Hemififths/Chords: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
This is 11-limit hemififths so 1-14/11-3/2 and 1-11/9-3/2-16/9 aren't dyadic chords. These lists should be complete.
TallKite (talk | contribs)
autopatrolled, Sysops
3,992 edits
added more chord names, improved the interval chart
Line 3: Line 3:
A striking feature of these hemififths chords is that essentially just chords tend to be of higher complexity than essentially tempered chords. Hemififths has MOS of size 7, 10, 17 and 24, and even seven notes are well-supplied with chords, mostly but by no means entirely essentially tempered chords. Extending consideration to the 13-limit adds even more such chords.
A striking feature of these hemififths chords is that essentially just chords tend to be of higher complexity than essentially tempered chords. Hemififths has MOS of size 7, 10, 17 and 24, and even seven notes are well-supplied with chords, mostly but by no means entirely essentially tempered chords. Extending consideration to the 13-limit adds even more such chords.


Chords are named with ups and downs. The tilde ~ means mid, half-way between major and minor. The comma (the actual punctuation mark) is pronounced "add", thus C~,7 is "C mid add7".
Chords are named with ups and downs. The tilde ~ means mid, half-way between major and minor. ~4 = ^4 = vA4. The comma (the actual punctuation mark) is pronounced "add", thus C~,7 is "C mid add 7". To facilitate chord naming, lifts and drops are also used. Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ - 8.5c, where c = the amount in cents the tempered fifth exceeds 700¢. /1 = ~81/80 = ~64/63 and ^ = ~33/31.
{| class="wikitable"
{| class="wikitable"
|+Hemififth's genchain
|+Hemififth's genchain
Line 21: Line 21:
!12
!12
!13
!13
!14
!...
!15
!20
!21
!...
!23
!...
!25
|-
!Cents (41edo)
|0
|351
|702
|1054
|205
|556
|907
|59
|410
|761
|1112
|263
|615
|966
!
|1024
|176
!
|878
!
|380
|-
|-
!Ratio
!Ratio
|1/1
|1/1
|11/9
|11/9
16/13
|3/2
|3/2
|11/6
|11/6
24/13
|9/8
|9/8
|11/8
|11/8
18/13
|27/16
|27/16
|33/32
22/13
|28/27
33/32
|14/11
|14/11
|14/9
|14/9
|40/21
|40/21
21/11
|7/6
|7/6
|10/7
|10/7
|7/4
|7/4
|15/14
!
|21/16
|20/11
|10/9
!
|5/3
!
|5/4
|-
|-
!Interval
!Interval
Line 49: Line 88:
|M2
|M2
|~4
|~4
\b5
|M6
|M6
|^1
|^1
\m2
|M3
|M3
|^5
|^5
\m6
|M7
|M7
|^2
|^M2
\m3
|A4
|A4
|^6
\~5
|A1
|^M6
|^3
\m7
!
|A6
 
\~7
|^A1
 
\M2
!
|^A5
 
\M6
!
|^A2
 
\M3
|}
|}


Line 69: Line 127:
! Type
! Type
!Name
!Name
!Inversion
|-
|-
| 1
| 1
Line 75: Line 134:
| rastmic
| rastmic
|~
|~
|
|-
|-
| 2
| 2
Line 81: Line 141:
| utonal
| utonal
|~7no5
|~7no5
|
|-
|-
| 3
| 3
Line 87: Line 148:
| otonal
| otonal
|~7no3
|~7no3
|
|-
|-
| 4
| 4
Line 92: Line 154:
| 1-11/9-9/8
| 1-11/9-9/8
| rastmic
| rastmic
|~,9no5
|
|
|-
|-
Line 99: Line 162:
| ambitonal
| ambitonal
|2 <u>or</u> 4
|2 <u>or</u> 4
|1-4/3-9/8
|-
|-
| 6
| 6
Line 104: Line 168:
| 1-11/6-9/8
| 1-11/6-9/8
| rastmic
| rastmic
|
|
|
|-
|-
Line 110: Line 175:
| 1-11/9-11/8
| 1-11/9-11/8
| utonal
| utonal
|~6sus2
|~(\b5)
|
|-
|-
| 8
| 8
Line 116: Line 182:
| 1-3/2-11/8
| 1-3/2-11/8
| otonal
| otonal
|~7sus4
|~7sus4no5
|1-4/3-11/6
|-
|-
| 9
| 9
Line 123: Line 190:
| utonal
| utonal
|~2
|~2
|1-12/11-3/2
|-
|-
| 10
| 10
Line 129: Line 197:
| otonal
| otonal
|~,7no5
|~,7no5
|1-11/9-16/9
|-
|-
| 11
| 11
Line 135: Line 204:
| hemimin
| hemimin
|,~7no5
|,~7no5
|
|-
|-
| 12
| 12
Line 141: Line 211:
| pentacircle
| pentacircle
|,9no5
|,9no5
|
|-
|-
| 13
| 13
Line 146: Line 217:
| 1-11/8-14/11
| 1-11/8-14/11
| hemimin
| hemimin
|(\b5)
|
|
|-
|-
Line 152: Line 224:
| 1-11/9-14/9
| 1-11/9-14/9
| otonal
| otonal
|~(^5)
|
|
|-
|-
Line 159: Line 232:
| pentacircle
| pentacircle
|7~4no5
|7~4no5
|1-11/8-16/9
|-
|-
| 16
| 16
Line 164: Line 238:
| 1-11/8-14/9
| 1-11/8-14/9
| pentacircle
| pentacircle
|
|
|
|-
|-
Line 171: Line 246:
| utonal
| utonal
|(^5)
|(^5)
|
|-
|-
| 18
| 18
Line 176: Line 252:
| 1-3/2-7/6
| 1-3/2-7/6
| otonal
| otonal
|
|
|
|-
|-
Line 182: Line 259:
| 1-11/6-7/6
| 1-11/6-7/6
| otonal
| otonal
|
|
|
|-
|-
Line 188: Line 266:
| 1-14/11-7/6
| 1-14/11-7/6
| utonal
| utonal
|
|
|
|-
|-
Line 194: Line 273:
| 1-14/9-7/6
| 1-14/9-7/6
| utonal
| utonal
|
|/
|1-9/7-3/2
|-
|-
| 22
| 22
Line 200: Line 280:
| 1-11/9-10/7
| 1-11/9-10/7
| swetismic
| swetismic
|
|
|
|-
|-
Line 206: Line 287:
| 1-11/6-10/7
| 1-11/6-10/7
| swetismic
| swetismic
|v4(b5)
|/(b5)
|1-9/7-7/5
|-
|-
| 24
| 24
Line 212: Line 294:
| 1-9/8-10/7
| 1-9/8-10/7
| werckismic
| werckismic
|
|
|
|-
|-
Line 218: Line 301:
| 1-14/11-10/7
| 1-14/11-10/7
| werckismic
| werckismic
|7(b5)no3
|(\~5)
|
|-
|-
| 26
| 26
Line 225: Line 309:
| swetismic
| swetismic
|~2(b5)
|~2(b5)
|1-12/11-7/5
|-
|-
| 27
| 27
Line 230: Line 315:
| 1-7/6-10/7
| 1-7/6-10/7
| swetismic
| swetismic
|~6(b5)no3
|\m(\~5)
|
|-
|-
| 28
| 28
Line 236: Line 322:
| 1-11/9-7/4
| 1-11/9-7/4
| werckismic
| werckismic
|~,\7no5
|
|
|-
|-
Line 242: Line 329:
| 1-3/2-7/4
| 1-3/2-7/4
| otonal
| otonal
|\7no3
|
|
|-
|-
Line 248: Line 336:
| 1-9/8-7/4
| 1-9/8-7/4
| otonal
| otonal
|2\7
|
|
|-
|-
Line 254: Line 343:
| 1-11/8-7/4
| 1-11/8-7/4
| otonal
| otonal
|
|
|
|-
|-
Line 260: Line 350:
| 1-14/11-7/4
| 1-14/11-7/4
| utonal
| utonal
|
|
|
|-
|-
Line 266: Line 357:
| 1-14/9-7/4
| 1-14/9-7/4
| utonal
| utonal
|
|
|
|-
|-
Line 272: Line 364:
| 1-7/6-7/4
| 1-7/6-7/4
| utonal
| utonal
|
|
|
|-
|-
Line 278: Line 371:
| 1-10/7-7/4
| 1-10/7-7/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 284: Line 378:
| 1-14/11-20/11
| 1-14/11-20/11
| otonal
| otonal
|
|
|
|-
|-
Line 290: Line 385:
| 1-14/9-20/11
| 1-14/9-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 296: Line 392:
| 1-7/6-20/11
| 1-7/6-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 303: Line 400:
| utonal
| utonal
|(b5)
|(b5)
|
|-
|-
| 40
| 40
Line 308: Line 406:
| 1-11/9-10/9
| 1-11/9-10/9
| otonal
| otonal
|
|
|
|-
|-
Line 314: Line 413:
| 1-14/11-10/9
| 1-14/11-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 320: Line 420:
| 1-14/9-10/9
| 1-14/9-10/9
| otonal
| otonal
|
|
|
|-
|-
Line 326: Line 427:
| 1-10/7-10/9
| 1-10/7-10/9
| utonal
| utonal
|
|
|
|-
|-
Line 332: Line 434:
| 1-7/4-10/9
| 1-7/4-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 338: Line 441:
| 1-20/11-10/9
| 1-20/11-10/9
| utonal
| utonal
|
|
|
|-
|-
Line 344: Line 448:
| 1-3/2-5/3
| 1-3/2-5/3
| otonal
| otonal
|
|
|
|-
|-
Line 350: Line 455:
| 1-11/6-5/3
| 1-11/6-5/3
| otonal
| otonal
|
|
|
|-
|-
Line 356: Line 462:
| 1-7/6-5/3
| 1-7/6-5/3
| otonal
| otonal
|
|
|
|-
|-
Line 362: Line 469:
| 1-10/7-5/3
| 1-10/7-5/3
| utonal
| utonal
|
|
|
|-
|-
Line 368: Line 476:
| 1-20/11-5/3
| 1-20/11-5/3
| utonal
| utonal
|
|
|
|-
|-
Line 374: Line 483:
| 1-10/9-5/3
| 1-10/9-5/3
| utonal
| utonal
|
|/7no3
|1-3/2-9/5
|-
|-
| 52
| 52
Line 380: Line 490:
| 1-3/2-5/4
| 1-3/2-5/4
| otonal
| otonal
|\
|
|
|-
|-
Line 386: Line 497:
| 1-9/8-5/4
| 1-9/8-5/4
| otonal
| otonal
|\,9no5
|
|
|-
|-
Line 392: Line 504:
| 1-11/8-5/4
| 1-11/8-5/4
| otonal
| otonal
|
|
|
|-
|-
Line 398: Line 511:
| 1-10/7-5/4
| 1-10/7-5/4
| utonal
| utonal
|\(\~5)
|
|
|-
|-
Line 404: Line 518:
| 1-7/4-5/4
| 1-7/4-5/4
| otonal
| otonal
|\7no5
|
|
|-
|-
Line 410: Line 525:
| 1-20/11-5/4
| 1-20/11-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 416: Line 532:
| 1-10/9-5/4
| 1-10/9-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 422: Line 539:
| 1-5/3-5/4
| 1-5/3-5/4
| utonal
| utonal
|
|/m
|1-6/5-3/2
|}
|}


Line 433: Line 551:
! Type
! Type
!Name
!Name
!Inversion
|-
|-
| 1
| 1
Line 439: Line 558:
| rastmic
| rastmic
|~7
|~7
|
|-
|-
| 2
| 2
Line 445: Line 565:
| rastmic
| rastmic
|~,9
|~,9
|
|-
|-
| 3
| 3
Line 451: Line 572:
| rastmic
| rastmic
|~9no5
|~9no5
|
|-
|-
| 4
| 4
Line 457: Line 579:
| rastmic
| rastmic
|~9no3
|~9no3
|
|-
|-
| 5
| 5
Line 462: Line 585:
| 1-11/9-3/2-11/8
| 1-11/9-3/2-11/8
| rastmic
| rastmic
|~6,9
|~,~11
|
|-
|-
| 6
| 6
Line 468: Line 592:
| 1-11/9-11/6-11/8
| 1-11/9-11/6-11/8
| utonal
| utonal
|
|
|
|-
|-
Line 474: Line 599:
| 1-3/2-11/6-11/8
| 1-3/2-11/6-11/8
| ambitonal
| ambitonal
|
|
|
|-
|-
Line 480: Line 606:
| 1-11/9-9/8-11/8
| 1-11/9-9/8-11/8
| rastmic
| rastmic
|
|
|
|-
|-
Line 486: Line 613:
| 1-3/2-9/8-11/8
| 1-3/2-9/8-11/8
| otonal
| otonal
|
|
|
|-
|-
Line 493: Line 621:
| rastmic
| rastmic
|~11no35
|~11no35
|
|-
|-
| 11
| 11
Line 499: Line 628:
| parahemif
| parahemif
|9(~7)no5
|9(~7)no5
|
|-
|-
| 12
| 12
Line 504: Line 634:
| 1-11/6-11/8-14/11
| 1-11/6-11/8-14/11
| hemimin
| hemimin
|,~7~11no5
|~2~11
|1-12/11-11/8-3/2
|-
|-
| 13
| 13
Line 510: Line 641:
| 1-9/8-11/8-14/11
| 1-9/8-11/8-14/11
| parahemif
| parahemif
|
|
|
|-
|-
Line 516: Line 648:
| 1-11/9-9/8-14/9
| 1-11/9-9/8-14/9
| parahemif
| parahemif
|
|
|
|-
|-
Line 522: Line 655:
| 1-11/9-11/8-14/9
| 1-11/9-11/8-14/9
| pentacircle
| pentacircle
|,~6,9no5
|
|
|-
|-
| 16
| 16
Line 528: Line 662:
| 1-9/8-11/8-14/9
| 1-9/8-11/8-14/9
| pentacircle
| pentacircle
|
|
|
|-
|-
Line 534: Line 669:
| 1-9/8-14/11-14/9
| 1-9/8-14/11-14/9
| pentacircle
| pentacircle
|
|
|
|-
|-
Line 540: Line 676:
| 1-11/8-14/11-14/9
| 1-11/8-14/11-14/9
| parahemif
| parahemif
|
|
|
|-
|-
Line 546: Line 683:
| 1-3/2-11/6-7/6
| 1-3/2-11/6-7/6
| otonal
| otonal
|
|
|
|-
|-
Line 552: Line 690:
| 1-11/6-14/11-7/6
| 1-11/6-14/11-7/6
| hemimin
| hemimin
|
|
|
|-
|-
Line 558: Line 697:
| 1-14/11-14/9-7/6
| 1-14/11-14/9-7/6
| utonal
| utonal
|
|
|
|-
|-
Line 564: Line 704:
| 1-11/9-11/6-10/7
| 1-11/9-11/6-10/7
| swetismic
| swetismic
|
|
|
|-
|-
Line 570: Line 711:
| 1-11/9-9/8-10/7
| 1-11/9-9/8-10/7
| jove
| jove
|
|
|
|-
|-
Line 576: Line 718:
| 1-11/6-9/8-10/7
| 1-11/6-9/8-10/7
| jove
| jove
|
|
|
|-
|-
Line 582: Line 725:
| 1-11/6-14/11-10/7
| 1-11/6-14/11-10/7
| hemififths
| hemififths
|
|
|
|-
|-
Line 588: Line 732:
| 1-9/8-14/11-10/7
| 1-9/8-14/11-10/7
| pele
| pele
|
|
|
|-
|-
Line 594: Line 739:
| 1-11/9-14/9-10/7
| 1-11/9-14/9-10/7
| swetismic
| swetismic
|
|
|
|-
|-
Line 600: Line 746:
| 1-9/8-14/9-10/7
| 1-9/8-14/9-10/7
| hemififths
| hemififths
|
|
|
|-
|-
Line 606: Line 753:
| 1-14/11-14/9-10/7
| 1-14/11-14/9-10/7
| jove
| jove
|
|
|
|-
|-
Line 612: Line 760:
| 1-11/6-7/6-10/7
| 1-11/6-7/6-10/7
| swetismic
| swetismic
|
|
|
|-
|-
Line 618: Line 767:
| 1-14/11-7/6-10/7
| 1-14/11-7/6-10/7
| jove
| jove
|
|
|
|-
|-
Line 624: Line 774:
| 1-14/9-7/6-10/7
| 1-14/9-7/6-10/7
| swetismic
| swetismic
|
|
|
|-
|-
Line 630: Line 781:
| 1-11/9-3/2-7/4
| 1-11/9-3/2-7/4
| jove
| jove
|
|
|
|-
|-
Line 636: Line 788:
| 1-11/9-9/8-7/4
| 1-11/9-9/8-7/4
| jove
| jove
|
|
|
|-
|-
Line 642: Line 795:
| 1-3/2-9/8-7/4
| 1-3/2-9/8-7/4
| otonal
| otonal
|
|
|
|-
|-
Line 648: Line 802:
| 1-11/9-11/8-7/4
| 1-11/9-11/8-7/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 654: Line 809:
| 1-3/2-11/8-7/4
| 1-3/2-11/8-7/4
| otonal
| otonal
|
|
|
|-
|-
Line 660: Line 816:
| 1-9/8-11/8-7/4
| 1-9/8-11/8-7/4
| otonal
| otonal
|
|
|
|-
|-
Line 666: Line 823:
| 1-9/8-14/11-7/4
| 1-9/8-14/11-7/4
| pentacircle
| pentacircle
|
|
|
|-
|-
Line 672: Line 830:
| 1-11/8-14/11-7/4
| 1-11/8-14/11-7/4
| hemimin
| hemimin
|
|
|
|-
|-
Line 678: Line 837:
| 1-11/9-14/9-7/4
| 1-11/9-14/9-7/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 684: Line 844:
| 1-9/8-14/9-7/4
| 1-9/8-14/9-7/4
| pentacircle
| pentacircle
|
|
|
|-
|-
Line 690: Line 851:
| 1-11/8-14/9-7/4
| 1-11/8-14/9-7/4
| pentacircle
| pentacircle
|
|
|
|-
|-
Line 696: Line 858:
| 1-14/11-14/9-7/4
| 1-14/11-14/9-7/4
| utonal
| utonal
|
|
|
|-
|-
Line 702: Line 865:
| 1-3/2-7/6-7/4
| 1-3/2-7/6-7/4
| ambitonal
| ambitonal
|
|
|
|-
|-
Line 708: Line 872:
| 1-14/11-7/6-7/4
| 1-14/11-7/6-7/4
| utonal
| utonal
|
|
|
|-
|-
Line 714: Line 879:
| 1-14/9-7/6-7/4
| 1-14/9-7/6-7/4
| utonal
| utonal
|
|
|
|-
|-
Line 720: Line 886:
| 1-11/9-10/7-7/4
| 1-11/9-10/7-7/4
| jove
| jove
|
|
|
|-
|-
Line 726: Line 893:
| 1-9/8-10/7-7/4
| 1-9/8-10/7-7/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 732: Line 900:
| 1-14/11-10/7-7/4
| 1-14/11-10/7-7/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 738: Line 907:
| 1-14/9-10/7-7/4
| 1-14/9-10/7-7/4
| jove
| jove
|
|
|
|-
|-
Line 744: Line 914:
| 1-7/6-10/7-7/4
| 1-7/6-10/7-7/4
| jove
| jove
|
|
|
|-
|-
Line 750: Line 921:
| 1-14/11-14/9-20/11
| 1-14/11-14/9-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 756: Line 928:
| 1-14/11-7/6-20/11
| 1-14/11-7/6-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 762: Line 935:
| 1-14/9-7/6-20/11
| 1-14/9-7/6-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 768: Line 942:
| 1-14/11-10/7-20/11
| 1-14/11-10/7-20/11
| werckismic
| werckismic
|
|
|
|-
|-
Line 774: Line 949:
| 1-14/9-10/7-20/11
| 1-14/9-10/7-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 780: Line 956:
| 1-7/6-10/7-20/11
| 1-7/6-10/7-20/11
| swetismic
| swetismic
|
|
|
|-
|-
Line 786: Line 963:
| 1-11/9-14/9-10/9
| 1-11/9-14/9-10/9
| otonal
| otonal
|
|
|
|-
|-
Line 792: Line 970:
| 1-14/11-14/9-10/9
| 1-14/11-14/9-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 798: Line 977:
| 1-11/9-10/7-10/9
| 1-11/9-10/7-10/9
| swetismic
| swetismic
|
|
|
|-
|-
Line 804: Line 984:
| 1-14/11-10/7-10/9
| 1-14/11-10/7-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 810: Line 991:
| 1-14/9-10/7-10/9
| 1-14/9-10/7-10/9
| swetismic
| swetismic
|
|
|
|-
|-
Line 816: Line 998:
| 1-11/9-7/4-10/9
| 1-11/9-7/4-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 822: Line 1,005:
| 1-14/11-7/4-10/9
| 1-14/11-7/4-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 828: Line 1,012:
| 1-14/9-7/4-10/9
| 1-14/9-7/4-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 834: Line 1,019:
| 1-10/7-7/4-10/9
| 1-10/7-7/4-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 840: Line 1,026:
| 1-14/11-20/11-10/9
| 1-14/11-20/11-10/9
| werckismic
| werckismic
|
|
|
|-
|-
Line 846: Line 1,033:
| 1-14/9-20/11-10/9
| 1-14/9-20/11-10/9
| swetismic
| swetismic
|
|
|
|-
|-
Line 852: Line 1,040:
| 1-10/7-20/11-10/9
| 1-10/7-20/11-10/9
| utonal
| utonal
|
|
|
|-
|-
Line 858: Line 1,047:
| 1-3/2-11/6-5/3
| 1-3/2-11/6-5/3
| otonal
| otonal
|
|
|
|-
|-
Line 864: Line 1,054:
| 1-3/2-7/6-5/3
| 1-3/2-7/6-5/3
| otonal
| otonal
|
|
|
|-
|-
Line 870: Line 1,061:
| 1-11/6-7/6-5/3
| 1-11/6-7/6-5/3
| otonal
| otonal
|
|
|
|-
|-
Line 876: Line 1,068:
| 1-11/6-10/7-5/3
| 1-11/6-10/7-5/3
| swetismic
| swetismic
|
|
|
|-
|-
Line 882: Line 1,075:
| 1-7/6-10/7-5/3
| 1-7/6-10/7-5/3
| swetismic
| swetismic
|
|
|
|-
|-
Line 888: Line 1,082:
| 1-7/6-20/11-5/3
| 1-7/6-20/11-5/3
| swetismic
| swetismic
|
|
|
|-
|-
Line 894: Line 1,089:
| 1-10/7-20/11-5/3
| 1-10/7-20/11-5/3
| utonal
| utonal
|
|
|
|-
|-
Line 900: Line 1,096:
| 1-10/7-10/9-5/3
| 1-10/7-10/9-5/3
| utonal
| utonal
|
|
|
|-
|-
Line 906: Line 1,103:
| 1-20/11-10/9-5/3
| 1-20/11-10/9-5/3
| utonal
| utonal
|
|
|
|-
|-
Line 912: Line 1,110:
| 1-3/2-9/8-5/4
| 1-3/2-9/8-5/4
| otonal
| otonal
|\,9
|
|
|-
|-
Line 918: Line 1,117:
| 1-3/2-11/8-5/4
| 1-3/2-11/8-5/4
| otonal
| otonal
|
|
|
|-
|-
Line 924: Line 1,124:
| 1-9/8-11/8-5/4
| 1-9/8-11/8-5/4
| otonal
| otonal
|
|
|
|-
|-
Line 930: Line 1,131:
| 1-9/8-10/7-5/4
| 1-9/8-10/7-5/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 936: Line 1,138:
| 1-3/2-7/4-5/4
| 1-3/2-7/4-5/4
| otonal
| otonal
|\7
|
|
|-
|-
Line 942: Line 1,145:
| 1-9/8-7/4-5/4
| 1-9/8-7/4-5/4
| otonal
| otonal
|
|
|
|-
|-
Line 948: Line 1,152:
| 1-11/8-7/4-5/4
| 1-11/8-7/4-5/4
| otonal
| otonal
|
|
|
|-
|-
Line 954: Line 1,159:
| 1-10/7-7/4-5/4
| 1-10/7-7/4-5/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 960: Line 1,166:
| 1-10/7-20/11-5/4
| 1-10/7-20/11-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 966: Line 1,173:
| 1-10/7-10/9-5/4
| 1-10/7-10/9-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 972: Line 1,180:
| 1-7/4-10/9-5/4
| 1-7/4-10/9-5/4
| werckismic
| werckismic
|
|
|
|-
|-
Line 978: Line 1,187:
| 1-20/11-10/9-5/4
| 1-20/11-10/9-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 984: Line 1,194:
| 1-3/2-5/3-5/4
| 1-3/2-5/3-5/4
| ambitonal
| ambitonal
|
|
|
|-
|-
Line 990: Line 1,201:
| 1-10/7-5/3-5/4
| 1-10/7-5/3-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 996: Line 1,208:
| 1-20/11-5/3-5/4
| 1-20/11-5/3-5/4
| utonal
| utonal
|
|
|
|-
|-
Line 1,002: Line 1,215:
| 1-10/9-5/3-5/4
| 1-10/9-5/3-5/4
| utonal
| utonal
|
|
|
|}
|}
Line 1,024: Line 1,238:
| 1-11/9-3/2-11/6-11/8
| 1-11/9-3/2-11/6-11/8
| rastmic
| rastmic
|~7,~11
|~11no9
|-
|-
| 3
| 3
Line 1,030: Line 1,244:
| 1-11/9-3/2-9/8-11/8
| 1-11/9-3/2-9/8-11/8
| rastmic
| rastmic
|~,9~11 <u>or</u> ~11no7
|~11no7
|-
|-
| 4
| 4

Revision as of 14:33, 16 November 2024

Below are listed the dyadic chords of 11-limit hemififths temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 441/440 werckismic, by 896/891 pentacircle, by 243/242 rastmic, and by 1344/1331 hemimin. Those requiring tempering by any two of 540/539, 441/440 or 243/242 are labeled jove. Those requiring both 441/440 and 896/891 are labeled pele. Those requiring any two of 243/242, 896/891 or 1344/1331 are labeled parahemif. If the full hemififths is required because of the tempering out of three independent hemififths commas, the chord is labeled "hemififths".

A striking feature of these hemififths chords is that essentially just chords tend to be of higher complexity than essentially tempered chords. Hemififths has MOS of size 7, 10, 17 and 24, and even seven notes are well-supplied with chords, mostly but by no means entirely essentially tempered chords. Extending consideration to the 13-limit adds even more such chords.

Chords are named with ups and downs. The tilde ~ means mid, half-way between major and minor. ~4 = ^4 = vA4. The comma (the actual punctuation mark) is pronounced "add", thus C~,7 is "C mid add 7". To facilitate chord naming, lifts and drops are also used. Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ - 8.5c, where c = the amount in cents the tempered fifth exceeds 700¢. /1 = ~81/80 = ~64/63 and ^ = ~33/31.

Hemififth's genchain
Genspan 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ... 20 21 ... 23 ... 25
Cents (41edo) 0 351 702 1054 205 556 907 59 410 761 1112 263 615 966 1024 176 878 380
Ratio 1/1 11/9

16/13

3/2 11/6

24/13

9/8 11/8

18/13

27/16

22/13

28/27

33/32

14/11 14/9 40/21

21/11

7/6 10/7 7/4 20/11 10/9 5/3 5/4
Interval P1 ~3 P5 ~7 M2 ~4

\b5

M6 ^1

\m2

M3 ^5

\m6

M7 ^M2

\m3

A4

\~5

^M6

\m7

A6

\~7

^A1

\M2

^A5

\M6

^A2

\M3

Triads

# Chord Transversal Type Name Inversion
1 0-1-2 1-11/9-3/2 rastmic ~
2 0-1-3 1-11/9-11/6 utonal ~7no5
3 0-2-3 1-3/2-11/6 otonal ~7no3
4 0-1-4 1-11/9-9/8 rastmic ~,9no5
5 0-2-4 1-3/2-9/8 ambitonal 2 or 4 1-4/3-9/8
6 0-3-4 1-11/6-9/8 rastmic
7 0-1-5 1-11/9-11/8 utonal ~(\b5)
8 0-2-5 1-3/2-11/8 otonal ~7sus4no5 1-4/3-11/6
9 0-3-5 1-11/6-11/8 utonal ~2 1-12/11-3/2
10 0-4-5 1-9/8-11/8 otonal ~,7no5 1-11/9-16/9
11 0-3-8 1-11/6-14/11 hemimin ,~7no5
12 0-4-8 1-9/8-14/11 pentacircle ,9no5
13 0-5-8 1-11/8-14/11 hemimin (\b5)
14 0-1-9 1-11/9-14/9 otonal ~(^5)
15 0-4-9 1-9/8-14/9 pentacircle 7~4no5 1-11/8-16/9
16 0-5-9 1-11/8-14/9 pentacircle
17 0-8-9 1-14/11-14/9 utonal (^5)
18 0-2-11 1-3/2-7/6 otonal
19 0-3-11 1-11/6-7/6 otonal
20 0-8-11 1-14/11-7/6 utonal
21 0-9-11 1-14/9-7/6 utonal / 1-9/7-3/2
22 0-1-12 1-11/9-10/7 swetismic
23 0-3-12 1-11/6-10/7 swetismic /(b5) 1-9/7-7/5
24 0-4-12 1-9/8-10/7 werckismic
25 0-8-12 1-14/11-10/7 werckismic (\~5)
26 0-9-12 1-14/9-10/7 swetismic ~2(b5) 1-12/11-7/5
27 0-11-12 1-7/6-10/7 swetismic \m(\~5)
28 0-1-13 1-11/9-7/4 werckismic ~,\7no5
29 0-2-13 1-3/2-7/4 otonal \7no3
30 0-4-13 1-9/8-7/4 otonal 2\7
31 0-5-13 1-11/8-7/4 otonal
32 0-8-13 1-14/11-7/4 utonal
33 0-9-13 1-14/9-7/4 utonal
34 0-11-13 1-7/6-7/4 utonal
35 0-12-13 1-10/7-7/4 werckismic
36 0-8-20 1-14/11-20/11 otonal
37 0-9-20 1-14/9-20/11 swetismic
38 0-11-20 1-7/6-20/11 swetismic
39 0-12-20 1-10/7-20/11 utonal (b5)
40 0-1-21 1-11/9-10/9 otonal
41 0-8-21 1-14/11-10/9 werckismic
42 0-9-21 1-14/9-10/9 otonal
43 0-12-21 1-10/7-10/9 utonal
44 0-13-21 1-7/4-10/9 werckismic
45 0-20-21 1-20/11-10/9 utonal
46 0-2-23 1-3/2-5/3 otonal
47 0-3-23 1-11/6-5/3 otonal
48 0-11-23 1-7/6-5/3 otonal
49 0-12-23 1-10/7-5/3 utonal
50 0-20-23 1-20/11-5/3 utonal
51 0-21-23 1-10/9-5/3 utonal /7no3 1-3/2-9/5
52 0-2-25 1-3/2-5/4 otonal \
53 0-4-25 1-9/8-5/4 otonal \,9no5
54 0-5-25 1-11/8-5/4 otonal
55 0-12-25 1-10/7-5/4 utonal \(\~5)
56 0-13-25 1-7/4-5/4 otonal \7no5
57 0-20-25 1-20/11-5/4 utonal
58 0-21-25 1-10/9-5/4 utonal
59 0-23-25 1-5/3-5/4 utonal /m 1-6/5-3/2

Tetrads

# Chord Transversal Type Name Inversion
1 0-1-2-3 1-11/9-3/2-11/6 rastmic ~7
2 0-1-2-4 1-11/9-3/2-9/8 rastmic ~,9
3 0-1-3-4 1-11/9-11/6-9/8 rastmic ~9no5
4 0-2-3-4 1-3/2-11/6-9/8 rastmic ~9no3
5 0-1-2-5 1-11/9-3/2-11/8 rastmic ~,~11
6 0-1-3-5 1-11/9-11/6-11/8 utonal
7 0-2-3-5 1-3/2-11/6-11/8 ambitonal
8 0-1-4-5 1-11/9-9/8-11/8 rastmic
9 0-2-4-5 1-3/2-9/8-11/8 otonal
10 0-3-4-5 1-11/6-9/8-11/8 rastmic ~11no35
11 0-3-4-8 1-11/6-9/8-14/11 parahemif 9(~7)no5
12 0-3-5-8 1-11/6-11/8-14/11 hemimin ~2~11 1-12/11-11/8-3/2
13 0-4-5-8 1-9/8-11/8-14/11 parahemif
14 0-1-4-9 1-11/9-9/8-14/9 parahemif
15 0-1-5-9 1-11/9-11/8-14/9 pentacircle
16 0-4-5-9 1-9/8-11/8-14/9 pentacircle
17 0-4-8-9 1-9/8-14/11-14/9 pentacircle
18 0-5-8-9 1-11/8-14/11-14/9 parahemif
19 0-2-3-11 1-3/2-11/6-7/6 otonal
20 0-3-8-11 1-11/6-14/11-7/6 hemimin
21 0-8-9-11 1-14/11-14/9-7/6 utonal
22 0-1-3-12 1-11/9-11/6-10/7 swetismic
23 0-1-4-12 1-11/9-9/8-10/7 jove
24 0-3-4-12 1-11/6-9/8-10/7 jove
25 0-3-8-12 1-11/6-14/11-10/7 hemififths
26 0-4-8-12 1-9/8-14/11-10/7 pele
27 0-1-9-12 1-11/9-14/9-10/7 swetismic
28 0-4-9-12 1-9/8-14/9-10/7 hemififths
29 0-8-9-12 1-14/11-14/9-10/7 jove
30 0-3-11-12 1-11/6-7/6-10/7 swetismic
31 0-8-11-12 1-14/11-7/6-10/7 jove
32 0-9-11-12 1-14/9-7/6-10/7 swetismic
33 0-1-2-13 1-11/9-3/2-7/4 jove
34 0-1-4-13 1-11/9-9/8-7/4 jove
35 0-2-4-13 1-3/2-9/8-7/4 otonal
36 0-1-5-13 1-11/9-11/8-7/4 werckismic
37 0-2-5-13 1-3/2-11/8-7/4 otonal
38 0-4-5-13 1-9/8-11/8-7/4 otonal
39 0-4-8-13 1-9/8-14/11-7/4 pentacircle
40 0-5-8-13 1-11/8-14/11-7/4 hemimin
41 0-1-9-13 1-11/9-14/9-7/4 werckismic
42 0-4-9-13 1-9/8-14/9-7/4 pentacircle
43 0-5-9-13 1-11/8-14/9-7/4 pentacircle
44 0-8-9-13 1-14/11-14/9-7/4 utonal
45 0-2-11-13 1-3/2-7/6-7/4 ambitonal
46 0-8-11-13 1-14/11-7/6-7/4 utonal
47 0-9-11-13 1-14/9-7/6-7/4 utonal
48 0-1-12-13 1-11/9-10/7-7/4 jove
49 0-4-12-13 1-9/8-10/7-7/4 werckismic
50 0-8-12-13 1-14/11-10/7-7/4 werckismic
51 0-9-12-13 1-14/9-10/7-7/4 jove
52 0-11-12-13 1-7/6-10/7-7/4 jove
53 0-8-9-20 1-14/11-14/9-20/11 swetismic
54 0-8-11-20 1-14/11-7/6-20/11 swetismic
55 0-9-11-20 1-14/9-7/6-20/11 swetismic
56 0-8-12-20 1-14/11-10/7-20/11 werckismic
57 0-9-12-20 1-14/9-10/7-20/11 swetismic
58 0-11-12-20 1-7/6-10/7-20/11 swetismic
59 0-1-9-21 1-11/9-14/9-10/9 otonal
60 0-8-9-21 1-14/11-14/9-10/9 werckismic
61 0-1-12-21 1-11/9-10/7-10/9 swetismic
62 0-8-12-21 1-14/11-10/7-10/9 werckismic
63 0-9-12-21 1-14/9-10/7-10/9 swetismic
64 0-1-13-21 1-11/9-7/4-10/9 werckismic
65 0-8-13-21 1-14/11-7/4-10/9 werckismic
66 0-9-13-21 1-14/9-7/4-10/9 werckismic
67 0-12-13-21 1-10/7-7/4-10/9 werckismic
68 0-8-20-21 1-14/11-20/11-10/9 werckismic
69 0-9-20-21 1-14/9-20/11-10/9 swetismic
70 0-12-20-21 1-10/7-20/11-10/9 utonal
71 0-2-3-23 1-3/2-11/6-5/3 otonal
72 0-2-11-23 1-3/2-7/6-5/3 otonal
73 0-3-11-23 1-11/6-7/6-5/3 otonal
74 0-3-12-23 1-11/6-10/7-5/3 swetismic
75 0-11-12-23 1-7/6-10/7-5/3 swetismic
76 0-11-20-23 1-7/6-20/11-5/3 swetismic
77 0-12-20-23 1-10/7-20/11-5/3 utonal
78 0-12-21-23 1-10/7-10/9-5/3 utonal
79 0-20-21-23 1-20/11-10/9-5/3 utonal
80 0-2-4-25 1-3/2-9/8-5/4 otonal \,9
81 0-2-5-25 1-3/2-11/8-5/4 otonal
82 0-4-5-25 1-9/8-11/8-5/4 otonal
83 0-4-12-25 1-9/8-10/7-5/4 werckismic
84 0-2-13-25 1-3/2-7/4-5/4 otonal \7
85 0-4-13-25 1-9/8-7/4-5/4 otonal
86 0-5-13-25 1-11/8-7/4-5/4 otonal
87 0-12-13-25 1-10/7-7/4-5/4 werckismic
88 0-12-20-25 1-10/7-20/11-5/4 utonal
89 0-12-21-25 1-10/7-10/9-5/4 utonal
90 0-13-21-25 1-7/4-10/9-5/4 werckismic
91 0-20-21-25 1-20/11-10/9-5/4 utonal
92 0-2-23-25 1-3/2-5/3-5/4 ambitonal
93 0-12-23-25 1-10/7-5/3-5/4 utonal
94 0-20-23-25 1-20/11-5/3-5/4 utonal
95 0-21-23-25 1-10/9-5/3-5/4 utonal

Pentads

# Chord Transversal Type Name
1 0-1-2-3-4 1-11/9-3/2-11/6-9/8 rastmic ~9
2 0-1-2-3-5 1-11/9-3/2-11/6-11/8 rastmic ~11no9
3 0-1-2-4-5 1-11/9-3/2-9/8-11/8 rastmic ~11no7
4 0-1-3-4-5 1-11/9-11/6-9/8-11/8 rastmic ~11no5
5 0-2-3-4-5 1-3/2-11/6-9/8-11/8 rastmic ~11no3
6 0-3-4-5-8 1-11/6-9/8-11/8-14/11 parahemif
7 0-1-4-5-9 1-11/9-9/8-11/8-14/9 parahemif
8 0-4-5-8-9 1-9/8-11/8-14/11-14/9 parahemif
9 0-1-3-4-12 1-11/9-11/6-9/8-10/7 jove
10 0-3-4-8-12 1-11/6-9/8-14/11-10/7 hemififths
11 0-1-4-9-12 1-11/9-9/8-14/9-10/7 hemififths
12 0-4-8-9-12 1-9/8-14/11-14/9-10/7 hemififths
13 0-3-8-11-12 1-11/6-14/11-7/6-10/7 hemififths
14 0-8-9-11-12 1-14/11-14/9-7/6-10/7 jove
15 0-1-2-4-13 1-11/9-3/2-9/8-7/4 jove
16 0-1-2-5-13 1-11/9-3/2-11/8-7/4 jove
17 0-1-4-5-13 1-11/9-9/8-11/8-7/4 jove
18 0-2-4-5-13 1-3/2-9/8-11/8-7/4 otonal
19 0-4-5-8-13 1-9/8-11/8-14/11-7/4 parahemif
20 0-1-4-9-13 1-11/9-9/8-14/9-7/4 hemififths
21 0-1-5-9-13 1-11/9-11/8-14/9-7/4 pele
22 0-4-5-9-13 1-9/8-11/8-14/9-7/4 pentacircle
23 0-4-8-9-13 1-9/8-14/11-14/9-7/4 pentacircle
24 0-5-8-9-13 1-11/8-14/11-14/9-7/4 parahemif
25 0-8-9-11-13 1-14/11-14/9-7/6-7/4 utonal
26 0-1-4-12-13 1-11/9-9/8-10/7-7/4 jove
27 0-4-8-12-13 1-9/8-14/11-10/7-7/4 pele
28 0-1-9-12-13 1-11/9-14/9-10/7-7/4 jove
29 0-4-9-12-13 1-9/8-14/9-10/7-7/4 hemififths
30 0-8-9-12-13 1-14/11-14/9-10/7-7/4 jove
31 0-8-11-12-13 1-14/11-7/6-10/7-7/4 jove
32 0-9-11-12-13 1-14/9-7/6-10/7-7/4 jove
33 0-8-9-11-20 1-14/11-14/9-7/6-20/11 swetismic
34 0-8-9-12-20 1-14/11-14/9-10/7-20/11 jove
35 0-8-11-12-20 1-14/11-7/6-10/7-20/11 jove
36 0-9-11-12-20 1-14/9-7/6-10/7-20/11 swetismic
37 0-1-9-12-21 1-11/9-14/9-10/7-10/9 swetismic
38 0-8-9-12-21 1-14/11-14/9-10/7-10/9 jove
39 0-1-9-13-21 1-11/9-14/9-7/4-10/9 werckismic
40 0-8-9-13-21 1-14/11-14/9-7/4-10/9 werckismic
41 0-1-12-13-21 1-11/9-10/7-7/4-10/9 jove
42 0-8-12-13-21 1-14/11-10/7-7/4-10/9 werckismic
43 0-9-12-13-21 1-14/9-10/7-7/4-10/9 jove
44 0-8-9-20-21 1-14/11-14/9-20/11-10/9 jove
45 0-8-12-20-21 1-14/11-10/7-20/11-10/9 werckismic
46 0-9-12-20-21 1-14/9-10/7-20/11-10/9 swetismic
47 0-2-3-11-23 1-3/2-11/6-7/6-5/3 otonal
48 0-3-11-12-23 1-11/6-7/6-10/7-5/3 swetismic
49 0-11-12-20-23 1-7/6-10/7-20/11-5/3 swetismic
50 0-12-20-21-23 1-10/7-20/11-10/9-5/3 utonal
51 0-2-4-5-25 1-3/2-9/8-11/8-5/4 otonal
52 0-2-4-13-25 1-3/2-9/8-7/4-5/4 otonal
53 0-2-5-13-25 1-3/2-11/8-7/4-5/4 otonal
54 0-4-5-13-25 1-9/8-11/8-7/4-5/4 otonal
55 0-4-12-13-25 1-9/8-10/7-7/4-5/4 werckismic
56 0-12-13-21-25 1-10/7-7/4-10/9-5/4 werckismic
57 0-12-20-21-25 1-10/7-20/11-10/9-5/4 utonal
58 0-12-20-23-25 1-10/7-20/11-5/3-5/4 utonal
59 0-12-21-23-25 1-10/7-10/9-5/3-5/4 utonal
60 0-20-21-23-25 1-20/11-10/9-5/3-5/4 utonal

Hexads

# Chord Transversal Type Name
1 0-1-2-3-4-5 1-11/9-3/2-11/6-9/8-11/8 rastmic ~11
2 0-1-2-4-5-13 1-11/9-3/2-9/8-11/8-7/4 jove
3 0-1-4-5-9-13 1-11/9-9/8-11/8-14/9-7/4 hemififths
4 0-4-5-8-9-13 1-9/8-11/8-14/11-14/9-7/4 nofives
5 0-1-4-9-12-13 1-11/9-9/8-14/9-10/7-7/4 hemififths
6 0-4-8-9-12-13 1-9/8-14/11-14/9-10/7-7/4 hemififths
7 0-8-9-11-12-13 1-14/11-14/9-7/6-10/7-7/4 jove
8 0-8-9-11-12-20 1-14/11-14/9-7/6-10/7-20/11 jove
9 0-1-9-12-13-21 1-11/9-14/9-10/7-7/4-10/9 jove
10 0-8-9-12-13-21 1-14/11-14/9-10/7-7/4-10/9 jove
11 0-8-9-12-20-21 1-14/11-14/9-10/7-20/11-10/9 jove
12 0-2-4-5-13-25 1-3/2-9/8-11/8-7/4-5/4 otonal
13 0-12-20-21-23-25 1-10/7-20/11-10/9-5/3-5/4 utonal
Retrieved from "https://en.xen.wiki/index.php?title=Hemififths/Chords&oldid=166362"