240edo

← 239edo

240edo

241edo →

Prime factorization	24 × 3 × 5 (highly composite)
Step size	5 ¢
Fifth	140\240 (700 ¢) (→ 7\12)
Semitones (A1:m2)	20:20 (100 ¢ : 100 ¢)
Dual sharp fifth	141\240 (705 ¢) (→ 47\80)
Dual flat fifth	140\240 (700 ¢) (→ 7\12)
Dual major 2nd	41\240 (205 ¢)
Consistency limit	5
Distinct consistency limit	5

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 2^1/240, or the 240th root of 2.

Theory

240edo notably provides the optimal patent val for the 5-limit compton temperament, the rank-2 temperament associated with the Pythagorean comma. However, it is only consistent in the 5-odd-limit. Its mapping for harmonic 3 is not well approximated, meaning it is a dual-fifth system; its alternative mapping for 3/2 is the 705 ¢ sharp fifth inherited from 80edo.

Although no longer consistent to the higher limits, 240edo's patent val tempers out the 225/224 in the 7-limit, supporting marvel 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.

240edo is similar to 197edo in terms of intonation in the 7-limit. 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 rank-3 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.)

Odd harmonics

Approximation of odd harmonics in 240edo

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)

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.

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

Table of rank-2 temperaments by generator

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	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 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

Instruments

A Lumatone mapping for 240edo is now available.

Music

Chris Charles (via Microtonal Guitar - Tolgahan Çoğulu)

• Balinese Gamelan Music on Microtonal Guitar - Chris Charles (2017) (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".)

Bryan Deister

• microtonal improvisation in 240edo (2025)

Trivia

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.