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Chords of magic: Difference between revisions

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TallKite (talk | contribs)
autopatrolled, Sysops
4,192 edits
fixed 2 errors, added a table of cents for all accidentals, added the vM7 and 7;6;5;4 chords
TallKite (talk | contribs)
autopatrolled, Sysops
4,192 edits
major rewrite of chord names: a lift has been redefined from -41 generators to -22 generators. Advantages: makes shorter chord names, also makes a less "tippy" notation, lifts are no longer negative cents for 22edo.
Line 5: Line 5:
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]. 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##. To simplify the chord names, slashes (lifts and drops) are also used. One lift is -41 generators, octave-reduced. (41 because 41edo is a near-optimal tuning of magic.) Thus {{nowrap|/1 {{=}} ^d2 {{=}} 20¢ &minus; 8.2c}}. Thus a lift equals an up minus a tempered pythagorean comma. {{nowrap|/C {{=}} ^Dbb|\C {{=}} vB#|^C {{=}} /B#|and vC {{=}} \Dbb}}. This notation is "tippy" in the sense that if the fifth is wider than 41edo's fifth, a lift will actually be descending. Furthermore if the fifth is narrower than 19edo's, an up will be descending.
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##.  
 
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.
 
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"
{| class="wikitable"
|+Cents values of Magic accidentals
|+Cents values of Magic accidentals in various tunings
!
!
!sharp
!sharp
!up
!up
!lift
!lift
!how to convert the score to an edo
!how to convert the notation to an edo
|-
|-
!19edo
!19edo
Line 23: Line 27:
|3\22 = 164¢
|3\22 = 164¢
|1\22 = 55¢
|1\22 = 55¢
| -1\22 = -55¢
|0\22 = 0¢
|treat lifts as downs and drops as ups
|ignore the slashes
|-
|-
!41edo
!41edo
|4\41 = 117¢
|4\41 = 117¢
|1\41 = 29¢
|1\41 = 29¢
|0\41 = 0¢
|1\41 = 29¢
|ignore the slashes
|treat slashes as arrows
|-
|-
!60edo
!60edo
|5\60 = 100¢
|5\60 = 100¢
|1\60 = 20¢
|1\60 = 20¢
|1\60 = 20¢
|2\60 = 40¢
|treat slashes as arrows
|treat slashes as double arrows
|-
|-
!rank-2
!rank-2
|100¢ + 7c
|100¢ + 7c
|20¢ + 3.8c
|20¢ + 3.8c
|20¢ - 8.2c
|40¢ - 4.4c
!
!
|}
|}
{| class="wikitable center-all"
{| class="wikitable center-all"
|+Magic's genchain
|+Magic's genchain
|-
!Genspan
! Genspan
!0
| 0
!1
| 1
!2
| 2
!3
| 3
!4
| 4
!5
| 5
!6
| 6
!7
| 7
!8
| 8
!9
| 9
!10
| 10
!11
| 11
!12
| 12
!13
| 13
|-
|-
! Cents (41edo)
! Cents (41edo)
Line 96: Line 99:
|-
|-
! Interval
! Interval
| '''P1'''
|'''P1'''
| vM3
| vM3
| vvA5
| vvA5
v\m6
\m6
| ^^d8
| ^^d8
^/M7
/M7
| ^m3
| ^m3
| '''P5'''
|'''P5'''
| vM7
| vM7
| vvA2
| vvA2
v\m3
\m3
| ^^d5
| ^^d5
^/A4
/A4
| ^m7
| ^m7
| '''M2'''
|'''M2'''
| vA4
| vA4
\d5
^\d5
| vvA6
| vvA6
v\m7
\m7
| ^^m2
| ^^m2
^/A1
/A1
|-
|-
!Note
!Note
Line 123: Line 126:
|vE
|vE
|vvG#
|vvG#
v\Ab
\Ab
|^^Cb
|^^Cb
^/B
/B
|^Eb
|^Eb
|'''G'''
|'''G'''
|vB
|vB
|vvD#
|vvD#
v\Eb
\Eb
|^^Gb
|^^Gb
^/F#
/F#
|^Bb
|^Bb
|'''D'''
|'''D'''
|vF#
|vF#
\Gb
^\Gb
|vvA#
|vvA#
v\Bb
\Bb
|^^Db
|^^Db
^/C#
/C#
|}
|}
'''''TODO: complete the tables'''''
'''''TODO: complete the tables'''''
Line 186: Line 189:
| 1-14/9-7/6
| 1-14/9-7/6
| Utonal
| Utonal
| ^/
| /
| 1-9/7-3/2
| 1-9/7-3/2
|-
|-
Line 193: Line 196:
| 1-3/2-7/6
| 1-3/2-7/6
| Otonal
| Otonal
| v\m
| \m
|
|
|-
|-
Line 214: Line 217:
| 1-7/6-16/11
| 1-7/6-16/11
| Keenanismic
| Keenanismic
| v\m(^^b5)
| \m(^^b5)
|
|
|-
|-
Line 249: Line 252:
| 1-7/6-9/5
| 1-7/6-9/5
| Sensamagic
| Sensamagic
| v\m,v7no5
| \m,v7no5
|
|
|-
|-
Line 256: Line 259:
| 1-16/11-20/11
| 1-16/11-20/11
| Otonal
| Otonal
| v(^^^bb5)
| v(\b5)
| 1-5/4-11/8
| 1-5/4-11/8
|-
|-
Line 298: Line 301:
| 1-5/4-7/5
| 1-5/4-7/5
| Marvel
| Marvel
| v(\b5)
| v(^\b5)
|
|
|-
|-
Line 305: Line 308:
| 1-14/9-7/5
| 1-14/9-7/5
| Utonal
| Utonal
| ^/,^7no5
| /,^7no5
| 1-9/7-9/5
| 1-9/7-9/5
|-
|-
Line 312: Line 315:
| 1-6/5-7/5
| 1-6/5-7/5
| Otonal
| Otonal
| ^m(\b5)
| ^m(^\b5)
|
|
|-
|-
Line 319: Line 322:
| 1-7/6-7/5
| 1-7/6-7/5
| Utonal
| Utonal
| v\m(\b5)
| \m(^\b5)
|
|
|-
|-
Line 326: Line 329:
| 1-9/5-7/5
| 1-9/5-7/5
| Otonal
| Otonal
| ^/(^b5)
| /(^b5)
| 1-9/7-10/7
| 1-9/7-10/7
|-
|-
Line 340: Line 343:
| 1-5/4-7/4
| 1-5/4-7/4
| Otonal
| Otonal
| v,v/7
| v,\7no5
|
|
|-
|-
Line 347: Line 350:
| 1-14/9-7/4
| 1-14/9-7/4
| Utonal
| Utonal
| ^/,9no5
| /,9no5
| 1-9/8-9/7
| 1-9/8-9/7
|-
|-
Line 354: Line 357:
| 1-6/5-7/4
| 1-6/5-7/4
| Keenanismic
| Keenanismic
| ^m,v\7
| ^m\7
|
|
|-
|-
Line 361: Line 364:
| 1-3/2-7/4
| 1-3/2-7/4
| Otonal
| Otonal
| v\m7no3
| \7no3
|
|
|-
|-
Line 368: Line 371:
| 1-7/6-7/4
| 1-7/6-7/4
| Utonal
| Utonal
| v\m7no5
| \m7no5
|
|
|-
|-
Line 375: Line 378:
| 1-16/11-7/4
| 1-16/11-7/4
| Keenanismic
| Keenanismic
| ^m(v\b5)
| ^m(\b5)
| 1-6/5-11/8
| 1-6/5-11/8
|-
|-
Line 382: Line 385:
| 1-9/8-7/4
| 1-9/8-7/4
| Otonal
| Otonal
| v\m7sus2
| \7sus2
|
|
|-
|-
Line 389: Line 392:
| 1-7/5-7/4
| 1-7/5-7/4
| Utonal
| Utonal
| v\7(\b5)no3
| \7(^\b5)no3
|
|
|-
|-
Line 403: Line 406:
| 1-14/9-12/11
| 1-14/9-12/11
| Swetismic
| Swetismic
|
|/(^\b5)
| 1-9/7-7/5
| 1-9/7-7/5
|-
|-
Line 610: Line 613:
| 1-14/9-7/6-9/5
| 1-14/9-7/6-9/5
| Sensamagic
| Sensamagic
| ^/,vv#9
| /,vv#9
| 1-9/7-3/2-7/3
| 1-9/7-3/2-7/3
|-
|-
Line 617: Line 620:
| 1-3/2-7/6-9/5
| 1-3/2-7/6-9/5
| Sensamagic
| Sensamagic
| v\m^7
| \m^7
|  
|  
|-
|-
Line 638: Line 641:
| 1-7/6-16/11-9/5
| 1-7/6-16/11-9/5
| Magic
| Magic
| v\m,^7(^^b5)
| \m,^7(^^b5)
|  
|  
|-
|-
Line 729: Line 732:
| 1-6/5-9/5-7/5
| 1-6/5-9/5-7/5
| Otonal
| Otonal
| ^m7(\b5)
| ^m7(^\b5)
|  
|  
|-
|-
Line 778: Line 781:
| 1-5/4-3/2-7/4
| 1-5/4-3/2-7/4
| Otonal
| Otonal
| v,v\7
| v,\7
|  
|  
|-
|-
Line 785: Line 788:
| 1-6/5-3/2-7/4
| 1-6/5-3/2-7/4
| Keenanismic
| Keenanismic
| ^mv\7
| ^m\7
|  
|  
|-
|-
Line 792: Line 795:
| 1-14/9-7/6-7/4
| 1-14/9-7/6-7/4
| Utonal
| Utonal
| v\m7(vv#5)
| \m7(vv#5)
|  
|  
|-
|-
Line 799: Line 802:
| 1-3/2-7/6-7/4
| 1-3/2-7/6-7/4
| Ambitonal
| Ambitonal
| v\m7
| \m7
|  
|  
|-
|-
Line 820: Line 823:
| 1-7/6-16/11-7/4
| 1-7/6-16/11-7/4
| Keenanismic
| Keenanismic
| v\m7(^^b5)
| \m7(^^b5)
|  
|  
|-
|-
Line 827: Line 830:
|1-7/6-7/5-7/4
|1-7/6-7/5-7/4
|Otonal
|Otonal
|v\m7(\b5)
|\m7(^\b5)
|<u>or</u> 1-6/5-3/2-12/7 = ^m^/6
|<u>or</u> 1-6/5-3/2-12/7 = ^m/6
|-
|-
| 38
| 38
Line 848: Line 851:
| 1-3/2-9/8-7/4
| 1-3/2-9/8-7/4
| Otonal
| Otonal
| v\7sus2
| \7sus2
|<u>or</u> 2,v\7 <u>or</u> v\9no3
|<u>or</u> 2,\7 <u>or</u> \9no3
|-
|-
| 41
| 41
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