240edo
| ← 239edo | 240edo | 241edo → |
240 equal divisions of the octave (abbreviated 240edo or 240ed2), also called 240-tone equal temperament (240tet) or 240 equal temperament (240et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 240 equal parts of exactly 5 ¢ each. Each step represents a frequency ratio of 21/240, or the 240th root of 2.
Theory
240edo is consistent in the 5-odd-limit and notably provides the optimal patent val for the 5-limit compton temperament, the rank-2 temperament associated with the Pythagorean comma. However, its mapping for 3 is not well approximated, meaning it is a dual-fifth system, with alternate mapping for 3/2 is the 705-cent sharp fifth inherited from 80edo.
Although no longer consistent to to the higher limits, 240edo's patent val tempers out the 225/224 in the 7-limit, supporting marvel temperaments with harmonics 3, 5, 7 having less than two cents of error. Retuning 5-limit scales to 240edo is a simple way to to make them function as 7-limit scales while retaining very accurate tuning.
From a regular temperament theory perspective in the 7-limit, 240edo is similar to 197edo. The main difference is that 197edo, despite a flatter third, gives generally better results and may be preferred, whitherfore a compromise between good results and an accurate 5 may be worked out by means of retuning 5-limit scales to the 43 & 197 temperament, which has a comma basis {225/224, [-49 19 -10 15⟩} in the 7-limit.
For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two undecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 is the 350-cent interval often employed in 24edo versions of Arabic music.)
Subsets and supersets
240edo is the 12th highly composite edo, with subset edos 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120.
In addition, as every fifth step of 1200edo, it is the largest highly composite edo expressible in integer cents.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | -1.31 | +1.17 | +1.09 | -1.32 | -0.53 | +1.73 | +0.04 | +2.49 | -0.78 | +1.73 |
| Relative (%) | -39.1 | -26.3 | +23.5 | +21.8 | -26.4 | -10.6 | +34.6 | +0.9 | +49.7 | -15.6 | +34.5 | |
| Steps (reduced) |
380 (140) |
557 (77) |
674 (194) |
761 (41) |
830 (110) |
888 (168) |
938 (218) |
981 (21) |
1020 (60) |
1054 (94) |
1086 (126) | |
Interval table
See Table of 240edo intervals.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 531441/524288, [-29 -11 20⟩ | [⟨240 380 557]] | 0.5998 | 0.5044 | 10.09 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 7\240 | 35.00 | 45/44 | Gammy |
| 1 | 101\240 | 505.00 | 104976/78125 | Countermeantone |
| 12 | 1\240 | 5.00 | ? | Romcom |
| 12 | 77\240 (3\240) |
385.00 (15.00) |
5/4 (81/80) |
Compton |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
- Scales derived from marvel and spectacle temperaments
- 23 17 23 14 23 17 23 23 14 26 14 23 – Ellis's Duodene genus [33355] retuned to 240edo
- 23 17 14 23 23 17 23 23 14 17 23 23 – Carl Lumma's scale
- 14 9 14 17 23 23 23 17 14 9 14 23 17 23 – Pum[14] scale
- 16 10 7 7 16 7 7 16 7 10 7 16 7 7 16 7 7 10 16 7 7 16 7 – Ellis duodene union 11/9 times the duodene
Other scales
- 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 17 3 – Compton[24]
- 23 31 80 23 83 – Balinese pentatonic pelog scale; Tolgahan Çoğulu's tuning
Music
The video Balinese Gamelan Music on Microtonal Guitar - Chris Charles on the YouTube channel Microtonal Guitar - Tolgahan Çoğulu uses a 5-tone subset of 240edo for all three pieces performed in the recording. As explained in the video description: "The scale we used in the piece: Pelog Selisir: D, Eb +30 F -15 A -30 Bb -15".
Links
Shaahin Mohajeri, an Iranian Tombak player and composer, calls his personal Google site "240edo", where he makes the point that five cents is a size close to the just noticeable difference between pitches.