Hemififths/Chords: Difference between revisions
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[[Category:Lists of chords]] | [[Category:Lists of chords]] | ||
[[Category:Hemififths]] | [[Category:Hemififths]] | ||
[[Category: | [[Category:Triads]] | ||
[[Category: | [[Category:Tetrads]] | ||
[[Category: | [[Category:Pentads]] | ||
[[Category: | [[Category:Hexads]] | ||
Revision as of 11:36, 23 May 2023
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.
Triads
| # | Chord | Transversal | Type |
|---|---|---|---|
| 1 | 0-1-2 | 1-11/9-3/2 | rastmic |
| 2 | 0-1-3 | 1-11/9-11/6 | utonal |
| 3 | 0-2-3 | 1-3/2-11/6 | otonal |
| 4 | 0-1-4 | 1-11/9-9/8 | rastmic |
| 5 | 0-2-4 | 1-3/2-9/8 | ambitonal |
| 6 | 0-3-4 | 1-11/6-9/8 | rastmic |
| 7 | 0-1-5 | 1-11/9-11/8 | utonal |
| 8 | 0-2-5 | 1-3/2-11/8 | otonal |
| 9 | 0-3-5 | 1-11/6-11/8 | utonal |
| 10 | 0-4-5 | 1-9/8-11/8 | otonal |
| 11 | 0-3-8 | 1-11/6-14/11 | hemimin |
| 12 | 0-4-8 | 1-9/8-14/11 | pentacircle |
| 13 | 0-5-8 | 1-11/8-14/11 | hemimin |
| 14 | 0-1-9 | 1-11/9-14/9 | otonal |
| 15 | 0-4-9 | 1-9/8-14/9 | pentacircle |
| 16 | 0-5-9 | 1-11/8-14/9 | pentacircle |
| 17 | 0-8-9 | 1-14/11-14/9 | utonal |
| 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 |
| 22 | 0-1-12 | 1-11/9-10/7 | swetismic |
| 23 | 0-3-12 | 1-11/6-10/7 | swetismic |
| 24 | 0-4-12 | 1-9/8-10/7 | werckismic |
| 25 | 0-8-12 | 1-14/11-10/7 | werckismic |
| 26 | 0-9-12 | 1-14/9-10/7 | swetismic |
| 27 | 0-11-12 | 1-7/6-10/7 | swetismic |
| 28 | 0-1-13 | 1-11/9-7/4 | werckismic |
| 29 | 0-2-13 | 1-3/2-7/4 | otonal |
| 30 | 0-4-13 | 1-9/8-7/4 | otonal |
| 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 |
| 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 |
| 52 | 0-2-25 | 1-3/2-5/4 | otonal |
| 53 | 0-4-25 | 1-9/8-5/4 | otonal |
| 54 | 0-5-25 | 1-11/8-5/4 | otonal |
| 55 | 0-12-25 | 1-10/7-5/4 | utonal |
| 56 | 0-13-25 | 1-7/4-5/4 | otonal |
| 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 |
Tetrads
| # | Chord | Transversal | Type |
|---|---|---|---|
| 1 | 0-1-2-3 | 1-11/9-3/2-11/6 | rastmic |
| 2 | 0-1-2-4 | 1-11/9-3/2-9/8 | rastmic |
| 3 | 0-1-3-4 | 1-11/9-11/6-9/8 | rastmic |
| 4 | 0-2-3-4 | 1-3/2-11/6-9/8 | rastmic |
| 5 | 0-1-2-5 | 1-11/9-3/2-11/8 | rastmic |
| 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 |
| 11 | 0-3-4-8 | 1-11/6-9/8-14/11 | parahemif |
| 12 | 0-3-5-8 | 1-11/6-11/8-14/11 | hemimin |
| 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 |
| 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 |
| 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 |
|---|---|---|---|
| 1 | 0-1-2-3-4 | 1-11/9-3/2-11/6-9/8 | rastmic |
| 2 | 0-1-2-3-5 | 1-11/9-3/2-11/6-11/8 | rastmic |
| 3 | 0-1-2-4-5 | 1-11/9-3/2-9/8-11/8 | rastmic |
| 4 | 0-1-3-4-5 | 1-11/9-11/6-9/8-11/8 | rastmic |
| 5 | 0-2-3-4-5 | 1-3/2-11/6-9/8-11/8 | rastmic |
| 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 |
|---|---|---|---|
| 1 | 0-1-2-3-4-5 | 1-11/9-3/2-11/6-9/8-11/8 | rastmic |
| 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 |