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

|-

|-

| 1\19 = 61{{c}}

| 0\19 = 0{{c}}

| 1\19 = 61{{c}}

| Ignore the arrows, treat slashes as sharps/flats

|-

| 3\22 = 164{{c}}

| 1\22 = 55{{c}}

| 0\22 = 0{{c}}

| Ignore the slashes

|-

| 4\41 = 117{{c}}

| 1\41 = 29{{c}}

| 1\41 = 29{{c}}

| Treat slashes as arrows

|-

| 5\60 = 100{{c}}

| 1\60 = 20{{c}}

| 2\60 = 40{{c}}

| Treat slashes as double arrows

|-

! Rank-2

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

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

| 40{{c}} − 4.4''c''

| N/a

|}

In magic, 5/4 = vM3, 7/4 = \m7 and 11/8 = vvA4. Thus an up is ~81/80 and a lift is ~64/63. This may not be true for other (P8, P12/5) temperaments. Therefore, the ratios in the following table are specific to magic, but the chord names apply to any (P8, P12/5) temperament.

{| class="wikitable center-1"

! Transversal

! Type

| 1

| 1–5/4–14/9

| 2

| 1–6/5–14/9

| 3

| 1–5/4–3/2

| [[4:5:6]]

|-

| 4

| 1–6/5–3/2

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

| 1–7/6–14/9

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

| C/

| 6

| 1–7/6–3/2

| 7

| 1–5/4–16/11

| 8

| 1–6/5–16/11

| 9

| 1–7/6–16/11

| 10

| 1–5/4–20/11

| 11

| 1–14/9–9/5

| 1–6/5–9/5

| [[6:9:10]]

| 1–3/2–9/5

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

| 14

| 1–7/6–9/5

| 15

| 1–16/11–20/11

| 1–5/4–11/8

| 16

| 1–9/8–5/4

| 17

| 1–9/8–14/9

| 1–9/8–3/2

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

| 19

| 1–9/8–16/11

| 20

| 1–9/8–9/5

| 21

| 1–5/4–7/5

| 22

| 1–7/5–14/9

| 1–9/7–9/5

| C/,^7no5

| 23

| 1–6/5–7/5

| [[5:6:7]]

| 1–7/6–7/5

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

| 25

| 1–7/5–9/5

| 1–9/7–10/7

| C/(^b5)

| 26

| 1–9/8–7/5

| 1–5/4–16/9

| 27

| 1–5/4–7/4

| [[4:5:7]]

| 28

| 1–14/9–7/4

| 1–9/8–9/7

| C/,9no5

| 29

| 1–6/5–7/4

| 30

| 1–3/2–7/4

| 1–7/6–7/4

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

| 32

| 1–16/11–7/4

| 1–6/5–11/8

| 33

| 1–9/8–7/4

| 1–7/5–7/4

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

| 35

| 1–12/11–5/4

| 36

| 1–12/11–14/9

| 1–9/7–7/5

| C/(^\b5)

| 37

| 1–12/11–6/5

| 38

| 1–12/11–3/2

| 39

| 1–12/11–16/11

| 1–11/8–3/2

| 40

| 1–12/11–20/11

| 41

| 1–12/11–7/5

| 42

| 1–12/11–7/4

| 43

| 1–3/2–18/11

| 44

| 1–7/6–18/11

| 45

| 1–16/11–18/11

| 46

| 1–18/11–9/5

| 47

| 1–9/8–18/11

| 48

| 1–7/5–18/11

| 49

| 1–12/11–18/11

| 50

| 1–14/11–14/9

| 51

| 1–7/6–14/11

| 52

| 1–14/11–16/11

| 53

| 1–14/11–20/11

| 54

| 1–9/8–14/11

| 55

| 1–14/11–7/5

| 56

| 1–14/11–7/4

| 57

| 1–12/11–14/11

| 58

| 1–14/11–18/11