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

Theory

240edo is a dual-fifth system. Nonetheless, it is consistent in the 5-limit and notably provides the optimal patent val for the compton temperament, the rank-2 temperament associated with the Pythagorean comma. 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 and spectacle 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 197 & 240 temperament, whhich 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 unidecimal 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.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. 3/2 is equated with two 11/9 as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9s and two 5/4s, 11/8 is equated with a stack of five 11/9s, 13/8 is equated with a stack of two 18/11s and four 5/4s, and 17/16 is equated with three 18/11s and three 5/4s. Every harmonic is reached with help of other intervals at most with three 5/4s.

Subsets and supersets

240edo is the 12th highly composite EDO, with subset EDOs 1, 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 compatible with cents.

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)

Interval table

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 140\240)	Ups and downs notation (Dual sharp fifth 141\240)
0	0	1/1	D	D
1	5		^D, ^E♭♭	^D, ^13E♭♭
2	10		^^D, ^^E♭♭	^^D, v13E♭
3	15		^3D, ^3E♭♭	^3D, v12E♭
4	20		^4D, ^4E♭♭	^4D, v11E♭
5	25	69/68, 70/69	^5D, ^5E♭♭	^5D, v10E♭
6	30	57/56, 58/57	^6D, ^6E♭♭	^6D, v9E♭
7	35	51/50	^7D, ^7E♭♭	^7D, v8E♭
8	40	43/42, 44/43	^8D, ^8E♭♭	^8D, v7E♭
9	45		^9D, ^9E♭♭	^9D, v6E♭
10	50	35/34	^10D, v10E♭	^10D, v5E♭
11	55	32/31	v9D♯, v9E♭	^11D, v4E♭
12	60	29/28	v8D♯, v8E♭	^12D, v3E♭
13	65		v7D♯, v7E♭	^13D, vvE♭
14	70	25/24	v6D♯, v6E♭	v13D♯, vE♭
15	75		v5D♯, v5E♭	v12D♯, E♭
16	80	22/21	v4D♯, v4E♭	v11D♯, ^E♭
17	85	21/20	v3D♯, v3E♭	v10D♯, ^^E♭
18	90		vvD♯, vvE♭	v9D♯, ^3E♭
19	95		vD♯, vE♭	v8D♯, ^4E♭
20	100		D♯, E♭	v7D♯, ^5E♭
21	105	17/16	^D♯, ^E♭	v6D♯, ^6E♭
22	110	49/46	^^D♯, ^^E♭	v5D♯, ^7E♭
23	115	31/29, 47/44	^3D♯, ^3E♭	v4D♯, ^8E♭
24	120		^4D♯, ^4E♭	v3D♯, ^9E♭
25	125	43/40	^5D♯, ^5E♭	vvD♯, ^10E♭
26	130	55/51, 69/64	^6D♯, ^6E♭	vD♯, ^11E♭
27	135	40/37	^7D♯, ^7E♭	D♯, ^12E♭
28	140		^8D♯, ^8E♭	^D♯, ^13E♭
29	145	62/57	^9D♯, ^9E♭	^^D♯, v13E
30	150	12/11	^10D♯, v10E	^3D♯, v12E
31	155	35/32	v9D𝄪, v9E	^4D♯, v11E
32	160	34/31	v8D𝄪, v8E	^5D♯, v10E
33	165	11/10	v7D𝄪, v7E	^6D♯, v9E
34	170	32/29	v6D𝄪, v6E	^7D♯, v8E
35	175	52/47	v5D𝄪, v5E	^8D♯, v7E
36	180		v4D𝄪, v4E	^9D♯, v6E
37	185	69/62	v3D𝄪, v3E	^10D♯, v5E
38	190	48/43, 77/69	vvD𝄪, vvE	^11D♯, v4E
39	195	47/42	vD𝄪, vE	^12D♯, v3E
40	200	46/41, 64/57	E	^13D♯, vvE
41	205		^E, ^F♭	v13D𝄪, vE
42	210	35/31	^^E, ^^F♭	E
43	215	77/68	^3E, ^3F♭	^E, ^13F♭
44	220	42/37	^4E, ^4F♭	^^E, v13F
45	225	74/65	^5E, ^5F♭	^3E, v12F
46	230		^6E, ^6F♭	^4E, v11F
47	235	55/48, 63/55	^7E, ^7F♭	^5E, v10F
48	240		^8E, ^8F♭	^6E, v9F
49	245		^9E, ^9F♭	^7E, v8F
50	250		^10E, v10F	^8E, v7F
51	255	51/44	v9E♯, v9F	^9E, v6F
52	260	43/37	v8E♯, v8F	^10E, v5F
53	265		v7E♯, v7F	^11E, v4F
54	270		v6E♯, v6F	^12E, v3F
55	275	34/29	v5E♯, v5F	^13E, vvF
56	280	47/40	v4E♯, v4F	v13E♯, vF
57	285		v3E♯, v3F	F
58	290	13/11	vvE♯, vvF	^F, ^13G♭♭
59	295	51/43	vE♯, vF	^^F, v13G♭
60	300	44/37, 69/58	F	^3F, v12G♭
61	305	31/26, 68/57	^F, ^G♭♭	^4F, v11G♭
62	310		^^F, ^^G♭♭	^5F, v10G♭
63	315	6/5	^3F, ^3G♭♭	^6F, v9G♭
64	320	77/64	^4F, ^4G♭♭	^7F, v8G♭
65	325	35/29, 41/34	^5F, ^5G♭♭	^8F, v7G♭
66	330	23/19	^6F, ^6G♭♭	^9F, v6G♭
67	335		^7F, ^7G♭♭	^10F, v5G♭
68	340	28/23	^8F, ^8G♭♭	^11F, v4G♭
69	345		^9F, ^9G♭♭	^12F, v3G♭
70	350		^10F, v10G♭	^13F, vvG♭
71	355	70/57	v9F♯, v9G♭	v13F♯, vG♭
72	360	16/13	v8F♯, v8G♭	v12F♯, G♭
73	365	21/17	v7F♯, v7G♭	v11F♯, ^G♭
74	370	26/21	v6F♯, v6G♭	v10F♯, ^^G♭
75	375	77/62	v5F♯, v5G♭	v9F♯, ^3G♭
76	380		v4F♯, v4G♭	v8F♯, ^4G♭
77	385		v3F♯, v3G♭	v7F♯, ^5G♭
78	390		vvF♯, vvG♭	v6F♯, ^6G♭
79	395		vF♯, vG♭	v5F♯, ^7G♭
80	400	63/50	F♯, G♭	v4F♯, ^8G♭
81	405		^F♯, ^G♭	v3F♯, ^9G♭
82	410		^^F♯, ^^G♭	vvF♯, ^10G♭
83	415	47/37	^3F♯, ^3G♭	vF♯, ^11G♭
84	420	51/40, 65/51	^4F♯, ^4G♭	F♯, ^12G♭
85	425		^5F♯, ^5G♭	^F♯, ^13G♭
86	430	50/39	^6F♯, ^6G♭	^^F♯, v13G
87	435		^7F♯, ^7G♭	^3F♯, v12G
88	440	49/38	^8F♯, ^8G♭	^4F♯, v11G
89	445		^9F♯, ^9G♭	^5F♯, v10G
90	450	48/37	^10F♯, v10G	^6F♯, v9G
91	455	13/10	v9F𝄪, v9G	^7F♯, v8G
92	460		v8F𝄪, v8G	^8F♯, v7G
93	465	17/13	v7F𝄪, v7G	^9F♯, v6G
94	470	21/16	v6F𝄪, v6G	^10F♯, v5G
95	475		v5F𝄪, v5G	^11F♯, v4G
96	480	33/25, 62/47	v4F𝄪, v4G	^12F♯, v3G
97	485		v3F𝄪, v3G	^13F♯, vvG
98	490	69/52, 77/58	vvF𝄪, vvG	v13F𝄪, vG
99	495		vF𝄪, vG	G
100	500		G	^G, ^13A♭♭
101	505		^G, ^A♭♭	^^G, v13A♭
102	510	47/35	^^G, ^^A♭♭	^3G, v12A♭
103	515	35/26	^3G, ^3A♭♭	^4G, v11A♭
104	520	77/57	^4G, ^4A♭♭	^5G, v10A♭
105	525	42/31, 65/48	^5G, ^5A♭♭	^6G, v9A♭
106	530		^6G, ^6A♭♭	^7G, v8A♭
107	535	64/47	^7G, ^7A♭♭	^8G, v7A♭
108	540	56/41	^8G, ^8A♭♭	^9G, v6A♭
109	545		^9G, ^9A♭♭	^10G, v5A♭
110	550		^10G, v10A♭	^11G, v4A♭
111	555	51/37	v9G♯, v9A♭	^12G, v3A♭
112	560	47/34	v8G♯, v8A♭	^13G, vvA♭
113	565		v7G♯, v7A♭	v13G♯, vA♭
114	570	57/41	v6G♯, v6A♭	v12G♯, A♭
115	575		v5G♯, v5A♭	v11G♯, ^A♭
116	580		v4G♯, v4A♭	v10G♯, ^^A♭
117	585		v3G♯, v3A♭	v9G♯, ^3A♭
118	590	52/37	vvG♯, vvA♭	v8G♯, ^4A♭
119	595	55/39	vG♯, vA♭	v7G♯, ^5A♭
120	600	41/29, 58/41	G♯, A♭	v6G♯, ^6A♭
121	605	78/55	^G♯, ^A♭	v5G♯, ^7A♭
122	610	37/26	^^G♯, ^^A♭	v4G♯, ^8A♭
123	615		^3G♯, ^3A♭	v3G♯, ^9A♭
124	620		^4G♯, ^4A♭	vvG♯, ^10A♭
125	625		^5G♯, ^5A♭	vG♯, ^11A♭
126	630		^6G♯, ^6A♭	G♯, ^12A♭
127	635		^7G♯, ^7A♭	^G♯, ^13A♭
128	640	68/47	^8G♯, ^8A♭	^^G♯, v13A
129	645	74/51	^9G♯, ^9A♭	^3G♯, v12A
130	650		^10G♯, v10A	^4G♯, v11A
131	655		v9G𝄪, v9A	^5G♯, v10A
132	660	41/28	v8G𝄪, v8A	^6G♯, v9A
133	665	47/32, 69/47	v7G𝄪, v7A	^7G♯, v8A
134	670		v6G𝄪, v6A	^8G♯, v7A
135	675	31/21, 65/44	v5G𝄪, v5A	^9G♯, v6A
136	680	77/52	v4G𝄪, v4A	^10G♯, v5A
137	685	52/35	v3G𝄪, v3A	^11G♯, v4A
138	690	70/47	vvG𝄪, vvA	^12G♯, v3A
139	695		vG𝄪, vA	^13G♯, vvA
140	700		A	v13G𝄪, vA
141	705		^A, ^B♭♭	A
142	710		^^A, ^^B♭♭	^A, ^13B♭♭
143	715	65/43	^3A, ^3B♭♭	^^A, v13B♭
144	720	47/31, 50/33	^4A, ^4B♭♭	^3A, v12B♭
145	725		^5A, ^5B♭♭	^4A, v11B♭
146	730	32/21	^6A, ^6B♭♭	^5A, v10B♭
147	735	26/17	^7A, ^7B♭♭	^6A, v9B♭
148	740		^8A, ^8B♭♭	^7A, v8B♭
149	745	20/13	^9A, ^9B♭♭	^8A, v7B♭
150	750	37/24	^10A, v10B♭	^9A, v6B♭
151	755		v9A♯, v9B♭	^10A, v5B♭
152	760	76/49	v8A♯, v8B♭	^11A, v4B♭
153	765		v7A♯, v7B♭	^12A, v3B♭
154	770	39/25	v6A♯, v6B♭	^13A, vvB♭
155	775		v5A♯, v5B♭	v13A♯, vB♭
156	780	80/51	v4A♯, v4B♭	v12A♯, B♭
157	785	74/47	v3A♯, v3B♭	v11A♯, ^B♭
158	790		vvA♯, vvB♭	v10A♯, ^^B♭
159	795		vA♯, vB♭	v9A♯, ^3B♭
160	800		A♯, B♭	v8A♯, ^4B♭
161	805		^A♯, ^B♭	v7A♯, ^5B♭
162	810		^^A♯, ^^B♭	v6A♯, ^6B♭
163	815		^3A♯, ^3B♭	v5A♯, ^7B♭
164	820		^4A♯, ^4B♭	v4A♯, ^8B♭
165	825		^5A♯, ^5B♭	v3A♯, ^9B♭
166	830	21/13	^6A♯, ^6B♭	vvA♯, ^10B♭
167	835	34/21	^7A♯, ^7B♭	vA♯, ^11B♭
168	840	13/8	^8A♯, ^8B♭	A♯, ^12B♭
169	845	57/35	^9A♯, ^9B♭	^A♯, ^13B♭
170	850		^10A♯, v10B	^^A♯, v13B
171	855	77/47	v9A𝄪, v9B	^3A♯, v12B
172	860	23/14	v8A𝄪, v8B	^4A♯, v11B
173	865		v7A𝄪, v7B	^5A♯, v10B
174	870	38/23	v6A𝄪, v6B	^6A♯, v9B
175	875	58/35, 68/41	v5A𝄪, v5B	^7A♯, v8B
176	880		v4A𝄪, v4B	^8A♯, v7B
177	885	5/3	v3A𝄪, v3B	^9A♯, v6B
178	890		vvA𝄪, vvB	^10A♯, v5B
179	895	52/31, 57/34	vA𝄪, vB	^11A♯, v4B
180	900	37/22	B	^12A♯, v3B
181	905		^B, ^C♭	^13A♯, vvB
182	910	22/13	^^B, ^^C♭	v13A𝄪, vB
183	915		^3B, ^3C♭	B
184	920	80/47	^4B, ^4C♭	^B, ^13C♭
185	925	29/17	^5B, ^5C♭	^^B, v13C
186	930		^6B, ^6C♭	^3B, v12C
187	935		^7B, ^7C♭	^4B, v11C
188	940	74/43	^8B, ^8C♭	^5B, v10C
189	945		^9B, ^9C♭	^6B, v9C
190	950		^10B, v10C	^7B, v8C
191	955		v9B♯, v9C	^8B, v7C
192	960		v8B♯, v8C	^9B, v6C
193	965		v7B♯, v7C	^10B, v5C
194	970		v6B♯, v6C	^11B, v4C
195	975	65/37	v5B♯, v5C	^12B, v3C
196	980	37/21	v4B♯, v4C	^13B, vvC
197	985		v3B♯, v3C	v13B♯, vC
198	990	62/35	vvB♯, vvC	C
199	995		vB♯, vC	^C, ^13D♭♭
200	1000	41/23, 57/32	C	^^C, v13D♭
201	1005		^C, ^D♭♭	^3C, v12D♭
202	1010	43/24	^^C, ^^D♭♭	^4C, v11D♭
203	1015		^3C, ^3D♭♭	^5C, v10D♭
204	1020		^4C, ^4D♭♭	^6C, v9D♭
205	1025	47/26	^5C, ^5D♭♭	^7C, v8D♭
206	1030	29/16	^6C, ^6D♭♭	^8C, v7D♭
207	1035	20/11	^7C, ^7D♭♭	^9C, v6D♭
208	1040	31/17	^8C, ^8D♭♭	^10C, v5D♭
209	1045	64/35	^9C, ^9D♭♭	^11C, v4D♭
210	1050	11/6	^10C, v10D♭	^12C, v3D♭
211	1055	57/31	v9C♯, v9D♭	^13C, vvD♭
212	1060		v8C♯, v8D♭	v13C♯, vD♭
213	1065	37/20	v7C♯, v7D♭	v12C♯, D♭
214	1070		v6C♯, v6D♭	v11C♯, ^D♭
215	1075	80/43	v5C♯, v5D♭	v10C♯, ^^D♭
216	1080		v4C♯, v4D♭	v9C♯, ^3D♭
217	1085	58/31	v3C♯, v3D♭	v8C♯, ^4D♭
218	1090		vvC♯, vvD♭	v7C♯, ^5D♭
219	1095	32/17	vC♯, vD♭	v6C♯, ^6D♭
220	1100		C♯, D♭	v5C♯, ^7D♭
221	1105		^C♯, ^D♭	v4C♯, ^8D♭
222	1110		^^C♯, ^^D♭	v3C♯, ^9D♭
223	1115	40/21	^3C♯, ^3D♭	vvC♯, ^10D♭
224	1120	21/11	^4C♯, ^4D♭	vC♯, ^11D♭
225	1125		^5C♯, ^5D♭	C♯, ^12D♭
226	1130	48/25	^6C♯, ^6D♭	^C♯, ^13D♭
227	1135		^7C♯, ^7D♭	^^C♯, v13D
228	1140	56/29	^8C♯, ^8D♭	^3C♯, v12D
229	1145	31/16	^9C♯, ^9D♭	^4C♯, v11D
230	1150	68/35	^10C♯, v10D	^5C♯, v10D
231	1155		v9C𝄪, v9D	^6C♯, v9D
232	1160	43/22	v8C𝄪, v8D	^7C♯, v8D
233	1165		v7C𝄪, v7D	^8C♯, v7D
234	1170	57/29	v6C𝄪, v6D	^9C♯, v6D
235	1175	69/35	v5C𝄪, v5D	^10C♯, v5D
236	1180		v4C𝄪, v4D	^11C♯, v4D
237	1185		v3C𝄪, v3D	^12C♯, v3D
238	1190		vvC𝄪, vvD	^13C♯, vvD
239	1195		vC𝄪, vD	v13C𝄪, vD
240	1200	2/1	D	D

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	531441/524288	⟨240 380]	0.6167	0.6171	12.34
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 (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	7\240	35.00	1990656/1953125	Gammic
1	101\240	505.00	104976/78125	Countermeantone
12	77\240 (3\240)	385.00 (15.00)	5/4 (81/80)	Compton

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.