224edo

← 223edo 224edo 225edo →
Prime factorization	25 × 7
Step size	5.35714 ¢
Fifth	131\224 (701.786 ¢)
Semitones (A1:m2)	21:17 (112.5 ¢ : 91.07 ¢)
Consistency limit	15
Distinct consistency limit	15

224 equal divisions of the octave (abbreviated 224edo or 224ed2), also called 224-tone equal temperament (224tet) or 224 equal temperament (224et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 224 equal parts of about 5.36 ¢ each. Each step represents a frequency ratio of 2^1/224, or the 224th root of 2.

Theory

224edo is a very strong 13-limit system. It is the twelfth zeta integral edo and is the second-smallest edo after 87 to approximate all of the first 16 harmonics of the harmonic series with no greater than 25% relative error.

As an equal temperament, 224et tempering out 32805/32768 in the 5-limit; 4375/4374, 16875/16807 and 65625/65536 in the 7-limit; 540/539, 1375/1372, 4000/3993 and notably, the quartisma in the 11-limit; and 625/624, 729/728, 1575/1573 and 2200/2197 in the 13-limit, leading to an abundance of precisely-tuned essentially tempered chords, including swetismic chords, squbemic chords, and petrmic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit. It defines the optimal patent val for the octoid in the 7-, 11- and 13-limit, and for mirkwai, the 7-limit planar temperament tempering out 16875/16807. It also provides an excellent tuning for indra and shibi temperaments. 217edo, only a bit smaller, has a worse 13-limit, but it achieves a much higher consistency limit, almost 31-odd.

224edo tempers the syntonic comma to 1/56th of the octave (4 steps) and as a corollary supports the barium temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for meantone and is very close (0.15 cents) to the quarter-comma meantone fifth. The generator however reduces to 112edo, being 65\112.

Prime harmonics

Approximation of prime harmonics in 224edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.17	-0.60	+0.82	+0.47	+0.54	+2.19	+2.49	-1.49	-1.01	+1.39
Error	Relative (%)	+0.0	-3.2	-11.2	+15.2	+8.7	+10.2	+40.8	+46.4	-27.8	-18.8	+26.0
Steps (reduced)		224 (0)	355 (131)	520 (72)	629 (181)	775 (103)	829 (157)	916 (20)	952 (56)	1013 (117)	1088 (192)	1110 (214)

Approximation of prime harmonics in 224edo (continued)
Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+0.44	-0.49	-2.59	-1.22	-0.29	+1.54	-2.60	+1.05	+2.45	+2.57	-0.25
Error	Relative (%)	+8.2	-9.2	-48.3	-22.8	-5.4	+28.8	-48.5	+19.6	+45.7	+47.9	-4.7
Steps (reduced)		1167 (47)	1200 (80)	1215 (95)	1244 (124)	1283 (163)	1318 (198)	1328 (208)	1359 (15)	1378 (34)	1387 (43)	1412 (68)

Subsets and supersets

Since 224 factors into 2⁵ × 7, 224edo has subset edos 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	5.36		^D, ^⁵E♭♭
2	10.71		^^D, ^⁶E♭♭
3	16.07		^³D, ^⁷E♭♭
4	21.43		^⁴D, ^⁸E♭♭
5	26.79	63/62, 64/63, 65/64, 66/65	^⁵D, ^⁹E♭♭
6	32.14	55/54, 56/55	^⁶D, ^¹⁰E♭♭
7	37.5	46/45, 47/46	^⁷D, v¹⁰E♭
8	42.86	41/40	^⁸D, v⁹E♭
9	48.21	36/35, 37/36	^⁹D, v⁸E♭
10	53.57	33/32, 65/63	^¹⁰D, v⁷E♭
11	58.93	30/29	v¹⁰D♯, v⁶E♭
12	64.29		v⁹D♯, v⁵E♭
13	69.64	51/49	v⁸D♯, v⁴E♭
14	75	47/45	v⁷D♯, v³E♭
15	80.36	22/21	v⁶D♯, vvE♭
16	85.71	41/39	v⁵D♯, vE♭
17	91.07	39/37, 58/55	v⁴D♯, E♭
18	96.43	37/35, 55/52	v³D♯, ^E♭
19	101.79	35/33	vvD♯, ^^E♭
20	107.14	50/47	vD♯, ^³E♭
21	112.5	16/15	D♯, ^⁴E♭
22	117.86		^D♯, ^⁵E♭
23	123.21	29/27, 44/41	^^D♯, ^⁶E♭
24	128.57	14/13	^³D♯, ^⁷E♭
25	133.93	27/25	^⁴D♯, ^⁸E♭
26	139.29	13/12	^⁵D♯, ^⁹E♭
27	144.64	25/23, 62/57	^⁶D♯, ^¹⁰E♭
28	150	12/11	^⁷D♯, v¹⁰E
29	155.36	35/32	^⁸D♯, v⁹E
30	160.71	34/31, 45/41	^⁹D♯, v⁸E
31	166.07		^¹⁰D♯, v⁷E
32	171.43		v¹⁰D𝄪, v⁶E
33	176.79	31/28, 41/37, 72/65	v⁹D𝄪, v⁵E
34	182.14	10/9	v⁸D𝄪, v⁴E
35	187.5	39/35	v⁷D𝄪, v³E
36	192.86	19/17	v⁶D𝄪, vvE
37	198.21	37/33, 65/58	v⁵D𝄪, vE
38	203.57	9/8	E
39	208.93	44/39	^E, ^⁵F♭
40	214.29		^^E, ^⁶F♭
41	219.64	42/37	^³E, ^⁷F♭
42	225	41/36, 74/65	^⁴E, ^⁸F♭
43	230.36	8/7	^⁵E, ^⁹F♭
44	235.71	47/41, 55/48, 63/55	^⁶E, ^¹⁰F♭
45	241.07	23/20, 54/47	^⁷E, v¹⁰F
46	246.43		^⁸E, v⁹F
47	251.79	37/32	^⁹E, v⁸F
48	257.14	29/25, 65/56	^¹⁰E, v⁷F
49	262.5	57/49, 64/55	v¹⁰E♯, v⁶F
50	267.86		v⁹E♯, v⁵F
51	273.21	41/35, 48/41	v⁸E♯, v⁴F
52	278.57	47/40, 74/63	v⁷E♯, v³F
53	283.93	33/28	v⁶E♯, vvF
54	289.29	13/11	v⁵E♯, vF
55	294.64	32/27	F
56	300	44/37, 69/58	^F, ^⁵G♭♭
57	305.36	31/26, 37/31, 68/57	^^F, ^⁶G♭♭
58	310.71		^³F, ^⁷G♭♭
59	316.07	6/5	^⁴F, ^⁸G♭♭
60	321.43	65/54	^⁵F, ^⁹G♭♭
61	326.79	29/24	^⁶F, ^¹⁰G♭♭
62	332.14	40/33, 63/52	^⁷F, v¹⁰G♭
63	337.5	62/51	^⁸F, v⁹G♭
64	342.86	39/32, 50/41	^⁹F, v⁸G♭
65	348.21	11/9	^¹⁰F, v⁷G♭
66	353.57		v¹⁰F♯, v⁶G♭
67	358.93	16/13	v⁹F♯, v⁵G♭
68	364.29	58/47	v⁸F♯, v⁴G♭
69	369.64	26/21	v⁷F♯, v³G♭
70	375	36/29, 41/33	v⁶F♯, vvG♭
71	380.36		v⁵F♯, vG♭
72	385.71	5/4	v⁴F♯, G♭
73	391.07		v³F♯, ^G♭
74	396.43	44/35	vvF♯, ^^G♭
75	401.79	29/23	vF♯, ^³G♭
76	407.14	62/49	F♯, ^⁴G♭
77	412.5	33/26	^F♯, ^⁵G♭
78	417.86	14/11	^^F♯, ^⁶G♭
79	423.21	60/47	^³F♯, ^⁷G♭
80	428.57	41/32	^⁴F♯, ^⁸G♭
81	433.93		^⁵F♯, ^⁹G♭
82	439.29	49/38, 58/45	^⁶F♯, ^¹⁰G♭
83	444.64	75/58	^⁷F♯, v¹⁰G
84	450	35/27, 48/37	^⁸F♯, v⁹G
85	455.36		^⁹F♯, v⁸G
86	460.71	30/23, 47/36	^¹⁰F♯, v⁷G
87	466.07	55/42, 72/55	v¹⁰F𝄪, v⁶G
88	471.43	21/16	v⁹F𝄪, v⁵G
89	476.79	54/41	v⁸F𝄪, v⁴G
90	482.14	37/28	v⁷F𝄪, v³G
91	487.5		v⁶F𝄪, vvG
92	492.86		v⁵F𝄪, vG
93	498.21	4/3	G
94	503.57		^G, ^⁵A♭♭
95	508.93	51/38, 55/41	^^G, ^⁶A♭♭
96	514.29	35/26, 74/55	^³G, ^⁷A♭♭
97	519.64	27/20	^⁴G, ^⁸A♭♭
98	525	42/31, 65/48	^⁵G, ^⁹A♭♭
99	530.36		^⁶G, ^¹⁰A♭♭
100	535.71		^⁷G, v¹⁰A♭
101	541.07	41/30	^⁸G, v⁹A♭
102	546.43	37/27, 48/35	^⁹G, v⁸A♭
103	551.79	11/8	^¹⁰G, v⁷A♭
104	557.14	40/29, 69/50	v¹⁰G♯, v⁶A♭
105	562.5	18/13	v⁹G♯, v⁵A♭
106	567.86	25/18, 68/49	v⁸G♯, v⁴A♭
107	573.21	39/28	v⁷G♯, v³A♭
108	578.57		v⁶G♯, vvA♭
109	583.93		v⁵G♯, vA♭
110	589.29	45/32, 52/37	v⁴G♯, A♭
111	594.64	31/22, 55/39	v³G♯, ^A♭
112	600	41/29, 58/41	vvG♯, ^^A♭
113	605.36	44/31	vG♯, ^³A♭
114	610.71	37/26, 64/45	G♯, ^⁴A♭
115	616.07		^G♯, ^⁵A♭
116	621.43	63/44	^^G♯, ^⁶A♭
117	626.79	56/39	^³G♯, ^⁷A♭
118	632.14	36/25, 49/34	^⁴G♯, ^⁸A♭
119	637.5	13/9	^⁵G♯, ^⁹A♭
120	642.86	29/20	^⁶G♯, ^¹⁰A♭
121	648.21	16/11	^⁷G♯, v¹⁰A
122	653.57	35/24, 54/37	^⁸G♯, v⁹A
123	658.93	60/41	^⁹G♯, v⁸A
124	664.29	69/47	^¹⁰G♯, v⁷A
125	669.64		v¹⁰G𝄪, v⁶A
126	675	31/21, 65/44	v⁹G𝄪, v⁵A
127	680.36	40/27	v⁸G𝄪, v⁴A
128	685.71	52/35, 55/37	v⁷G𝄪, v³A
129	691.07	76/51	v⁶G𝄪, vvA
130	696.43		v⁵G𝄪, vA
131	701.79	3/2	A
132	707.14		^A, ^⁵B♭♭
133	712.5		^^A, ^⁶B♭♭
134	717.86	56/37	^³A, ^⁷B♭♭
135	723.21	41/27	^⁴A, ^⁸B♭♭
136	728.57	32/21	^⁵A, ^⁹B♭♭
137	733.93	55/36	^⁶A, ^¹⁰B♭♭
138	739.29	23/15, 72/47	^⁷A, v¹⁰B♭
139	744.64		^⁸A, v⁹B♭
140	750	37/24, 54/35	^⁹A, v⁸B♭
141	755.36	65/42	^¹⁰A, v⁷B♭
142	760.71	45/29, 76/49	v¹⁰A♯, v⁶B♭
143	766.07		v⁹A♯, v⁵B♭
144	771.43	64/41	v⁸A♯, v⁴B♭
145	776.79	47/30	v⁷A♯, v³B♭
146	782.14	11/7	v⁶A♯, vvB♭
147	787.5	52/33	v⁵A♯, vB♭
148	792.86	49/31	v⁴A♯, B♭
149	798.21	46/29, 65/41	v³A♯, ^B♭
150	803.57	35/22	vvA♯, ^^B♭
151	808.93	75/47	vA♯, ^³B♭
152	814.29	8/5	A♯, ^⁴B♭
153	819.64	69/43	^A♯, ^⁵B♭
154	825	29/18, 66/41	^^A♯, ^⁶B♭
155	830.36	21/13	^³A♯, ^⁷B♭
156	835.71	47/29	^⁴A♯, ^⁸B♭
157	841.07	13/8	^⁵A♯, ^⁹B♭
158	846.43	75/46	^⁶A♯, ^¹⁰B♭
159	851.79	18/11	^⁷A♯, v¹⁰B
160	857.14	41/25, 64/39	^⁸A♯, v⁹B
161	862.5	51/31	^⁹A♯, v⁸B
162	867.86	33/20	^¹⁰A♯, v⁷B
163	873.21	48/29	v¹⁰A𝄪, v⁶B
164	878.57		v⁹A𝄪, v⁵B
165	883.93	5/3	v⁸A𝄪, v⁴B
166	889.29		v⁷A𝄪, v³B
167	894.64	52/31, 57/34, 62/37	v⁶A𝄪, vvB
168	900	37/22	v⁵A𝄪, vB
169	905.36	27/16	B
170	910.71	22/13	^B, ^⁵C♭
171	916.07	56/33	^^B, ^⁶C♭
172	921.43	63/37	^³B, ^⁷C♭
173	926.79	41/24, 70/41	^⁴B, ^⁸C♭
174	932.14		^⁵B, ^⁹C♭
175	937.5	55/32	^⁶B, ^¹⁰C♭
176	942.86	50/29	^⁷B, v¹⁰C
177	948.21	64/37	^⁸B, v⁹C
178	953.57		^⁹B, v⁸C
179	958.93	40/23, 47/27	^¹⁰B, v⁷C
180	964.29		v¹⁰B♯, v⁶C
181	969.64	7/4	v⁹B♯, v⁵C
182	975	65/37, 72/41	v⁸B♯, v⁴C
183	980.36	37/21	v⁷B♯, v³C
184	985.71		v⁶B♯, vvC
185	991.07	39/22	v⁵B♯, vC
186	996.43	16/9	C
187	1001.79	66/37	^C, ^⁵D♭♭
188	1007.14	34/19	^^C, ^⁶D♭♭
189	1012.5	70/39	^³C, ^⁷D♭♭
190	1017.86	9/5	^⁴C, ^⁸D♭♭
191	1023.21	56/31, 65/36, 74/41	^⁵C, ^⁹D♭♭
192	1028.57		^⁶C, ^¹⁰D♭♭
193	1033.93		^⁷C, v¹⁰D♭
194	1039.29	31/17	^⁸C, v⁹D♭
195	1044.64	64/35, 75/41	^⁹C, v⁸D♭
196	1050	11/6	^¹⁰C, v⁷D♭
197	1055.36	46/25, 57/31	v¹⁰C♯, v⁶D♭
198	1060.71	24/13	v⁹C♯, v⁵D♭
199	1066.07	50/27	v⁸C♯, v⁴D♭
200	1071.43	13/7	v⁷C♯, v³D♭
201	1076.79	41/22, 54/29	v⁶C♯, vvD♭
202	1082.14		v⁵C♯, vD♭
203	1087.5	15/8	v⁴C♯, D♭
204	1092.86	47/25	v³C♯, ^D♭
205	1098.21	66/35	vvC♯, ^^D♭
206	1103.57	70/37	vC♯, ^³D♭
207	1108.93	55/29, 74/39	C♯, ^⁴D♭
208	1114.29		^C♯, ^⁵D♭
209	1119.64	21/11	^^C♯, ^⁶D♭
210	1125		^³C♯, ^⁷D♭
211	1130.36		^⁴C♯, ^⁸D♭
212	1135.71		^⁵C♯, ^⁹D♭
213	1141.07	29/15	^⁶C♯, ^¹⁰D♭
214	1146.43	64/33	^⁷C♯, v¹⁰D
215	1151.79	35/18, 72/37	^⁸C♯, v⁹D
216	1157.14		^⁹C♯, v⁸D
217	1162.5	45/23	^¹⁰C♯, v⁷D
218	1167.86	55/28	v¹⁰C𝄪, v⁶D
219	1173.21	63/32, 65/33	v⁹C𝄪, v⁵D
220	1178.57		v⁸C𝄪, v⁴D
221	1183.93		v⁷C𝄪, v³D
222	1189.29		v⁶C𝄪, vvD
223	1194.64		v⁵C𝄪, vD
224	1200	2/1	D

Notation

Sagittal

224edo can be written in Sagittal using almost the entire Athenian extension, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian^[1]. It is identical to 217edo's Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.^[2]

Sagittal notation
Steps		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
Symbol	Evo																						
Symbol	Revo																						

Because it uses the entire Athenian system (except for     since it tempers 1240029/1239040), it allows no accidental enharmonic respellings

Ups-and-downs notation

The 4-up (quup) alteration maps to the pythagorean/syntonic comma.

Ups-and-downs notation
0	1	2	3	4	5	6	7	8	9	10
h	^	^^	^^^	v>	>	^>	^^>	^^^>	v>>	>>
h	<<<<#	^<<<<#	vvv<<<#	vv<<<#	v<<<#	<<<#	^<<<#	vvv<<#	vv<<#	v<<#
11	12	13	14	15	16	17	18	19	20	21
^>>	^^>>	^^^>>	v>>>	>>>	^>>>	^^>>>	^^^>>>	v>>>>	>>>>	#
<<#	^<<#	vvv<#	vv<#	v<#	<#	^<#	vvv#	vv#	v#	#

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 224edo. Prime harmonics are in bold.

As 224edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 224edo
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/11, 22/13	0.076	1.4
3/2, 4/3	0.169	3.2
9/5, 10/9	0.261	4.9
13/7, 14/13	0.273	5.1
9/8, 16/9	0.339	6.3
11/7, 14/11	0.349	6.5
5/3, 6/5	0.430	8.0
11/8, 16/11	0.468	8.7
13/8, 16/13	0.544	10.2
5/4, 8/5	0.599	11.2
11/6, 12/11	0.637	11.9
13/12, 24/13	0.713	13.3
15/8, 16/15	0.769	14.3
11/9, 18/11	0.806	15.1
7/4, 8/7	0.817	15.2
13/9, 18/13	0.882	16.5
7/6, 12/7	0.986	18.4
11/10, 20/11	1.067	19.9
13/10, 20/13	1.143	21.3
9/7, 14/9	1.156	21.6
15/11, 22/15	1.236	23.1
15/13, 26/15	1.312	24.5
7/5, 10/7	1.416	26.4
15/14, 28/15	1.586	29.6

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-355 224⟩	[⟨224 355]]	+0.053	0.0534	1.00
2.3.5	32805/32768, [-5 -32 24⟩	[⟨224 355 520]]	+0.122	0.1059	1.98
2.3.5.7	4375/4374, 16875/16807, 32805/32768	[⟨224 355 520 629]]	+0.018	0.2009	3.75
2.3.5.7.11	540/539, 1375/1372, 4000/3993, 32805/32768	[⟨224 355 520 629 775]]	−0.012	0.1899	3.54
2.3.5.7.11.13	540/539, 625/624, 729/728, 1375/1372, 2200/2197	[⟨224 355 520 629 775 829]]	−0.035	0.1805	3.37
2.3.5.7.11.13.17	375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197	[⟨224 355 520 629 775 829 916]]	−0.106	0.2420	4.52

224et has a lower relative error than any previous equal temperaments in the 13-limit, being the first to beat 72. The next equal temperament that does better in terms of either absolute or relative error is 270.
It is also notable in the 11- and 17-limit, with lower absolute errors than any previous equal temperaments. In the 11-limit it is the first to beat 152 and is superseded by 239. In the 17-limit it is the first to beat 217 and is superseded by 270.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	43\224	230.36	8/7	Gamera
1	59\224	316.07	6/5	Counterkleismic / counterlytic
1	65\224	348.21	11/9	Eris
1	71\224	380.36	56/45	Quanharuk
1	87\224	466.07	55/42	Hemiseptisix
1	93\224	498.21	4/3	Pontiac / ponta
1	103\224	551.79	11/8	Emkay
1	111\224	594.64	55/39	Gaster
2	93\224 (19\224)	498.21 (101.79)	4/3 (35/33)	Bipont
2	31\224	166.07	11/10	Pogo
2	33\224	176.79	195/176	Quatracot
2	39\224	208.93	44/39	Abigail
2	43\224	230.36	8/7	Hemigamera
4	71\224 (15\224)	380.36 (80.36)	81/65 (22/21)	Quasithird
4	93\224 (19\224)	498.21 (101.79)	4/3 (35/33)	Quadrant
7	97\224 (1\224)	519.64 (5.36)	27/20 (325/324)	Brahmagupta
7	93\224 (3\224)	498.21 (16.07)	4/3 (99/98)	Septant
8	93\224 (9\224)	498.21 (48.21)	4/3 (36/35)	Octant
8	109\224 (3\224)	583.93 (16.07)	7/5 (100/99)	Octoid
14	93\224 (3\224)	498.21 (16.07)	4/3 (105/104)	Silicon
28	93\224 (3\224)	498.21 (16.07)	4/3 (126/125)	Oquatonic
32	50\224 (1\224)	267.86 (5.36)	245/143 (???)	Germanium
32	93\224 (2\224)	498.21 (10.71)	4/3 (???)	Bezique
56	93\224 (3\224)	498.21 (16.07)	4/3 (126/125)	Barium

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

Music

Gene Ward Smith

Dreyfus (archived 2010) – SoundCloud | details | play – Octoid[72] in 224edo tuning

Mercury Amalgam

Kindness Is A Weakness (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning

[:0-1] Ragismic microtemperaments#Brahmagupta

[2] ttps://sagittal.org/sagittal.pdf p. 11

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