16edf

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16edf

17edf →

Prime factorization	24
Step size	43.8722 ¢
Octave	27\16edf (1184.55 ¢)
Twelfth	43\16edf (1886.5 ¢)
Consistency limit	3
Distinct consistency limit	3

16EDF is the equal division of the just perfect fifth into 16 parts of 43.8722 cents each, corresponding to 27.3522 edo (similar to every third step of 82edo). 16edf contains good approximations of the 7th and 13th harmonics.

It serves as a good approximation to halftone temperament, containing the ~7/5 generator at 13 steps.

Lookalikes: 27edo, 43edt

Intervals

degree	cents value	corresponding JI intervals	Halftone[6] notation	comments
0	0.0000	1/1	C
1	43.8722	40/39, 39/38	^C
2	87.7444	20/19	Db
3	131.6166	55/51, (27/25)	vD
4	175.4888	(21/19)	D
5	219.3609		vE
6	263.2331	(7/6)	E
7	307.1053		Fb
8	350.9775	60/49, 49/40	vf
9	394.8497	(44/35)	F
10	438.7219	(9/7)	Ab
11	482.5941		vA
12	526.4663	(19/14)	A
13	570.3384	(25/18), 153/110, 112/81	B
14	614.2106	(10/7)	Cb
15	658.0828	19/13	vC
16	701.9550	3/2 (exact)	C	just perfect fifth
17	745.8272	20/13
18	789.6994	30/19
19	833.5716	55/34
20	877.4438
21	921.3159
22	965.1881		7/4
23	1009.0603
24	1052.9325	90/49, (11/6)
25	1096.8047	(66/35)
26	1140.6769
27	1184.5491
28	1228.4213	128/63
29	1272.2934	25/12
30	1316.1656	15/7
31	1360.0378	57/26
32	1403.9100	9/4 (exact)		pythagorean ninth

Scale tree

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

Generator range: 42.85714 cents (4\7/16 = 1\28) to 45 cents (3\5/16 = 3\80)

Fifth							Cents	Comments
4\7							42.8571
						27\47	43.0851
					23\40		43.1250
						42\73	43.1507
				19\33			43.18
						53\92	43.2065
					34\59		43.2203
						49\85	43.2353
			15\26				43.2692
						56\97	43.2990
					41\71		43.3099
						67\116	43.3190
				26\45			43.3	Flattone is in this region
						63\109	43.3486
					37\64		43.3594
						48\83	43.3735
		11\19					43.42105
						51\88	43.46590
					40\69		43.4783
						69\119	43.4874
				29\50			43.5000
						76\131	43.51145	Golden meantone (696.2145¢) The generator closest to a just 9/7 for EDOs less than 800
					47\81		43.518
						65\112	43.5268
			18\31				43.5484	Meantone is in this region
						61\105	43.5714
					43\74		43.5810
						68\117	43.5897
				25\43			43.60465
						57\98	43.62245
					32\55		43.63
						39\67	43.6567
	7\12						43.7500
						38\65	43.84615
					31\53		43.8679	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	43.8830	Garibaldi / Cassandra
				24\41			43.9024
						65\111	43.918
					41\70		43.9286
						58\99	43.93
			17\29				43.9655
						61\104	43.9904
					44\75		44.0000
						71\121	44.0083	Golden neogothic (704.0956¢)
				27\46			44.0217	Neogothic is in this region
						64\109	44.0367
					37\63		44.0476
						47\80	44.0625
		10\17					44.11765
						43\73	44.1781
					33\56		44.1964
						56\95	44.2105
				23\39			44.3208
						59\100	43.2500
					36\61		44.2623
						49\83	44.2771
			13\22				44.318	Archy is in this region
						42\71	44.3662
					29\49		44.3878
						45\76	44.4079
				16\27			44.4
						35\59	44.4915
					19\32		44.53125
						22\37	44.594
3\5							45.0000

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

Compositions

schizophrenic lullaby fugue