User:BudjarnLambeth/Sooty fox scale

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A sooty fox scale^{[idiosyncratic term]} (ed343/338 or syfx^{[idiosyncratic term]}) is an equal-step tuning in which 343/338 is justly tuned and is divided in a given number of equal steps.

This type of scale is named after the Aleutian sooty fox sparrow, taxa #343388 on iNaturalist.

The first sooty fox scale

← 0ed343/338

1ed343/338

2ed343/338 →

Prime factorization	n/a
Step size	25.4224 ¢
Octave	47\1ed343/338 (1194.85 ¢)
Twelfth	75\1ed343/338 (1906.68 ¢)
Consistency limit	3
Distinct consistency limit	3

1ed343/338 or 1syfx for short.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	25.4
2	50.8	31/30
3	76.3
4	101.7	18/17
5	127.1	14/13
6	152.5	12/11, 23/21
7	178	10/9, 21/19, 31/28
8	203.4
9	228.8
10	254.2
11	279.6	20/17, 27/23
12	305.1	25/21, 31/26
13	330.5	17/14, 23/19, 29/24
14	355.9
15	381.3
16	406.7	19/15
17	432.2	9/7
18	457.6	13/10, 30/23
19	483	29/22
20	508.4
21	533.9	19/14
22	559.3	18/13
23	584.7	7/5
24	610.1	27/19
25	635.5	13/9
26	661	19/13
27	686.4
28	711.8
29	737.2	23/15, 26/17
30	762.6	14/9, 31/20
31	788.1	30/19
32	813.5
33	838.9
34	864.3	23/14, 28/17
35	889.8	5/3
36	915.2	17/10
37	940.6	31/18
38	966
39	991.4	23/13
40	1016.9	9/5
41	1042.3	31/17
42	1067.7	13/7
43	1093.1
44	1118.5
45	1144
46	1169.4
47	1194.8	2/1

Harmonics

Approximation of harmonics in 1syfx

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-5.1	+4.7	-10.3	+10.1	-0.4	+12.4	+10.0	+9.4	+5.0	-7.5	-5.6
	Relative (%)	-20.2	+18.6	-40.5	+39.9	-1.7	+48.6	+39.3	+37.2	+19.7	-29.4	-21.9
Step		47	75	94	110	122	133	142	150	157	163	169

47edo, 109ed5 for comparison:

Approximation of harmonics in 47edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.0	-12.6	+0.0	-3.3	-12.6	+1.4	+0.0	+0.3	-3.3	+10.4	-12.6
	Relative (%)	+0.0	-49.3	+0.0	-13.1	-49.3	+5.4	+0.0	+1.4	-13.1	+40.7	-49.3
Steps (reduced)		47 (0)	74 (27)	94 (0)	109 (15)	121 (27)	132 (38)	141 (0)	149 (8)	156 (15)	163 (22)	168 (27)

Approximation of harmonics in 109ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.4	-10.3	+2.9	+0.0	-8.9	+5.4	+4.3	+4.9	+1.4	-10.2	-7.5
	Relative (%)	+5.6	-40.4	+11.3	+0.0	-34.8	+21.2	+16.9	+19.2	+5.6	-39.9	-29.2
Steps (reduced)		47 (47)	74 (74)	94 (94)	109 (0)	121 (12)	132 (23)	141 (32)	149 (40)	156 (47)	162 (53)	168 (59)

The second sooty fox scale

← 1ed343/338

2ed343/338

3ed343/338 →

Prime factorization	2 (prime) (highly composite)
Step size	12.7112 ¢
Octave	94\2ed343/338 (1194.85 ¢) (→ 47\1ed343/338)
Twelfth	150\2ed343/338 (1906.68 ¢) (→ 75\1ed343/338)
Consistency limit	2
Distinct consistency limit	2

2ed343/338 or 2syfx for short.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	12.7
2	25.4
3	38.1	43/42
4	50.8	35/34, 36/35
5	63.6
6	76.3	23/22
7	89	39/37, 41/39
8	101.7
9	114.4	31/29
10	127.1	14/13
11	139.8	13/12, 38/35
12	152.5
13	165.2
14	178	41/37
15	190.7	19/17
16	203.4
17	216.1	17/15, 43/38
18	228.8
19	241.5
20	254.2	22/19
21	266.9	7/6
22	279.6
23	292.3
24	305.1	37/31, 43/36
25	317.8	6/5
26	330.5	23/19
27	343.2
28	355.9	43/35
29	368.6
30	381.3
31	394
32	406.7	19/15, 43/34
33	419.5	37/29
34	432.2
35	444.9	22/17
36	457.6	30/23
37	470.3
38	483	41/31
39	495.7
40	508.4
41	521.1	23/17
42	533.9	34/25
43	546.6
44	559.3	29/21
45	572
46	584.7	7/5
47	597.4	41/29
48	610.1
49	622.8	43/30
50	635.5
51	648.3
52	661	22/15
53	673.7	31/21
54	686.4
55	699.1
56	711.8
57	724.5	35/23, 38/25
58	737.2
59	749.9
60	762.6
61	775.4	36/23
62	788.1
63	800.8
64	813.5
65	826.2	29/18
66	838.9
67	851.6	18/11
68	864.3
69	877
70	889.8
71	902.5	37/22
72	915.2	39/23
73	927.9
74	940.6	31/18, 43/25
75	953.3
76	966
77	978.7	37/21
78	991.4	39/22
79	1004.2	25/14
80	1016.9
81	1029.6
82	1042.3	31/17, 42/23
83	1055	35/19
84	1067.7
85	1080.4
86	1093.1
87	1105.8	36/19
88	1118.5	21/11
89	1131.3	25/13
90	1144	29/15
91	1156.7	41/21
92	1169.4
93	1182.1
94	1194.8

Harmonics

Approximation of harmonics in 2syfx

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-5.15	+4.72	+2.42	-2.56	-0.42	-0.36	-2.73	-3.26	+5.00	+5.24	-5.57
	Relative (%)	-40.5	+37.2	+19.0	-20.2	-3.3	-2.8	-21.5	-25.7	+39.4	+41.3	-43.8
Steps (reduced)		94 (0)	150 (0)	189 (1)	219 (1)	244 (0)	265 (1)	283 (1)	299 (1)	314 (0)	327 (1)	338 (0)

94edo, 218ed5 for comparison:

Approximation of harmonics in 94edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.00	+0.17	+0.00	-3.33	+0.17	+1.39	+0.00	+0.35	-3.33	-2.38	+0.17
	Relative (%)	+0.0	+1.4	+0.0	-26.1	+1.4	+10.9	+0.0	+2.7	-26.1	-18.7	+1.4
Steps (reduced)		94 (0)	149 (55)	188 (0)	218 (30)	243 (55)	264 (76)	282 (0)	298 (16)	312 (30)	325 (43)	337 (55)

Approximation of harmonics in 218ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.44	+2.45	+2.88	+0.00	+3.89	+5.43	+4.31	+4.90	+1.44	+2.59	+5.33
	Relative (%)	+11.3	+19.2	+22.5	+0.0	+30.4	+42.4	+33.8	+38.4	+11.3	+20.3	+41.7
Steps (reduced)		94 (94)	149 (149)	188 (188)	218 (0)	243 (25)	264 (46)	282 (64)	298 (80)	312 (94)	325 (107)	337 (119)

The third sooty fox scale

← 2ed343/338

3ed343/338

4ed343/338 →

Prime factorization	3 (prime)
Step size	8.47413 ¢
Octave	142\3ed343/338 (1203.33 ¢)
Twelfth	224\3ed343/338 (1898.21 ¢) (semiconvergent)
Consistency limit	2
Distinct consistency limit	2

3ed343/338 or 3syfx for short.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	8.5
2	16.9
3	25.4
4	33.9	50/49
5	42.4	42/41
6	50.8	35/34
7	59.3	30/29
8	67.8	26/25
9	76.3	23/22
10	84.7
11	93.2
12	101.7	52/49
13	110.2	49/46
14	118.6
15	127.1
16	135.6
17	144.1	25/23
18	152.5
19	161	34/31
20	169.5	43/39
21	178	41/37
22	186.4	49/44
23	194.9	47/42
24	203.4
25	211.8	26/23
26	220.3	25/22, 42/37
27	228.8
28	237.3	47/41
29	245.7
30	254.2	22/19
31	262.7
32	271.2	55/47
33	279.6
34	288.1	13/11
35	296.6	51/43
36	305.1	31/26, 37/31
37	313.5
38	322
39	330.5	23/19
40	339
41	347.4
42	355.9
43	364.4	21/17, 37/30
44	372.8	31/25, 36/29
45	381.3
46	389.8
47	398.3
48	406.7
49	415.2	47/37
50	423.7
51	432.2
52	440.6	49/38
53	449.1
54	457.6	43/33
55	466.1	17/13, 55/42
56	474.5	25/19, 46/35
57	483	41/31
58	491.5
59	500
60	508.4	55/41
61	516.9	31/23
62	525.4	42/31
63	533.9	34/25
64	542.3	26/19, 41/30
65	550.8
66	559.3	29/21, 47/34
67	567.7
68	576.2
69	584.7
70	593.2	31/22
71	601.6	17/12
72	610.1	37/26
73	618.6	10/7
74	627.1
75	635.5
76	644
77	652.5	35/24
78	661
79	669.4
80	677.9	34/23, 37/25
81	686.4	52/35, 55/37
82	694.9
83	703.3
84	711.8
85	720.3	47/31
86	728.8
87	737.2	26/17
88	745.7
89	754.2	17/11
90	762.6
91	771.1
92	779.6
93	788.1	41/26
94	796.5	19/12
95	805	35/22
96	813.5
97	822	37/23
98	830.4	21/13
99	838.9
100	847.4	31/19
101	855.9	41/25
102	864.3
103	872.8
104	881.3
105	889.8
106	898.2	42/25
107	906.7
108	915.2
109	923.7	29/17
110	932.1	12/7
111	940.6
112	949.1
113	957.5
114	966
115	974.5
116	983	30/17
117	991.4	55/31
118	999.9	41/23
119	1008.4	34/19
120	1016.9
121	1025.3	47/26
122	1033.8
123	1042.3	42/23
124	1050.8	11/6
125	1059.2	35/19
126	1067.7
127	1076.2	41/22
128	1084.7
129	1093.1	47/25
130	1101.6
131	1110.1	19/10
132	1118.5	21/11
133	1127	23/12
134	1135.5
135	1144
136	1152.4	37/19
137	1160.9
138	1169.4
139	1177.9
140	1186.3
141	1194.8

Harmonics

Approximation of harmonics in 3syfx

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+3.33	-3.75	-1.82	+1.68	-0.42	+3.88	+1.51	+0.98	-3.47	+1.01	+2.90
	Relative (%)	+39.3	-44.2	-21.5	+19.8	-5.0	+45.8	+17.8	+11.5	-41.0	+11.9	+34.3
Steps (reduced)		142 (1)	224 (2)	283 (1)	329 (2)	366 (0)	398 (2)	425 (2)	449 (2)	470 (2)	490 (1)	508 (1)

141edo, 327ed5 for comparison:

Approximation of harmonics in 141edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.00	-4.08	+0.00	-3.33	-4.08	+1.39	+0.00	+0.35	-3.33	+1.87	-4.08
	Relative (%)	+0.0	-48.0	+0.0	-39.2	-48.0	+16.3	+0.0	+4.1	-39.2	+22.0	-48.0
Steps (reduced)		141 (0)	223 (82)	282 (0)	327 (45)	364 (82)	396 (114)	423 (0)	447 (24)	468 (45)	488 (65)	505 (82)

Approximation of harmonics in 327ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.44	-1.81	+2.88	+0.00	-0.37	-3.10	-4.21	-3.62	+1.44	-1.67	+1.07
	Relative (%)	+16.9	-21.2	+33.8	+0.0	-4.3	-36.3	-49.4	-42.4	+16.9	-19.6	+12.5
Steps (reduced)		141 (141)	223 (223)	282 (282)	327 (0)	364 (37)	395 (68)	422 (95)	446 (119)	468 (141)	487 (160)	505 (178)

The fourth sooty fox scale

← 3ed343/338

4ed343/338

5ed343/338 →

Prime factorization	22 (highly composite)
Step size	6.3556 ¢
Octave	189\4ed343/338 (1201.21 ¢) (convergent)
Twelfth	299\4ed343/338 (1900.32 ¢) (semiconvergent)
Consistency limit	3
Distinct consistency limit	3

4ed343/338 or 4syfx for short.

Harmonics

Approximation of harmonics in 4syfx

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.21	-1.63	+2.42	-2.56	-0.42	-0.36	-2.73	+3.09	-1.35	-1.11	+0.79
	Relative (%)	+19.0	-25.7	+38.0	-40.3	-6.7	-5.6	-43.0	+48.7	-21.3	-17.5	+12.4
Steps (reduced)		189 (1)	299 (3)	378 (2)	438 (2)	488 (0)	530 (2)	566 (2)	599 (3)	627 (3)	653 (1)	677 (1)

188edo, 436ed5 for comparison:

Approximation of harmonics in 188edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.00	+0.17	+0.00	+3.05	+0.17	+1.39	+0.00	+0.35	+3.05	-2.38	+0.17
	Relative (%)	+0.0	+2.7	+0.0	+47.8	+2.7	+21.7	+0.0	+5.4	+47.8	-37.3	+2.7
Steps (reduced)		188 (0)	298 (110)	376 (0)	437 (61)	486 (110)	528 (152)	564 (0)	596 (32)	625 (61)	650 (86)	674 (110)

Approximation of harmonics in 436ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.44	+2.45	+2.88	+0.00	-2.50	-0.97	-2.08	-1.49	+1.44	+2.59	-1.06
	Relative (%)	+22.5	+38.4	+45.0	+0.0	-39.1	-15.1	-32.5	-23.3	+22.5	+40.5	-16.6
Steps (reduced)		188 (188)	298 (298)	376 (376)	436 (0)	485 (49)	527 (91)	563 (127)	595 (159)	624 (188)	650 (214)	673 (237)

The fifth sooty fox scale

← 4ed343/338

5ed343/338

6ed343/338 →

Prime factorization	5 (prime)
Step size	5.08448 ¢
Octave	236\5ed343/338 (1199.94 ¢) (convergent)
Twelfth	374\5ed343/338 (1901.6 ¢) (convergent)
Consistency limit	6
Distinct consistency limit	6

5ed343/338 or 5syfx for short.

Harmonics

Approximation of harmonics in 5syfx

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.06	-0.36	-0.13	-0.02	-0.42	+2.18	-0.19	-0.72	-0.08	-2.38	-0.49
	Relative (%)	-1.2	-7.1	-2.5	-0.4	-8.3	+43.0	-3.7	-14.2	-1.6	-46.9	-9.6
Steps (reduced)		236 (1)	374 (4)	472 (2)	548 (3)	610 (0)	663 (3)	708 (3)	748 (3)	784 (4)	816 (1)	846 (1)

235edo, 545ed5 for comparison:

Approximation of harmonics in 235edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.00	-2.38	+0.00	+1.77	-2.38	+1.39	+0.00	+0.35	+1.77	+0.17	-2.38
	Relative (%)	+0.0	-46.6	+0.0	+34.7	-46.6	+27.2	+0.0	+6.8	+34.7	+3.4	-46.6
Steps (reduced)		235 (0)	372 (137)	470 (0)	546 (76)	607 (137)	660 (190)	705 (0)	745 (40)	781 (76)	813 (108)	842 (137)

Approximation of harmonics in 545ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.44	-0.10	-2.24	+0.00	+1.33	+0.31	-0.80	-0.21	+1.44	+0.03	-2.34
	Relative (%)	+28.1	-2.0	-43.7	+0.0	+26.1	+6.1	-15.6	-4.1	+28.1	+0.7	-45.8
Steps (reduced)		235 (235)	372 (372)	469 (469)	545 (0)	607 (62)	659 (114)	704 (159)	744 (199)	780 (235)	812 (267)	841 (296)