7 - P7 - A7/d8 - P8
+**__22-edo__:** (generator = 3\22 = perfect 2nd)
+D * * E * * F * * G * * * A * * B * * C * * D
+P1 - A1 - d2 - P2 - A2 - d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6 - d7 - P7 - A7 - d8 - P8
+etc.
-For 22-edo, it's a 3\22 2nd
+__**Chroma-4 edos**__**:**
-genchain: ...D# E# F# G# A B C D E F G Ab Bb Cb Db...
+genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.
-scale: D * * E * * F * * G * * * A * * B * * C * * D
+Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.
+**__20-edo__:** (generator = 3\20 = perfect 2nd)
+D * * E * * F * * G * A * * B * * C * * D
+P1 - A1 - d2 - P2 - A2/d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6/d7 - P7 - A7 - d8 - P8
-For 17-edo, the natural generator is the 5\17 3rd.
+**__27-edo__:** (generator = 4\27 = perfect 2nd)
+D * * * E * * * F * * * G * * A * * * B * * * C * * * D
+P1 - A1 - AA1/dd2 - d2 - P2 - A2 - d3 - m3 - M3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 etc.
-For 15-edo, it's the 2\15 2nd.
+etc.
-For 12-edo, it's the 7\12 5th, which generates standard notation.
-For 16-edo, it's the 9\16 5th.
-For 21-edo, it's the 3\21 2nd, the 6\21 3rd, the 9\21 4th, the 12\21 5th, etc. All these intervals generate the same scale, and the same notation.
-You can easily find the natural generator using the scale tree. It's also related to the chroma. But it doesn't intrinsically depend on any ratios or commas, or on the prime limit.
 =__Ups and downs solfege__=
@@ Line 1,596: / Line 1,626: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;G7 = G + M3 + P5 + m7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Gm7 = G + m3 + P5 + m7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;G + m3 + P5 + M7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;G + M3 + P5 + M7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;G Bb D F#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;G B D F#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;G B# D Fb&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;G B D Fb&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;Ab aug = Ab + M3 + A5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ab7aug = Ab + M3 + A5 + m7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;A# + m3 + d5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;A# + m3 + d5 + M7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;A# C# E&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;A# C# E G##&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;Ab Cb E&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ab Cb E Gbb&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,630: / Line 1,660: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;C# minor scale = C# + M2 + m3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C minor scale = C + M2 + m3&lt;br /&gt;
 + P4 + P5 + m6 + m7 + P8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;Cb + m2 + M3 + P4 &lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C + m2 + M3 + P4 &lt;br /&gt;
 + P5 + M6 + M7 + P8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;Cb Dbb Eb Fb&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C Db E F&lt;br /&gt;
-Gb Ab Bb Cb&lt;br /&gt;
+G A B C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C# D## E# F#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C D# E F&lt;br /&gt;
-G# A# B# C#&lt;br /&gt;
+G A B C&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,798: / Line 3,828: @@
 Natural generators can be used for other EDOs as well. For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.&lt;br /&gt;
 &lt;br /&gt;
-&lt;u&gt;&lt;strong&gt;Chroma-2 and chroma-5 edos:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;Chroma-2 edos:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
-genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - d6 etc.&lt;br /&gt;
+genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.&lt;br /&gt;
-Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C&lt;br /&gt;
+Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#&lt;br /&gt;
 &amp;quot;Every Good Boy Deserves Fudge And Candy&amp;quot; &lt;br /&gt;
 &lt;br /&gt;
@@ Line 3,807: / Line 3,837: @@
 P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8&lt;br /&gt;
 &lt;br /&gt;
-&lt;u&gt;&lt;strong&gt;17-edo&lt;/strong&gt;&lt;/u&gt;: (generator = 3\17 = perfect 3rd)&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;17-edo&lt;/strong&gt;&lt;/u&gt;: (generator = 5\17 = perfect 3rd)&lt;br /&gt;
 D * E * * F * G * * A * B * * C * D&lt;br /&gt;
 P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
+&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;24-edo&lt;/strong&gt;&lt;/u&gt;: (generator = 7\24 = perfect 3rd)&lt;br /&gt;
+D * * E * * * F * * G * * * A * * B * * * C * * D&lt;br /&gt;
+P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4/d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8&lt;br /&gt;
+&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;31-edo&lt;/strong&gt;&lt;/u&gt;: (generator = 9\31 = perfect 3rd)&lt;br /&gt;
+D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D&lt;br /&gt;
+P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...&lt;br /&gt;
+&lt;br /&gt;
+etc.&lt;br /&gt;
+&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;Chroma-5 edos:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
+genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.&lt;br /&gt;
+E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
+&amp;quot;Every Good Boy Deserves Fudge And Candy&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;18-edo&lt;/strong&gt;&lt;/u&gt;: see above&lt;br /&gt;
 &lt;br /&gt;
-&lt;u&gt;&lt;strong&gt;17-edo&lt;/strong&gt;&lt;/u&gt;: (generator = 3\17 = perfect 3rd)&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;25-edo&lt;/strong&gt;&lt;/u&gt;: (generator = 7\25 = perfect 3rd)&lt;br /&gt;
-D * E * * F * G * * A * B * * C * D&lt;br /&gt;
+D * * * E * * F * * * G * * A * * * B * * C * * * D&lt;br /&gt;
-P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
+P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8&lt;br /&gt;
+&lt;br /&gt;
+etc.&lt;br /&gt;
+&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;Chroma-3 edos&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt;&lt;br /&gt;
+genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.&lt;br /&gt;
+D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.&lt;br /&gt;
+&lt;br /&gt;
+&lt;strong&gt;&lt;u&gt;15-edo&lt;/u&gt;:&lt;/strong&gt; (generator = 2\15 = perfect 2nd)&lt;br /&gt;
+D * E * F * G * * A * B * C * D&lt;br /&gt;
+P1 - A1/d2 - P2 - A2/d3 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - A6/d7 - P7 - A7/d8 - P8&lt;br /&gt;
 &lt;br /&gt;
+&lt;strong&gt;&lt;u&gt;22-edo&lt;/u&gt;:&lt;/strong&gt; (generator = 3\22 = perfect 2nd)&lt;br /&gt;
+D * * E * * F * * G * * * A * * B * * C * * D&lt;br /&gt;
+P1 - A1 - d2 - P2 - A2 - d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6 - d7 - P7 - A7 - d8 - P8&lt;br /&gt;
 &lt;br /&gt;
+etc.&lt;br /&gt;
 &lt;br /&gt;
-For 22-edo, it's a 3\22 2nd&lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;Chroma-4 edos&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt;&lt;br /&gt;
-genchain: ...D# E# F# G# A B C D E F G Ab Bb Cb Db...&lt;br /&gt;
+genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.&lt;br /&gt;
-scale: D * * E * * F * * G * * * A * * B * * C * * D&lt;br /&gt;
+Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.&lt;br /&gt;
 &lt;br /&gt;
+&lt;strong&gt;&lt;u&gt;20-edo&lt;/u&gt;:&lt;/strong&gt; (generator = 3\20 = perfect 2nd)&lt;br /&gt;
+D * * E * * F * * G * A * * B * * C * * D&lt;br /&gt;
+P1 - A1 - d2 - P2 - A2/d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6/d7 - P7 - A7 - d8 - P8&lt;br /&gt;
 &lt;br /&gt;
-For 17-edo, the natural generator is the 5\17 3rd.&lt;br /&gt;
+&lt;strong&gt;&lt;u&gt;27-edo&lt;/u&gt;:&lt;/strong&gt; (generator = 4\27 = perfect 2nd)&lt;br /&gt;
+D * * * E * * * F * * * G * * A * * * B * * * C * * * D&lt;br /&gt;
+P1 - A1 - AA1/dd2 - d2 - P2 - A2 - d3 - m3 - M3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 etc.&lt;br /&gt;
 &lt;br /&gt;
-For 15-edo, it's the 2\15 2nd.&lt;br /&gt;
+etc.&lt;br /&gt;
-For 12-edo, it's the 7\12 5th, which generates standard notation.&lt;br /&gt;
-For 16-edo, it's the 9\16 5th.&lt;br /&gt;
-For 21-edo, it's the 3\21 2nd, the 6\21 3rd, the 9\21 4th, the 12\21 5th, etc. All these intervals generate the same scale, and the same notation.&lt;br /&gt;
 &lt;br /&gt;
-You can easily find the natural generator using the scale tree. It's also related to the chroma. But it doesn't intrinsically depend on any ratios or commas, or on the prime limit.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Ups and downs solfege"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;&lt;u&gt;Ups and downs solfege&lt;/u&gt;&lt;/h1&gt;