205edo

← 204edo 205edo 206edo →
Prime factorization	5 × 41
Step size	5.85366 ¢
Fifth	120\205 (702.439 ¢) (→ 24\41)
Semitones (A1:m2)	20:15 (117.1 ¢ : 87.8 ¢)
Consistency limit	9
Distinct consistency limit	9

205 equal divisions of the octave (abbreviated 205edo or 205ed2), also called 205-tone equal temperament (205tet) or 205 equal temperament (205et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 205 equal parts of about 5.85 ¢ each. Each step represents a frequency ratio of 2^1/205, or the 205th root of 2.

205edo's step size is called a mem when used as an interval size unit.

Theory

Since 205 = 5 × 41, 205edo shares its fifth with 41edo. It can serve as a tuning for various temperaments, such as amity or laka, and supplies the optimal patent val for quanic in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments tempering out the huntma, 640/637, the rank-5 temperament for which it also supplies the optimal patent val.

In the 5-limit it tempers out 1600000/1594323, the amity comma, and [38 -2 -15⟩, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val ⟨205 325 476 576 709 759] tempers out 4375/4374, 5120/5103, 6144/6125 in the 7-limit; 540/539, 1331/1323, and 2420/2401 in the 11-limit; 352/351, 640/637, 729/728, 847/845, and 1188/1183 in the 13-limit.

Using its alternative mapping ⟨205 325 476 575] (205d) it can also be used for hemithirds temperament. This extension tempers out 385/384, 441/440, and 3388/3375 in the 11-limit. The 13-limit version of this, ⟨205 325 476 575 709 759] (205d), is especially noteworthy, where it tempers out 196/195 and 1001/1000. Another 13-limit extension is ⟨205 325 476 575 709 758] (205df), where it adds 325/324 and 364/363 to the comma list.

Anyway, assuming the patent val, 205et tempers out 540/539, so that it allows swetismic chords; 729/728, so that it allows squbemic chords; 640/637, so that it allows huntmic chords; 352/351, so that it allows minthmic chords; 1188/1183, so that it allows kestrel chords; and 847/845, so that it allows the cuthbert triad. In the alternative 205df val, it allows marveltwin chords, keenanismic chords, gentle chords, and werckismic chords. This makes it a tuning of exceptional fludity for its degree of accuracy.

Odd harmonics

Approximation of odd harmonics in 205edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.48	+0.03	+2.88	+0.97	-1.07	+2.40	+0.51	+0.41	+1.02	-2.49	-1.93
Error	Relative (%)	+8.3	+0.5	+49.2	+16.5	-18.3	+41.0	+8.7	+7.0	+17.5	-42.5	-33.0
Steps (reduced)		325 (120)	476 (66)	576 (166)	650 (35)	709 (94)	759 (144)	801 (186)	838 (18)	871 (51)	900 (80)	927 (107)

Approximation of odd harmonics in 205edo (continued)
Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	+0.06	+1.45	+0.67	+2.28	-0.59	+2.91	+0.36	+2.88	-1.75	-2.25	+1.00
Error	Relative (%)	+0.9	+24.8	+11.4	+39.0	-10.1	+49.7	+6.2	+49.3	-29.8	-38.4	+17.0
Steps (reduced)		952 (132)	975 (155)	996 (176)	1016 (196)	1034 (9)	1052 (27)	1068 (43)	1084 (59)	1098 (73)	1112 (87)	1126 (101)

Structural properties

205edo contains a very accurate approximation of the 2.3.5.11 subgroup, inheriting the perfect fifth from 41edo. The patent val mappings of primes 7 and 13 can then be found by mapping 7/5 to the Pythagorean diminished fifth, and 13/11 at the Pythagorean minor third, thus tempering out 5120/5103 and 352/351, as well as 847/845 and 2080/2079. In fact, it is the last edo tempering out 5120/5103 to map both 7/5 and 1024/729 consistently. It also supports the counterpyth mapping of prime 19.

Its step size represents several important intervals, such as the septimal kleisma 225/224, and the keenanisma 385/384. Notably, the mappings of primes 5, 7, 11, 13, and 19 all differ from their nearest 41edo step by 1 step of 205edo, so 205edo can be considered as 41edo with fine-tuning, similarly to how 217edo can be considered as 31edo with fine-tuning. The intervals 11/10, 12/11, 13/12, 14/13, and 15/14 are mapped equidistant, corresponding to 121/120, 144/143, 169/168, and 196/195 all being mapped to 2 steps. The mappings of 17 and 19 are accurate, with 15/14, 16/15, 17/16, 18/17, 19/18, and 20/19 all spaced apart from each other by one step. Overall, despite the sharpness of its 7 and 13, 205edo does fairly well in a range of prime limits.

Temperament generators and Tonal Plexus

205edo is the default tuning for the Tonal Plexus midi controller. See the theory part on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent meantone fifth in 119\205, an excellent myna generator in 53\205, and a very good porcupine generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.

The 119\205 meantone fifth is extremely close to the 1/4-comma meantone fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing Nicola Vicentino's adaptive-JI scheme of 1555. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

Subsets and supersets

205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 minas.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	5.85		^D, ^⁶E♭♭
2	11.71		^^D, ^⁷E♭♭
3	17.56		^³D, ^⁸E♭♭
4	23.41		^⁴D, ^⁹E♭♭
5	29.27	58/57	^⁵D, v¹⁰E♭
6	35.12	51/50	^⁶D, v⁹E♭
7	40.98		^⁷D, v⁸E♭
8	46.83	37/36, 38/37	^⁸D, v⁷E♭
9	52.68	33/32, 34/33	^⁹D, v⁶E♭
10	58.54	30/29	^¹⁰D, v⁵E♭
11	64.39	27/26	v⁹D♯, v⁴E♭
12	70.24	25/24	v⁸D♯, v³E♭
13	76.1	23/22, 47/45	v⁷D♯, vvE♭
14	81.95	43/41, 65/62	v⁶D♯, vE♭
15	87.8	20/19	v⁵D♯, E♭
16	93.66	19/18	v⁴D♯, ^E♭
17	99.51	18/17	v³D♯, ^^E♭
18	105.37	17/16	vvD♯, ^³E♭
19	111.22	16/15	vD♯, ^⁴E♭
20	117.07	46/43	D♯, ^⁵E♭
21	122.93	29/27, 44/41	^D♯, ^⁶E♭
22	128.78	14/13	^^D♯, ^⁷E♭
23	134.63	40/37	^³D♯, ^⁸E♭
24	140.49	51/47	^⁴D♯, ^⁹E♭
25	146.34	37/34, 62/57	^⁵D♯, v¹⁰E
26	152.2		^⁶D♯, v⁹E
27	158.05	57/52	^⁷D♯, v⁸E
28	163.9		^⁸D♯, v⁷E
29	169.76	32/29	^⁹D♯, v⁶E
30	175.61	31/28, 52/47	^¹⁰D♯, v⁵E
31	181.46	10/9	v⁹D𝄪, v⁴E
32	187.32	39/35	v⁸D𝄪, v³E
33	193.17	19/17	v⁷D𝄪, vvE
34	199.02	37/33, 46/41	v⁶D𝄪, vE
35	204.88	9/8	E
36	210.73	35/31	^E, ^⁶F♭
37	216.59	17/15	^^E, ^⁷F♭
38	222.44	58/51	^³E, ^⁸F♭
39	228.29	65/57	^⁴E, ^⁹F♭
40	234.15		^⁵E, v¹⁰F
41	240	31/27, 54/47	^⁶E, v⁹F
42	245.85		^⁷E, v⁸F
43	251.71	37/32	^⁸E, v⁷F
44	257.56	29/25, 65/56	^⁹E, v⁶F
45	263.41	64/55	^¹⁰E, v⁵F
46	269.27		v⁹E♯, v⁴F
47	275.12	34/29	v⁸E♯, v³F
48	280.98	20/17	v⁷E♯, vvF
49	286.83		v⁶E♯, vF
50	292.68	45/38	F
51	298.54	19/16	^F, ^⁶G♭♭
52	304.39	31/26, 56/47	^^F, ^⁷G♭♭
53	310.24	55/46	^³F, ^⁸G♭♭
54	316.1	6/5	^⁴F, ^⁹G♭♭
55	321.95	47/39, 65/54	^⁵F, v¹⁰G♭
56	327.8	29/24	^⁶F, v⁹G♭
57	333.66	40/33, 57/47	^⁷F, v⁸G♭
58	339.51	45/37	^⁸F, v⁷G♭
59	345.37		^⁹F, v⁶G♭
60	351.22		^¹⁰F, v⁵G♭
61	357.07		v⁹F♯, v⁴G♭
62	362.93	37/30	v⁸F♯, v³G♭
63	368.78	26/21, 47/38	v⁷F♯, vvG♭
64	374.63	36/29	v⁶F♯, vG♭
65	380.49		v⁵F♯, G♭
66	386.34	5/4	v⁴F♯, ^G♭
67	392.2	64/51, 69/55	v³F♯, ^^G♭
68	398.05	34/27, 39/31	vvF♯, ^³G♭
69	403.9	24/19	vF♯, ^⁴G♭
70	409.76	19/15	F♯, ^⁵G♭
71	415.61		^F♯, ^⁶G♭
72	421.46	37/29, 51/40	^^F♯, ^⁷G♭
73	427.32	32/25	^³F♯, ^⁸G♭
74	433.17		^⁴F♯, ^⁹G♭
75	439.02	58/45	^⁵F♯, v¹⁰G
76	444.88		^⁶F♯, v⁹G
77	450.73	48/37	^⁷F♯, v⁸G
78	456.59		^⁸F♯, v⁷G
79	462.44	47/36	^⁹F♯, v⁶G
80	468.29	38/29	^¹⁰F♯, v⁵G
81	474.15	25/19	v⁹F𝄪, v⁴G
82	480	33/25, 62/47	v⁸F𝄪, v³G
83	485.85	45/34	v⁷F𝄪, vvG
84	491.71		v⁶F𝄪, vG
85	497.56	4/3	G
86	503.41		^G, ^⁶A♭♭
87	509.27	51/38, 55/41	^^G, ^⁷A♭♭
88	515.12	35/26	^³G, ^⁸A♭♭
89	520.98	50/37	^⁴G, ^⁹A♭♭
90	526.83		^⁵G, v¹⁰A♭
91	532.68	34/25	^⁶G, v⁹A♭
92	538.54		^⁷G, v⁸A♭
93	544.39		^⁸G, v⁷A♭
94	550.24		^⁹G, v⁶A♭
95	556.1	40/29, 51/37	^¹⁰G, v⁵A♭
96	561.95	65/47	v⁹G♯, v⁴A♭
97	567.8	25/18	v⁸G♯, v³A♭
98	573.66	39/28	v⁷G♯, vvA♭
99	579.51		v⁶G♯, vA♭
100	585.37		v⁵G♯, A♭
101	591.22	38/27, 45/32	v⁴G♯, ^A♭
102	597.07	24/17	v³G♯, ^^A♭
103	602.93	17/12	vvG♯, ^³A♭
104	608.78	27/19, 64/45	vG♯, ^⁴A♭
105	614.63		G♯, ^⁵A♭
106	620.49		^G♯, ^⁶A♭
107	626.34	56/39	^^G♯, ^⁷A♭
108	632.2	36/25	^³G♯, ^⁸A♭
109	638.05		^⁴G♯, ^⁹A♭
110	643.9	29/20	^⁵G♯, v¹⁰A
111	649.76		^⁶G♯, v⁹A
112	655.61		^⁷G♯, v⁸A
113	661.46		^⁸G♯, v⁷A
114	667.32	25/17	^⁹G♯, v⁶A
115	673.17		^¹⁰G♯, v⁵A
116	679.02	37/25	v⁹G𝄪, v⁴A
117	684.88	52/35	v⁸G𝄪, v³A
118	690.73		v⁷G𝄪, vvA
119	696.59		v⁶G𝄪, vA
120	702.44	3/2	A
121	708.29		^A, ^⁶B♭♭
122	714.15	68/45	^^A, ^⁷B♭♭
123	720	47/31, 50/33	^³A, ^⁸B♭♭
124	725.85	38/25	^⁴A, ^⁹B♭♭
125	731.71	29/19	^⁵A, v¹⁰B♭
126	737.56		^⁶A, v⁹B♭
127	743.41		^⁷A, v⁸B♭
128	749.27	37/24	^⁸A, v⁷B♭
129	755.12	65/42	^⁹A, v⁶B♭
130	760.98	45/29	^¹⁰A, v⁵B♭
131	766.83		v⁹A♯, v⁴B♭
132	772.68	25/16	v⁸A♯, v³B♭
133	778.54	58/37, 69/44	v⁷A♯, vvB♭
134	784.39		v⁶A♯, vB♭
135	790.24	30/19	v⁵A♯, B♭
136	796.1	19/12	v⁴A♯, ^B♭
137	801.95	27/17, 62/39	v³A♯, ^^B♭
138	807.8	51/32	vvA♯, ^³B♭
139	813.66	8/5	vA♯, ^⁴B♭
140	819.51	69/43	A♯, ^⁵B♭
141	825.37	29/18	^A♯, ^⁶B♭
142	831.22	21/13	^^A♯, ^⁷B♭
143	837.07	60/37	^³A♯, ^⁸B♭
144	842.93		^⁴A♯, ^⁹B♭
145	848.78		^⁵A♯, v¹⁰B
146	854.63		^⁶A♯, v⁹B
147	860.49		^⁷A♯, v⁸B
148	866.34	33/20	^⁸A♯, v⁷B
149	872.2	48/29	^⁹A♯, v⁶B
150	878.05		^¹⁰A♯, v⁵B
151	883.9	5/3	v⁹A𝄪, v⁴B
152	889.76		v⁸A𝄪, v³B
153	895.61	47/28, 52/31	v⁷A𝄪, vvB
154	901.46	32/19, 69/41	v⁶A𝄪, vB
155	907.32		B
156	913.17		^B, ^⁶C♭
157	919.02	17/10	^^B, ^⁷C♭
158	924.88	29/17	^³B, ^⁸C♭
159	930.73		^⁴B, ^⁹C♭
160	936.59	55/32	^⁵B, v¹⁰C
161	942.44	50/29	^⁶B, v⁹C
162	948.29	64/37	^⁷B, v⁸C
163	954.15		^⁸B, v⁷C
164	960	47/27, 54/31	^⁹B, v⁶C
165	965.85		^¹⁰B, v⁵C
166	971.71		v⁹B♯, v⁴C
167	977.56	51/29	v⁸B♯, v³C
168	983.41	30/17	v⁷B♯, vvC
169	989.27	62/35	v⁶B♯, vC
170	995.12	16/9	C
171	1000.98	41/23, 66/37	^C, ^⁶D♭♭
172	1006.83	34/19	^^C, ^⁷D♭♭
173	1012.68	70/39	^³C, ^⁸D♭♭
174	1018.54	9/5	^⁴C, ^⁹D♭♭
175	1024.39	47/26, 56/31	^⁵C, v¹⁰D♭
176	1030.24	29/16	^⁶C, v⁹D♭
177	1036.1		^⁷C, v⁸D♭
178	1041.95		^⁸C, v⁷D♭
179	1047.8		^⁹C, v⁶D♭
180	1053.66	57/31, 68/37	^¹⁰C, v⁵D♭
181	1059.51		v⁹C♯, v⁴D♭
182	1065.37	37/20	v⁸C♯, v³D♭
183	1071.22	13/7	v⁷C♯, vvD♭
184	1077.07	41/22, 54/29	v⁶C♯, vD♭
185	1082.93	43/23	v⁵C♯, D♭
186	1088.78	15/8	v⁴C♯, ^D♭
187	1094.63	32/17	v³C♯, ^^D♭
188	1100.49	17/9	vvC♯, ^³D♭
189	1106.34	36/19	vC♯, ^⁴D♭
190	1112.2	19/10	C♯, ^⁵D♭
191	1118.05		^C♯, ^⁶D♭
192	1123.9	44/23	^^C♯, ^⁷D♭
193	1129.76	48/25	^³C♯, ^⁸D♭
194	1135.61	52/27	^⁴C♯, ^⁹D♭
195	1141.46	29/15	^⁵C♯, v¹⁰D
196	1147.32	33/17, 64/33	^⁶C♯, v⁹D
197	1153.17	37/19	^⁷C♯, v⁸D
198	1159.02		^⁸C♯, v⁷D
199	1164.88		^⁹C♯, v⁶D
200	1170.73	57/29	^¹⁰C♯, v⁵D
201	1176.59		v⁹C𝄪, v⁴D
202	1182.44		v⁸C𝄪, v³D
203	1188.29		v⁷C𝄪, vvD
204	1194.15		v⁶C𝄪, vD
205	1200	2/1	D

Notation

Ups and downs

205edo can be notated with ups and downs representing 5\205 = 1\41, and lifts and drops (written as / and \) representing 1\205. Alternatively, ups and downs represent 1\205 and the quintuple-arrow symbols quip and quid (> and <) represent 5\205 = 1\41. Both notations have the advantage of building on a familiarity with 41edo The first notation is especially useful for Kite guitarists who want to notate microbends more precisely.

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	…
P1	/1	//1	^\\1	^\1	^1	^/1	^//1	v\\m2	v\m2	vm2	v/m2	v//m2	\\m2	\m2	m2	/m2	…
P1	^1	^^1	^^^1	v>1	>1	^>1	^^>1	vv<m2	v<m2	<m2	^<m2	vvvm2	vvm2	vm2	m2	^m2

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3.5	1600000/1594323, [38 -2 -15⟩	[⟨205 325 476]]	−0.106	0.141	2.41
2.3.5.11	5632/5625, 14641/14580, 1600000/1594323	[⟨181 287 420 508]]	−0.002	0.218	3.72

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	6\205	35.122	45/44	Gammic (205e)
1	24\205	140.488	13/12	Quanic (205)
1	33\205	193.171	28/25	Luna / lunatic (205) / hemithirds (205d)
1	58\205	339.512	128/105	Amity (205)
5	63\205 (19\205)	368.780 (111.220)	1024/891 (16/15)	Quintosec
41	66\205 (1\205)	386.341 (5.85)	5/4 (32805/32768)	Countercomp

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

Scales

Quanic (24\205) mos

17-note: 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13
26-note: 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2

Amity (58\205) mos

11-note: 27 27 4 27 27 4 27 27 4 27 4
18-note: 23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4
25-note: 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4

Hemithirds (33\205) mos

13-note: 26 7 26 7 26 7 26 7 26 7 26 7 7
19-note: 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7
25-note: 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7
31-notes: 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7

Meantone (119\205) mos

12-note: 13 20 13 20 13 20 20 13 20 13 20 20
19-note: 13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7
31-note: 6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7

Myna (53\205) mos

11-note: 7 7 39 7 7 39 7 7 39 7 39
15-note: 7 7 7 32 7 7 7 32 7 7 7 32 7 7 32
19-note: 7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25
23-note: 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18
27-note: 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11
31-note: 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4

Porcupine (28\205) mos

15-note: 19 9 19 9 19 9 19 9 19 9 19 9 19 9 9
22-note: 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9
29-note: 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 9

External links

mem, 205-edo on Tonalsoft Encyclopedia