Chords of magic: Difference between revisions

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Below are listed the [[dyadic chord]]s of 11-limit [[magic|magic temperament]]. 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 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 225/224, 100/99 or 896/891 are labeled apollo, any two of 100/99, 245/243 or 540/539 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|unimarvel]]. Chords requiring both 100/99 and 385/384 are labeled [[supermagic chords|supermagic]]. Finally, anything requiring three independent commas among those discussed above is labeled [[magic chords|magic]].

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 225/224, 100/99 or 896/891 are labeled apollo, any two of 100/99, 245/243 or 540/539 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|unimarvel]]. Chords requiring both 100/99 and 385/384 are labeled [[supermagic chords|supermagic]]. Finally, anything requiring three independent commas among those discussed above is labeled [[magic chords|magic]].

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

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

The chord names use arrows (ups and downs) as described on the [[Pergen|pergens]] page. The pergen is (P8, P12/5) fifth-of-a-12th, #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 vM3 = 380¢ + c/5, where c = the amount in cents the tempered fifth exceeds 700¢. The enharmonic interval is ^5dd2, thus ^5C = B##.

The chord names use arrows (ups and downs) as described on the [[Pergen|pergens]] page. The pergen is (P8, P12/5) fifth-of-a-12th, #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''/5}}, where ''c'' is the amount in cents the tempered fifth exceeds 700¢. The enharmonic interval is ^5dd2, thus {{nowrap|^5C {{=}} B##}}.

To simplify the chord names, slashes (lifts and drops) are also used. One lift is -22 generators, octave-reduced. Thus {{nowrap|1=/1 {{=}} -25G + 3G = m2 + ^^d8 = ^^d2}}. Thus a lift equals two ups minus a tempered pythagorean comma. {{nowrap|1=/C {{=}} ^^Dbb|2=\C {{=}} vvB#|3=^^C {{=}} /B#|4=and vvC {{=}} \Dbb}}. When naming 7-limit chords, note that an up is ~81/80 and a lift is ~64/63.

To simplify the chord names, slashes (lifts and drops) are also used. One lift is -22 generators, octave-reduced. Thus {{nowrap|/1 {{=}} −25G + 3G {{=}} m2 + ^^d8 {{=}} ^^d2}}. Thus a lift equals two ups minus a tempered pythagorean comma, so {{nowrap|/C {{=}} ^^Dbb|\C {{=}} vvB#|^^C {{=}} /B#|and vvC {{=}} \Dbb}}. When naming 7-limit chords, note that an up is ~81/80 and a lift is ~64/63.

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"

|+Cents values of Magic accidentals in various tunings

|+ style="font-size: 105%;" | Cents values of Magic accidentals in various tunings

!

|-

!~~sharp~~

!

!up

! Sharp

!~~lift~~

! Up

!~~how~~ to convert the notation to an edo

! Lift

! How to convert the notation to an edo

|-

!19edo

! 19edo

|1\19 = 61¢

| 1\19 = 61¢

|0\19 = 0¢

| 0\19 = 0¢

|1\19 = 61¢

| 1\19 = 61¢

|~~ignore~~ the arrows, treat slashes as sharps/flats

| Ignore the arrows, treat slashes as sharps/flats

|-

!22edo

! 22edo

|3\22 = 164¢

| 3\22 = 164¢

|1\22 = 55¢

| 1\22 = 55¢

|0\22 = 0¢

| 0\22 = 0¢

|~~ignore~~ the slashes

| Ignore the slashes

|-

!41edo

! 41edo

|4\41 = 117¢

| 4\41 = 117¢

|1\41 = 29¢

| 1\41 = 29¢

|1\41 = 29¢

| 1\41 = 29¢

|~~treat~~ slashes as arrows

| Treat slashes as arrows

|-

!60edo

! 60edo

|5\60 = 100¢

| 5\60 = 100¢

|1\60 = 20¢

| 1\60 = 20¢

|2\60 = 40¢

| 2\60 = 40¢

|~~treat~~ slashes as double arrows

| Treat slashes as double arrows

|-

!~~rank~~-2

! Rank-2

|100¢ + 7c

| 100¢ + 7c

|20¢ + 3.8c

| 20¢ + 3.8c

|40¢ - 4.4c

| 40¢ − 4.4c

!

|}

{| class="wikitable center-all"

|+Magic's genchain

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

!Genspan

|-

!0

! Genspan

!1

! 0

!2

! 1

!3

! 2

!4

! 3

!5

! 4

!6

! 5

!7

! 6

!8

! 7

!9

! 8

!10

! 9

!11

! 10

!12

! 11

!13

! 12

! 13

|-

! Cents (41edo)

Line 99:

Line 103:

|-

! Interval

|'''P1'''

| '''P1'''

| vM3

| vvA5

| vvA5 \m6

\m6

| ^^d8 /M7

| ^^d8

/M7

| ^m3

|'''P5'''

| '''P5'''

| vM7

| vvA2

| vvA2 \m3

\m3

| ^^d5 /A4

| ^^d5

/A4

| ^m7

|'''M2'''

| vA4

| vA4 ^\d5

^\d5

| vvA6 \m7

| vvA6

| ^^m2 /A1

\m7

| ^^m2

/A1

|-

!Note

! Note (in C)

(in C)

| '''C'''

|'''C'''

| vE

|vE

| vvG# \Ab

|vvG#

| ^^Cb /B

\Ab

| ^Eb

|^^Cb

| '''G'''

/B

| vB

|^Eb

| vvD# \Eb

|'''G'''

| ^^Gb /F#

|vB

| ^Bb

|vvD#

| '''D'''

\Eb

| vF# ^\Gb

|^^Gb

| vvA# \Bb

/F#

| ^^Db /C#

|^Bb

|'''D'''

|vF#

^\Gb

|vvA#

\Bb

|^^Db

/C#

|}

'''''TODO: complete the tables'''''

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{| class="wikitable center-1"

|-

! #

! Chord

! Transversal

@@ Line 1: / Line 1: @@
 Below are listed the [[dyadic chord]]s of 11-limit [[magic|magic temperament]]. 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 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 225/224, 100/99 or 896/891 are labeled apollo, any two of 100/99, 245/243 or 540/539 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|unimarvel]]. Chords requiring both 100/99 and 385/384 are labeled [[supermagic chords|supermagic]]. Finally, anything requiring three independent commas among those discussed above is labeled [[magic chords|magic]].
+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 225/224, 100/99 or 896/891 are labeled apollo, any two of 100/99, 245/243 or 540/539 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|unimarvel]]. Chords requiring both 100/99 and 385/384 are labeled [[supermagic chords|supermagic]]. Finally, anything requiring three independent commas among those discussed above is labeled [[magic chords|magic]].
-Magic has [[mos scale]]s of sizes 7, 10, 13, 16, 19 and 22 notes. It may be seen that even the seven-note mos is not without a few harmonic resources, and the larger ones do much better.
+Magic has [[MOS scale]]s of sizes 7, 10, 13, 16, 19, and 22 notes. It may be seen that even the seven-note mos is not without a few harmonic resources, and the larger ones do much better.
-The chord names use arrows (ups and downs) as described on the [[Pergen|pergens]] page. The pergen is (P8, P12/5) fifth-of-a-12th, #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 vM3 = 380¢ + c/5, where c = the amount in cents the tempered fifth exceeds 700¢. The enharmonic interval is ^<sup>5</sup>dd2, thus ^<sup>5</sup>C = B##.
+The chord names use arrows (ups and downs) as described on the [[Pergen|pergens]] page. The pergen is (P8, P12/5) fifth-of-a-12th, #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''/5}}, where ''c'' is the amount in cents the tempered fifth exceeds 700¢. The enharmonic interval is ^<sup>5</sup>dd2, thus {{nowrap|^<sup>5</sup>C {{=}} B##}}.
-To simplify the chord names, slashes (lifts and drops) are also used. One lift is -22 generators, octave-reduced. Thus {{nowrap|1=/1 {{=}} -25G + 3G = m2 + ^^d8 = ^^d2}}. Thus a lift equals two ups minus a tempered pythagorean comma. {{nowrap|1=/C {{=}} ^^Dbb|2=\C {{=}} vvB#|3=^^C {{=}} /B#|4=and vvC {{=}} \Dbb}}. When naming 7-limit chords, note that an up is ~81/80 and a lift is ~64/63.
+To simplify the chord names, slashes (lifts and drops) are also used. One lift is -22 generators, octave-reduced. Thus {{nowrap|/1 {{=}} &minus;25G + 3G {{=}} m2 + ^^d8 {{=}} ^^d2}}. Thus a lift equals two ups minus a tempered pythagorean comma, so {{nowrap|/C {{=}} ^^Dbb|\C {{=}} vvB#|^^C {{=}} /B#|and vvC {{=}} \Dbb}}. When naming 7-limit chords, note that an up is ~81/80 and a lift is ~64/63.
+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.
-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"
-|+Cents values of Magic accidentals in various tunings
+|+ style="font-size: 105%;" | Cents values of Magic accidentals in various tunings
-!
+|-
-!sharp
+!
-!up
+! Sharp
-!lift
+! Up
-!how to convert the notation to an edo
+! Lift
+! How to convert the notation to an edo
 |-
-!19edo
+! 19edo
-|1\19 = 61¢
+| 1\19 = 61¢
-|0\19 = 0¢
+| 0\19 = 0¢
-|1\19 = 61¢
+| 1\19 = 61¢
-|ignore the arrows, treat slashes as sharps/flats
+| Ignore the arrows, treat slashes as sharps/flats
 |-
-!22edo
+! 22edo
-|3\22 = 164¢
+| 3\22 = 164¢
-|1\22 = 55¢
+| 1\22 = 55¢
-|0\22 = 0¢
+| 0\22 = 0¢
-|ignore the slashes
+| Ignore the slashes
 |-
-!41edo
+! 41edo
-|4\41 = 117¢
+| 4\41 = 117¢
-|1\41 = 29¢
+| 1\41 = 29¢
-|1\41 = 29¢
+| 1\41 = 29¢
-|treat slashes as arrows
+| Treat slashes as arrows
 |-
-!60edo
+! 60edo
-|5\60 = 100¢
+| 5\60 = 100¢
-|1\60 = 20¢
+| 1\60 = 20¢
-|2\60 = 40¢
+| 2\60 = 40¢
-|treat slashes as double arrows
+| Treat slashes as double arrows
 |-
-!rank-2
+! Rank-2
-|100¢ + 7c
+| 100¢ + 7c
-|20¢ + 3.8c
+| 20¢ + 3.8c
-|40¢ - 4.4c
+| 40¢ &minus; 4.4c
 !
 |}
 {| class="wikitable center-all"
-|+Magic's genchain
+|+ style="font-size: 105%;" | Magic's genchain
-!Genspan
+|-
-!0
+! Genspan
-!1
+! 0
-!2
+! 1
-!3
+! 2
-!4
+! 3
-!5
+! 4
-!6
+! 5
-!7
+! 6
-!8
+! 7
-!9
+! 8
-!10
+! 9
-!11
+! 10
-!12
+! 11
-!13
+! 12
+! 13
 |-
 ! Cents (41edo)
@@ Line 99: / Line 103: @@
 |-
 ! Interval
-|'''P1'''
+| '''P1'''
 | vM3
-| vvA5
+| vvA5<br />\m6
-\m6
+| ^^d8<br />/M7
-| ^^d8
-/M7
 | ^m3
-|'''P5'''
+| '''P5'''
 | vM7
-| vvA2
+| vvA2<br />\m3
-\m3
+| ^^d5<br />/A4
-| ^^d5
-/A4
 | ^m7
 |'''M2'''
-| vA4
+| vA4<br />^\d5
-^\d5
+| vvA6<br />\m7
-| vvA6
+| ^^m2<br />/A1
-\m7
-| ^^m2
-/A1
 |-
-!Note
+! Note<br />(in C)
-(in C)
+| '''C'''
-|'''C'''
+| vE
-|vE
+| vvG#<br />\Ab
-|vvG#
+| ^^Cb<br />/B
-\Ab
+| ^Eb
-|^^Cb
+| '''G'''
-/B
+| vB
-|^Eb
+| vvD#<br />\Eb
-|'''G'''
+| ^^Gb<br />/F#
-|vB
+| ^Bb
-|vvD#
+| '''D'''
-\Eb
+| vF#<br />^\Gb
-|^^Gb
+| vvA#<br />\Bb
-/F#
+| ^^Db<br />/C#
-|^Bb
-|'''D'''
-|vF#
-^\Gb
-|vvA#
-\Bb
-|^^Db
-/C#
 |}
 '''''TODO: complete the tables'''''
@@ Line 150: / Line 139: @@
 {| class="wikitable center-1"
 |-
-! #
+! &#35;
 ! Chord
 ! Transversal