239edo

← 238edo

239edo

240edo →

Prime factorization	239 (prime)
Step size	5.02092 ¢
Fifth	140\239 (702.929 ¢)
Semitones (A1:m2)	24:17 (120.5 ¢ : 85.36 ¢)
Consistency limit	11
Distinct consistency limit	11

239 equal divisions of the octave (abbreviated 239edo or 239ed2), also called 239-tone equal temperament (239tet) or 239 equal temperament (239et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 239 equal parts of about 5.02 ¢ each. Each step represents a frequency ratio of 2^1/239, or the 239th root of 2.

Theory

239edo excels as an 11-limit system with a sharp tendency: prime harmonics 3 through 11 are all tuned sharp. The accuracy of 12/11 is particularly notable, as 30\239 represents a convergent to this interval, which forms a barely consistent circle, with the closing error being about 45% of a step.

239 is a convergent to the argent tuning, where the perfect fourth and fifth are in the logarithmic ratio of 1:√2, and with a fifth in this range, tempers out 5120/5103 with great accuracy. This implies that 81/80 and 64/63 are equated (to 5 steps), that three wholetones (9/8) stack to 10/7, and that the apotome, the limma, and the Pythagorean comma are equated with 15/14 (24 steps), 21/20 (17 steps), and 50/49 (7 steps) respectively.

Another notable feature of 239edo is that many of its 5-limit intervals are mapped to composite numbers of steps: 3/2 to 140, 4/3 to 99, 5/3 to 176, 5/4 to 77, 6/5 to 63, and 8/5 to 162. Not only that, but plenty of these form relatively simple logarithmic ratios with each other, as described by Don Page commas. In particular: gammic, where 9 units form 6/5, 11 form 5/4, and 20 form 3/2; quartonic, where 7 units form 6/5, 11 form 4/3, and 18 form 8/5; and escapade, where 7 units form 5/4, 9 form 4/3, and 16 form 5/3, are all supported by 239edo with generators 7\239, 9\239, and 11\239 respectively. 239edo is describable as the unique intersection of any two of these.

As a result, 239edo possesses many of the structures associated with temperaments such as tetracot, porcupine, bleu, orwell, and sensamagic, but with interpretations of their generators that are generally much more precise than those temperaments provide (and are thus distinguished from the intervals that those temperaments' simplest generators are mapped to in 239edo). This proves advantageous in terms of adeptness at representing different "flavors" of categories of interval: for example, in the category of submajor or equable heptatonic seconds, 239edo distinguishes 11/10 (the "pine" generator, 1/3 of a perfect fourth) at 33\239, from 32/29 (the diminished third on the chain of fifths) at 34\239, from 31/28 (the "tetracot" generator, 1/4 of a perfect fifth) at 35\239, from 10/9 (the sesquiaugmented unison) at 36\239.

In addition to its 11-limit, 239edo also encompasses a large variety of higher primes, and is specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup. Primes 23, 67, and 71 are for the most part usable as well, though one should be cautious about pitting 23 against sharp odds (like 9 or 33).

The equal temperament tempers out 2401/2400, 5120/5103, 10976/10935, and 29360128/29296875 in the 7-limit, supporting the hemififths temperament and providing an excellent tuning. It also supports and provides a good tuning for quasiorwell, neptune, and alphaquarter. In the 11-limit, it tempers out 3025/3024, 4000/3993, 5632/5625, 12005/11979, and 41503/41472, supporting quadrafifths and unthirds.

Prime harmonics

Approximation of prime harmonics in 239edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.97	+0.30	+0.21	+0.98	-2.03	+0.48	-1.28	-0.66	-0.29	-0.27
	Relative (%)	+0.0	+19.4	+5.9	+4.2	+19.6	-40.5	+9.6	-25.5	-13.1	-5.7	-5.3
Steps (reduced)		239 (0)	379 (140)	555 (77)	671 (193)	827 (110)	884 (167)	977 (21)	1015 (59)	1081 (125)	1161 (205)	1184 (228)

Approximation of prime harmonics in 239edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-0.30	-2.28	+0.62	+2.28	+0.14	+0.24	-2.24	+1.03	+1.06	-1.85	+1.99
	Relative (%)	-5.9	-45.5	+12.3	+45.3	+2.7	+4.8	-44.6	+20.5	+21.0	-36.8	+39.6
Steps (reduced)		1245 (50)	1280 (85)	1297 (102)	1328 (133)	1369 (174)	1406 (211)	1417 (222)	1450 (16)	1470 (36)	1479 (45)	1507 (73)

Subsets and supersets

239edo is the 52nd prime edo.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	5.02		^D, ^8E♭♭
2	10.04		^^D, ^9E♭♭
3	15.06		^3D, ^10E♭♭
4	20.08		^4D, ^11E♭♭
5	25.1	69/68, 70/69	^5D, v12E♭
6	30.13	57/56, 58/57	^6D, v11E♭
7	35.15	49/48, 50/49, 51/50	^7D, v10E♭
8	40.17	43/42, 44/43	^8D, v9E♭
9	45.19	39/38, 77/75	^9D, v8E♭
10	50.21	35/34	^10D, v7E♭
11	55.23	32/31	^11D, v6E♭
12	60.25	29/28	^12D, v5E♭
13	65.27		v11D♯, v4E♭
14	70.29	25/24	v10D♯, v3E♭
15	75.31	47/45	v9D♯, vvE♭
16	80.33	22/21	v8D♯, vE♭
17	85.36	21/20	v7D♯, E♭
18	90.38	39/37	v6D♯, ^E♭
19	95.4	37/35	v5D♯, ^^E♭
20	100.42		v4D♯, ^3E♭
21	105.44	17/16	v3D♯, ^4E♭
22	110.46		vvD♯, ^5E♭
23	115.48	31/29, 77/72	vD♯, ^6E♭
24	120.5	74/69	D♯, ^7E♭
25	125.52	43/40	^D♯, ^8E♭
26	130.54	55/51, 69/64	^^D♯, ^9E♭
27	135.56	40/37	^3D♯, ^10E♭
28	140.59	51/47	^4D♯, ^11E♭
29	145.61	37/34, 62/57	^5D♯, v12E
30	150.63	12/11	^6D♯, v11E
31	155.65	35/32	^7D♯, v10E
32	160.67	34/31	^8D♯, v9E
33	165.69	11/10	^9D♯, v8E
34	170.71	32/29	^10D♯, v7E
35	175.73	31/28	^11D♯, v6E
36	180.75		^12D♯, v5E
37	185.77	49/44, 69/62	v11D𝄪, v4E
38	190.79	48/43, 77/69	v10D𝄪, v3E
39	195.82	28/25	v9D𝄪, vvE
40	200.84	55/49, 64/57	v8D𝄪, vE
41	205.86		E
42	210.88	35/31	^E, ^8F♭
43	215.9	17/15, 77/68	^^E, ^9F♭
44	220.92	25/22	^3E, ^10F♭
45	225.94	49/43	^4E, ^11F♭
46	230.96	8/7	^5E, v12F
47	235.98	55/48, 63/55	^6E, v11F
48	241	54/47	^7E, v10F
49	246.03		^8E, v9F
50	251.05	37/32	^9E, v8F
51	256.07	29/25, 51/44	^10E, v7F
52	261.09	50/43, 57/49	^11E, v6F
53	266.11	7/6	^12E, v5F
54	271.13	76/65	v11E♯, v4F
55	276.15	34/29	v10E♯, v3F
56	281.17	20/17	v9E♯, vvF
57	286.19	46/39	v8E♯, vF
58	291.21	58/49	F
59	296.23	51/43	^F, ^8G♭♭
60	301.26	25/21, 69/58	^^F, ^9G♭♭
61	306.28	37/31, 68/57	^3F, ^10G♭♭
62	311.3		^4F, ^11G♭♭
63	316.32	6/5	^5F, v12G♭
64	321.34		^6F, v11G♭
65	326.36	35/29	^7F, v10G♭
66	331.38	23/19	^8F, v9G♭
67	336.4	17/14	^9F, v8G♭
68	341.42	28/23	^10F, v7G♭
69	346.44		^11F, v6G♭
70	351.46	49/40, 60/49	^12F, v5G♭
71	356.49	43/35, 70/57	v11F♯, v4G♭
72	361.51	69/56	v10F♯, v3G♭
73	366.53	21/17, 68/55	v9F♯, vvG♭
74	371.55	31/25, 57/46	v8F♯, vG♭
75	376.57	46/37	v7F♯, G♭
76	381.59		v6F♯, ^G♭
77	386.61	5/4	v5F♯, ^^G♭
78	391.63		v4F♯, ^3G♭
79	396.65	39/31, 44/35	v3F♯, ^4G♭
80	401.67	29/23	vvF♯, ^5G♭
81	406.69	43/34, 62/49	vF♯, ^6G♭
82	411.72	52/41	F♯, ^7G♭
83	416.74	14/11	^F♯, ^8G♭
84	421.76	37/29	^^F♯, ^9G♭
85	426.78	32/25, 55/43	^3F♯, ^10G♭
86	431.8	77/60	^4F♯, ^11G♭
87	436.82		^5F♯, v12G
88	441.84	40/31	^6F♯, v11G
89	446.86	22/17	^7F♯, v10G
90	451.88	74/57	^8F♯, v9G
91	456.9	56/43	^9F♯, v8G
92	461.92	47/36, 64/49	^10F♯, v7G
93	466.95	55/42, 72/55	^11F♯, v6G
94	471.97		^12F♯, v5G
95	476.99		v11F𝄪, v4G
96	482.01	37/28	v10F𝄪, v3G
97	487.03	49/37	v9F𝄪, vvG
98	492.05		v8F𝄪, vG
99	497.07		G
100	502.09		^G, ^8A♭♭
101	507.11	63/47	^^G, ^9A♭♭
102	512.13	39/29, 43/32	^3G, ^10A♭♭
103	517.15	31/23	^4G, ^11A♭♭
104	522.18	50/37	^5G, v12A♭
105	527.2		^6G, v11A♭
106	532.22	34/25	^7G, v10A♭
107	537.24	15/11	^8G, v9A♭
108	542.26	26/19	^9G, v8A♭
109	547.28	48/35	^10G, v7A♭
110	552.3		^11G, v6A♭
111	557.32	40/29, 69/50	^12G, v5A♭
112	562.34		v11G♯, v4A♭
113	567.36	43/31, 68/49	v10G♯, v3A♭
114	572.38	32/23	v9G♯, vvA♭
115	577.41	60/43	v8G♯, vA♭
116	582.43	7/5	v7G♯, A♭
117	587.45	66/47	v6G♯, ^A♭
118	592.47	69/49	v5G♯, ^^A♭
119	597.49	24/17	v4G♯, ^3A♭
120	602.51	17/12	v3G♯, ^4A♭
121	607.53		vvG♯, ^5A♭
122	612.55	47/33, 57/40	vG♯, ^6A♭
123	617.57	10/7	G♯, ^7A♭
124	622.59	43/30	^G♯, ^8A♭
125	627.62	23/16	^^G♯, ^9A♭
126	632.64	49/34, 62/43	^3G♯, ^10A♭
127	637.66		^4G♯, ^11A♭
128	642.68	29/20	^5G♯, v12A
129	647.7		^6G♯, v11A
130	652.72	35/24	^7G♯, v10A
131	657.74	19/13	^8G♯, v9A
132	662.76	22/15	^9G♯, v8A
133	667.78	25/17	^10G♯, v7A
134	672.8		^11G♯, v6A
135	677.82	37/25	^12G♯, v5A
136	682.85	46/31	v11G𝄪, v4A
137	687.87	58/39, 64/43	v10G𝄪, v3A
138	692.89		v9G𝄪, vvA
139	697.91		v8G𝄪, vA
140	702.93		A
141	707.95		^A, ^8B♭♭
142	712.97	74/49, 77/51	^^A, ^9B♭♭
143	717.99	56/37	^3A, ^10B♭♭
144	723.01		^4A, ^11B♭♭
145	728.03		^5A, v12B♭
146	733.05	55/36	^6A, v11B♭
147	738.08	49/32, 72/47	^7A, v10B♭
148	743.1	43/28	^8A, v9B♭
149	748.12	57/37, 77/50	^9A, v8B♭
150	753.14	17/11	^10A, v7B♭
151	758.16	31/20	^11A, v6B♭
152	763.18		^12A, v5B♭
153	768.2		v11A♯, v4B♭
154	773.22	25/16	v10A♯, v3B♭
155	778.24	58/37, 69/44	v9A♯, vvB♭
156	783.26	11/7	v8A♯, vB♭
157	788.28	41/26	v7A♯, B♭
158	793.31	49/31, 68/43	v6A♯, ^B♭
159	798.33	46/29, 65/41	v5A♯, ^^B♭
160	803.35	35/22, 62/39	v4A♯, ^3B♭
161	808.37	75/47	v3A♯, ^4B♭
162	813.39	8/5	vvA♯, ^5B♭
163	818.41	69/43, 77/48	vA♯, ^6B♭
164	823.43	37/23	A♯, ^7B♭
165	828.45	50/31	^A♯, ^8B♭
166	833.47	34/21, 55/34	^^A♯, ^9B♭
167	838.49		^3A♯, ^10B♭
168	843.51	57/35, 70/43	^4A♯, ^11B♭
169	848.54	49/30	^5A♯, v12B
170	853.56		^6A♯, v11B
171	858.58	23/14	^7A♯, v10B
172	863.6	28/17	^8A♯, v9B
173	868.62	38/23	^9A♯, v8B
174	873.64	58/35	^10A♯, v7B
175	878.66		^11A♯, v6B
176	883.68	5/3	^12A♯, v5B
177	888.7		v11A𝄪, v4B
178	893.72	57/34, 62/37	v10A𝄪, v3B
179	898.74	42/25	v9A𝄪, vvB
180	903.77		v8A𝄪, vB
181	908.79	49/29	B
182	913.81	39/23	^B, ^8C♭
183	918.83	17/10	^^B, ^9C♭
184	923.85	29/17, 75/44	^3B, ^10C♭
185	928.87	65/38	^4B, ^11C♭
186	933.89	12/7	^5B, v12C
187	938.91	43/25	^6B, v11C
188	943.93	50/29, 69/40	^7B, v10C
189	948.95	64/37	^8B, v9C
190	953.97		^9B, v8C
191	959	47/27	^10B, v7C
192	964.02		^11B, v6C
193	969.04	7/4	^12B, v5C
194	974.06		v11B♯, v4C
195	979.08	44/25	v10B♯, v3C
196	984.1	30/17	v9B♯, vvC
197	989.12	62/35	v8B♯, vC
198	994.14		C
199	999.16	57/32	^C, ^8D♭♭
200	1004.18	25/14	^^C, ^9D♭♭
201	1009.21	43/24, 77/43	^3C, ^10D♭♭
202	1014.23		^4C, ^11D♭♭
203	1019.25		^5C, v12D♭
204	1024.27	56/31	^6C, v11D♭
205	1029.29	29/16	^7C, v10D♭
206	1034.31	20/11	^8C, v9D♭
207	1039.33	31/17	^9C, v8D♭
208	1044.35	64/35	^10C, v7D♭
209	1049.37	11/6	^11C, v6D♭
210	1054.39	57/31, 68/37	^12C, v5D♭
211	1059.41		v11C♯, v4D♭
212	1064.44	37/20	v10C♯, v3D♭
213	1069.46		v9C♯, vvD♭
214	1074.48		v8C♯, vD♭
215	1079.5	69/37	v7C♯, D♭
216	1084.52	58/31	v6C♯, ^D♭
217	1089.54		v5C♯, ^^D♭
218	1094.56	32/17	v4C♯, ^3D♭
219	1099.58		v3C♯, ^4D♭
220	1104.6	70/37	vvC♯, ^5D♭
221	1109.62	74/39	vC♯, ^6D♭
222	1114.64	40/21	C♯, ^7D♭
223	1119.67	21/11	^C♯, ^8D♭
224	1124.69		^^C♯, ^9D♭
225	1129.71	48/25	^3C♯, ^10D♭
226	1134.73	77/40	^4C♯, ^11D♭
227	1139.75	56/29	^5C♯, v12D
228	1144.77	31/16	^6C♯, v11D
229	1149.79	68/35	^7C♯, v10D
230	1154.81	76/39	^8C♯, v9D
231	1159.83	43/22	^9C♯, v8D
232	1164.85	49/25	^10C♯, v7D
233	1169.87	57/29	^11C♯, v6D
234	1174.9	69/35	^12C♯, v5D
235	1179.92		v11C𝄪, v4D
236	1184.94		v10C𝄪, v3D
237	1189.96		v9C𝄪, vvD
238	1194.98		v8C𝄪, vD
239	1200	2/1	D

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[379 -239⟩	[⟨239 379]]	−0.307	0.307	6.12
2.3.5	[3 -18 11⟩, [32 -7 -9⟩	[⟨239 379 555]]	−0.247	0.265	5.27
2.3.5.7	2401/2400, 5120/5103, 29360128/29296875	[⟨239 379 555 671]]	−0.204	0.241	4.80
2.3.5.7.11	2401/2400, 3025/3024, 4000/3993, 5120/5103	[⟨239 379 555 671 827]]	−0.220	0.218	4.34
2.3.5.7.11.17	595/594, 1156/1155, 2058/2057, 2401/2400, 5120/5103	[⟨239 379 555 671 827 977]]	−0.203	0.203	4.03
2.3.5.7.11.13	352/351, 625/624, 847/845, 1575/1573, 2401/2400	[⟨239 379 555 671 827 885]] (239f)	−0.318	0.296	5.89
2.3.5.7.11.13.17	352/351, 595/594, 625/624, 833/832, 1156/1155, 1575/1573	[⟨239 379 555 671 827 885 977]] (239f)	−0.290	0.282	5.63

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	3\239	15.06	121/120	Yarman I (239)
1	7\239	35.15	1990656/1953125	Gammic (5-limit)
1	9\239	45.19	250/243	Quartonic (5-limit)
1	11\239	55.23	33/32	Escapade / alphaquarter (239f)
1	35\239	175.73	72/65	Quadrafifths (239f)
1	54\239	271.13	90/77	Quasiorwell (239)
1	70\239	351.46	49/40	Hemififths (7-limit)
1	79\239	396.65	44/35	Squarschmidt
1	83\239	416.74	14/11	Unthirds (239f)
1	103\239	517.15	66/49	Cutefourths (239f)
1	116\239	582.43	7/5	Neptune (7-limit)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium

• "thatyounglighthouseboi" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube
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