Chords of magic: Difference between revisions

Below is a complete list of the [[11-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[magic|magic temperament]]. Note that there are many common chords, for example [[8:10:12:15]], which are not listed; in this case due to [[15/8]] not being in the 11-odd-limit. Every chord listed has multiple [[chord #Inversion|inversions]]; only one is listed, that being the inversion where all notes are a nonnegative number of major third [[generator]]s above the root.

Typing the chords requires consideration of the fact that magic conflates [[10/9]] and [[11/10]] and so also [[9/5]] and [[20/11]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as [[otonal]], [[utonal]], or [[ambitonal]]. If the chord is essentially tempered, it is analyzed in terms of the transversal that requires the minimum amount of commas to be tempered out; if there is a tie between multiple transversals, it is analyzed in terms of the transversal which employs 10/9 and 9/5.

Chords requiring tempering only by [[225/224]] are labeled [[marvel chords|marvel]], by [[245/243]] [[sensamagic chords|sensamagic]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[896/891]] [[pentacircle chords|pentacircle]], by [[385/384]] [[keenanismic chords|keenanismic]], and by [[540/539]] [[swetismic chords|swetismic]]. Those requiring any two of 100/99, 225/224 or 896/891 are labeled [[apollo chords|apollo]], any two of 100/99, 245/243 or 540/539 [[octarod chords|octarod]], any two of 245/243, 896/891 or 385/384 [[undecimal sensamagic chords|sensamagic11]], any two of 225/224, 385/384, or 540/539 [[undecimal marvel chords|marvel11]]. Chords requiring both 100/99 and 385/384 are labeled [[keemic chords|keemic]]. Finally, anything requiring three independent commas among those discussed above is labeled [[magic chords|magic]].

Magic has [[mos scale]]s of 7, 10, 13, 16, 19, and 22 notes. It may be seen that even the 7-note mos is not without a few harmonic resources, and the larger ones do much better.

[[Kite Giedraitis]] has named the chords using arrows (ups and downs), as described in [[Kite's thoughts on pergens]]. The pergen is (P8, P12/5) fifth-of-a-twelfth, #37 in the [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf list of pergens]. One up is 19 generators, octave-reduced. The generator is {{nowrap| vM3 {{=}} 380{{c}} + ''c''/5 }}, where ''c'' is the amount in cents the tempered fifth exceeds 700{{c}}. The [[Kite's thoughts on enharmonic unisons in ups and downs notation|enharmonic unison]] is ^⁵dd2, thus {{nowrap|^⁵C {{=}} Bx}}. To simplify the chord names, slashes (lifts and drops) are also used. One lift is -22 generators, octave-reduced. Thus {{nowrap|/1 {{=}} −25''G'' + 3''G'' {{=}} m2 + ^^d8 {{=}} ^^d2}}. Thus a lift equals two ups minus a tempered pythagorean comma, so {{nowrap| /C {{=}} ^^Dbb }}, {{nowrap| \C {{=}} vvB# }}, {{nowrap| ^^C {{=}} /B# }}, and {{nowrap| vvC {{=}} \Dbb }}. The cents values of sharps, ups and lifts vary greatly, as this table shows. Note that if the fifth is wider than 22edo's fifth, a lift will actually be descending. Furthermore, if the fifth is narrower than 19edo's, an up will be descending.

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Cents values of magic accidentals in various tunings

!  

! How to convert the notation to the edo

! 19edo

| 1\19 = 61{{c}}

| 0\19 = 0{{c}}

| 1\19 = 61{{c}}

| Ignore the arrows, treat slashes as sharps/flats

! 22edo

| 3\22 = 164{{c}}

| 1\22 = 55{{c}}

| 0\22 = 0{{c}}

| Ignore the slashes

! 41edo

| 4\41 = 117{{c}}

| 1\41 = 29{{c}}

| 1\41 = 29{{c}}

| Treat slashes as arrows

! 60edo

! Rank-2

| 100{{c}} + 7''c''

| 20{{c}} + 3.8''c''

| 40{{c}} − 4.4''c''

 

 

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Magic's genchain

! Genspan

! 0

! 1

! 2

! 3

! 4

! 5

! 6

! 7

! 8

! 9

! 10

! 11

! 12

! 13

| '''P1'''

| vvA5<br>\m6

| ^^d8<br>/M7

| '''P5'''

| vvA2<br>\m3

| ^^d5<br>/A4

| '''M2'''

| vA4<br>^\d5

| vvA6<br>\m7

| ^^m2<br>/A1

| ^^m6<br>/A5

! Note (in C)

| '''C'''

| vE

| vvG#<br>\Ab

| ^^Cb<br>/B

| ^Eb

| '''G'''

| vB

| vvD#<br>\Eb

| ^^Gb<br>/F#

| ^Bb

| vF#<br>^\Gb

| vvA#<br>\Bb

| ^^Db<br>/C#

| ^^Ab<br>/G#

{{Todo|inline=1|complete table}}

! Generators

| 0–1–2

| 1–5/4–14/9

|  

| Cv(vv#5)

| 0–2–4

| 1–6/5–14/9

|  

| C^m(vv#5)

| 0–1–5

| 1–5/4–3/2

| [[4:5:6]]

| Cv

| 0–4–5

| 1–6/5–3/2

| [[10:12:15|1/(6:5:4)]]

| C^m

| 0–2–7

| 1–7/6–14/9

| [[14:18:21|1/(9:7:6)]]

| C/

| 0–5–7

| 1–7/6–3/2

| [[6:7:9]]

| C\m

| 0–1–8

| 1–5/4–16/11

|  

| Cv(^^b5)

| 0–4–8

| 1–6/5–16/11

|  

| C^m(^^b5)

| 0–7–8

| 1–7/6–16/11

|  

| C\m(^^b5)

| 0–1–9

| 1–5/4–20/11

|  

| Cv^7no5

| 0–2–9

| 1–14/9–9/5

|  

| C^m7(vv#5)no3

| 0–4–9

| 1–6/5–9/5

| [[6:9:10]]

| C^m7no5 ''or'' Cv6no3

| 0–5–9

| 1–3/2–9/5

| [[10:15:18|1/(9:6:5)]]

| C^m7no3

| 0–7–9

| 1–7/6–9/5

|  

| C\mv7no5

| 0–8–9

| 1–16/11–20/11

| 1–5/4–11/8

| Cv(\b5)

| 0–1–10

| 1–9/8–5/4

|  

| Cv,9no5

| 0–2–10

| 1–9/8–14/9

|  

| C2(vv#5)

| 0–5–10

| 1–9/8–3/2

| [[6:8:9]], [[8:9:12]]

| C2

| 0–8–10

| 1–9/8–16/11

|  

| C2(^^b5)

| 0–9–10

| 1–9/8–9/5

|  

| C^9no35 ''or'' C^7sus2no5

| 0–1–11

| 1–5/4–7/5

|  

| Cv(^\b5)

| 0–2–11

| 1–7/5–14/9

| 1–9/7–9/5

| C/,^7no5

| 0–4–11

| 1–6/5–7/5

| [[5:6:7]]

| C^m(^\b5)

| 0–7–11

| 1–7/6–7/5

| [[30:35:42|1/(7:6:5)]]

| C\m(^\b5)

| 0–9–11

| 1–7/5–9/5

| 1–9/7–10/7

| C/(^b5)

| 0–10–11

| 1–9/8–7/5

| 1–5/4–16/9

| 0–1–12

| 1–5/4–7/4

| [[4:5:7]]

| Cv,\7no5

| 0–2–12

| 1–14/9–7/4

| 1–9/8–9/7

| C/,9no5

| 0–4–12

| 1–6/5–7/4

|  

| C^m\7

| 0–5–12

| 1–3/2–7/4

| [[4:6:7]]

| C\7no3

| 0–7–12

| 1–7/6–7/4

| [[14:18:21|1/(12:8:7)]]

| C\m7no5

| 0–8–12

| 1–16/11–7/4

| 1–6/5–11/8

| C^m(\b5)

| 0–10–12

| 1–9/8–7/4

|  

| C\7sus2

| 0–11–12

| 1–7/5–7/4

| [[28:35:40|1/(10:8:7)]]

| C\7(^\b5)no3

| 0–1–13

| 1–12/11–5/4

|  

|  

| 0–2–13

| 1–12/11–14/9

| 1–9/7–7/5

| C/(^\b5)

| 0–4–13

| 1–12/11–6/5

|  

|  

| 0–5–13

| 1–12/11–3/2

|  

| C^^b2

| 0–8–13

| 1–12/11–16/11

| 1–11/8–3/2

| Cvv#4

| 0–9–13

| 1–12/11–20/11

|  

|  

| 0–11–13

| 1–12/11–7/5

|  

|  

| 0–12–13

| 1–12/11–7/4

|  

|  

| 0–5–18

| 1–3/2–18/11

|  

|  

| 0–7–18

| 1–7/6–18/11

|  

|  

| 0–8–18

| 1–16/11–18/11

|  

|  

| 0–9–18

| 1–18/11–9/5

|  

|  

| 0–10–18

| 1–9/8–18/11

|  

|  

| 0–11–18

| 1–7/5–18/11

|  

|  

| 0–13–18

| 1–12/11–18/11

|  

|  

| 0–2–20

| 1–14/11–14/9

|  

|  

| 0–7–20

| 1–7/6–14/11

|  

|  

| 0–8–20

| 1–14/11–16/11

|  

|  

| 0–9–20

| 1–14/11–20/11

|  

|  

| 0–10–20

| 1–9/8–14/11

|  

|  

| 0–11–20

| 1–14/11–7/5

|  

|  

| 0–12–20

| 1–14/11–7/4

|  

|  

| 0–13–20

| 1–12/11–14/11

|  

|  

| 0–18–20

| 1–14/11–18/11

|  

|  

! Generators

| 1

| 0–1–2–9

| 1–5/4–14/9–9/5

| Magic

|  

| Cv^7(vv#5)

| 0–2–4–9

| 1–6/5–14/9–9/5

| Sensamagic

| 0–1–5–9

| 1–5/4–3/2–9/5

| Ptolemismic

| 0–4–5–9

| 1–6/5–3/2–9/5

| Ambitonal

| [[10:12:15:18]], [[12:15:18:20]]<br>[[9-odd-limit]] [[ASS]]

| C^m7 ''or'' Cv6

| 0–2–7–9

| 1–7/6–14/9–9/5

| Sensamagic

| 1–9/7–3/2–7/3

| C/,vv#9

| 0–5–7–9

| 1–7/6–3/2–9/5

|  

| C\m^7

| 0–1–8–9

| 1–5/4–16/11–9/5

| Keemic

| 0–4–8–9

| 1–6/5–16/11–9/5

| Ptolemismic

| 0–7–8–9

| 1–7/6–16/11–9/5

| Magic

| 0–1–2–10

| 1–9/8–5/4–14/9

| Apollo

| 0–1–5–10

| 1–9/8–5/4–3/2

| Otonal

| [[4:5:6:9]]

| Cv,9

| 0–1–8–10

| 1–9/8–5/4–16/11

| Sensamagic11

| 0–1–9–10

| 1–9/8–5/4–9/5

| Ptolemismic

| 0–2–9–10

| 1–9/8–14/9–9/5

| Sensamagic11

|  

| C^9(vv#5)no3 ''or'' C^7(vv#5)sus2

| 0–5–9–10

| 1–9/8–3/2–9/5

| Utonal

| [[20:30:36:45|1/(9:6:5:4)]]

| C^9no3 ''or'' C^7sus2 ''or'' C2,^7

| 0–8–9–10

| 1–9/8–16/11–9/5

| 17

| 0–1–2–11

| 1–5/4–7/5–14/9

| 18

| 0–2–4–11

| 1–6/5–7/5–14/9

| 19

| 0–2–7–11

| 1–7/6–7/5–14/9

| [[70:90:105:126|1/(9:7:6:5)]]

| 20

| 0–1–9–11

| 1–5/4–7/5–9/5

| 21

| 0–2–9–11

| 1–7/5–14/9–9/5

| 0–7–9–11

| 1–7/6–7/5–9/5

| Sensamagic

|  

| 0–1–10–11

| 1–9/8–5/4–7/5

| Marvel

| 0–2–10–11

| 1–9/8–7/5–14/9

| Apollo

| 0–9–10–11

| 1–9/8–7/5–9/5

| Marvel

| 0–1–2–12

| 1–5/4–14/9–7/4

| 0–2–4–12

| 1–6/5–14/9–7/4

| Sensamagic11

| 0–1–5–12

| 1–5/4–3/2–7/4

| Otonal

| [[4:5:6:7]]

| Cv,\7

| 0–4–5–12

| 1–6/5–3/2–7/4

| Keenanismic

| C^m\7

| 0–2–7–12

| 1–7/6–14/9–7/4

| Utonal

| 0–5–7–12

| 1–7/6–3/2–7/4

| Ambitonal

| [[12:14:18:21]], [[14:18:21:24]]<br>9-odd-limit ASS

| C\m7

| 0–1–8–12

| 1–5/4–16/11–7/4

| Keenanismic

|  

| 0–4–8–12

| 1–6/5–16/11–7/4

| Keemic

| 0–7–8–12

| 1–7/6–16/11–7/4

| Keenanismic

| 0–1–10–12

| 1–9/8–5/4–7/4

| Otonal

| [[4:5:7:9]]

| 37

| 0–2–10–12

| 1–9/8–14/9–7/4

| Pentacircle

|  

|  

| 0–5–10–12

| 1–9/8–3/2–7/4

| [[4:6:7:9]]

| C2\7 ''or'' C\7sus2 ''or'' C\9no3

| 0–8–10–12

| 1–9/8–16/11–7/4

| Sensamagic11

| 0–1–11–12

| 1–5/4–7/5–7/4

| Marvel

|  

|  

| 0–2–11–12

| 1–7/5–14/9–7/4

| Utonal

| [[140:180:252:315|1/(9:7:5:4)]]

| 0–4–11–12

| 1–6/5–7/5–7/4

| Keenanismic

| 0–7–11–12

| 1–7/6–7/5–7/4

| [[70:84:105:120|1/(12:10:8:7)]]

| C\m7(^\b5) ''or'' C^m/6

| 0–10–11–12

| 1–9/8–7/5–7/4

| Marvel

| 0–1–2–13

| 1–12/11–5/4–14/9

| Marvel11

| 0–2–4–13

| 1–12/11–6/5–14/9

| Octarod

| 0–1–5–13

| 1–12/11–5/4–3/2

| Keenanismic

| 0–4–5–13

| 1–12/11–6/5–3/2

| Utonal

| 0–1–8–13

| 1–12/11–5/4–16/11

| 0–4–8–13

| 1–12/11–6/5–16/11

| Ptolemismic

| 0–1–9–13

| 1–12/11–5/4–9/5

| Keemic

| 0–2–9–13

| 1–12/11–14/9–9/5