Trisected: Difference between revisions
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== Intervals == | == Intervals == | ||
{ | In the following table, odd harmonics 1–21 are in '''bold'''. | ||
{| class="wikitable center-1 right-2 right-4 right-6" | |||
|- | |||
! rowspan="2" | # | |||
! colspan="2" | Period 0 | |||
! colspan="2" | Period 1 | |||
! colspan="2" | Period 2 | |||
|- | |||
! Cents* | |||
! Approx. ratios | |||
! Cents* | |||
! Approx. ratios | |||
! Cents* | |||
! Approx. ratios | |||
|- | |||
| 0 | |||
| 0.0 | |||
| '''1/1''' | |||
| 400.0 | |||
| '''5/4''', 14/11 | |||
| 800.0 | |||
| '''8/5''', 11/7 | |||
|- | |||
| 1 | |||
| 235.0 | |||
| '''8/7''' | |||
| 635.0 | |||
| 10/7, '''16/11''', 13/9 | |||
| 1035.0 | |||
| 20/11 | |||
|- | |||
| 2 | |||
| 470.0 | |||
| '''21/16''' | |||
| 870.0 | |||
| 33/20 | |||
| 70.0 | |||
| 21/20, 33/32 | |||
|- | |||
| 3 | |||
| 705.0 | |||
| '''3/2''' | |||
| 1105.0 | |||
| '''15/8''', 21/11 | |||
| 305.0 | |||
| 6/5 | |||
|- | |||
| 4 | |||
| 940.0 | |||
| 12/7, 26/15 | |||
| 140.0 | |||
| 12/11, 13/12, 15/14 | |||
| 540.0 | |||
| 15/11 | |||
|- | |||
| 5 | |||
| 1175.0 | |||
| 63/32 | |||
| 375.0 | |||
| 26/21 | |||
| 775.0 | |||
| 52/33 | |||
|- | |||
| 6 | |||
| 209.9 | |||
| 9/8 | |||
| 609.9 | |||
| 45/32, 63/44 | |||
| 1009.9 | |||
| 9/5 | |||
|- | |||
| 7 | |||
| 444.9 | |||
| 9/7, 13/10 | |||
| 844.9 | |||
| 18/11, '''13/8''' | |||
| 44.9 | |||
| 36/35 | |||
|- | |||
| 8 | |||
| 679.9 | |||
| 52/35, 72/49 | |||
| 1079.9 | |||
| 13/7 | |||
| 279.9 | |||
| 13/11 | |||
|} | |||
<nowiki/>* In 13-limit CWE tuning, octave reduced | |||
== Tunings == | == Tunings == | ||
Revision as of 23:42, 19 March 2026
| Trisected |
56/55, 128/125, 1029/1000 (11-limit);
56/55, 91/90, 128/125, 1029/1000 (13-limit)
13-limit 21-odd-limit: 17.5 ¢
13-limit 21-odd-limit: 36 notes
Trisected is the rank-2 temperament tempering out 128/125, 1029/1000, and 1029/1024 in the 7-limit, making it a member of the augmented family, keegic temperaments, and gamelismic clan. Since it tempers out 128/125, the octave is split into 3 ~5/4's, each tuned to 400 ¢ if the octave is pure. Since it tempers out 1029/1024, the perfect fifth is split into three intervals of ~8/7.Since it tempers out 1029/1000, the tritave is split into three intervals of 10/7. This means that every Pythagorean interval is split into three equal parts.
In the 11-limit, the perfect fourth is split into three ~11/10's, thus tempering out 4000/3993. Additionally, the 1/3-octave period represents 14/11, tempering out 56/55 and 176/175. The 13-limit extension equates the ~10/7 with 13/9, tempering out 91/90 and 2197/2187.
For technical data, see Augmented family #Trisected.
Intervals
In the following table, odd harmonics 1–21 are in bold.
| # | Period 0 | Period 1 | Period 2 | |||
|---|---|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | 400.0 | 5/4, 14/11 | 800.0 | 8/5, 11/7 |
| 1 | 235.0 | 8/7 | 635.0 | 10/7, 16/11, 13/9 | 1035.0 | 20/11 |
| 2 | 470.0 | 21/16 | 870.0 | 33/20 | 70.0 | 21/20, 33/32 |
| 3 | 705.0 | 3/2 | 1105.0 | 15/8, 21/11 | 305.0 | 6/5 |
| 4 | 940.0 | 12/7, 26/15 | 140.0 | 12/11, 13/12, 15/14 | 540.0 | 15/11 |
| 5 | 1175.0 | 63/32 | 375.0 | 26/21 | 775.0 | 52/33 |
| 6 | 209.9 | 9/8 | 609.9 | 45/32, 63/44 | 1009.9 | 9/5 |
| 7 | 444.9 | 9/7, 13/10 | 844.9 | 18/11, 13/8 | 44.9 | 36/35 |
| 8 | 679.9 | 52/35, 72/49 | 1079.9 | 13/7 | 279.9 | 13/11 |
* In 13-limit CWE tuning, octave reduced
Tunings
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