10edf

← 9edf

10edf

11edf →

Prime factorization	2 × 5
Step size	70.1955¢
Octave	17\10edf (1193.32¢) (semiconvergent)
Twelfth	27\10edf (1895.28¢) (semiconvergent)
Consistency limit	7
Distinct consistency limit	4

Division of the just perfect fifth into 10 equal parts (10EDF) is related to 17 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the 7-integer-limit, but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit.

Lookalikes: 17edo, 27edt

Intervals

degree		Neptunian notation using 8\10edf	Neapolitan notation using 3/10edf
0		C	F
1	70.1955	^C, vDb	F^, Gb
2	140.391	C#, Db	F#, Gd
3	210.5865	vD	G
4	280.782	D	G^, Ab
5	350.9775	^D, vE	G#, Ad
6	421.173	E	A
7	491.3685	^E, vF	A^, Hb
8	561.564	F	A#, Hd
9	631.7595	^F, vC	H
10	701.955	C	B
11	772.1505	^C, vDb	B^, Cb
12	842.346	C#, Db	B#, Cd
13	912.5415	vD	C
14	982.737	D	C^, Db
15	1052.9325	^D, vE	C#, Dd
16	1123.128	E	D
17	1193.3235	^E, vF	D^, Eb
18	1263.519	F	D#, Eb
19	1333.7145	^F, vC	E
20	1403.91	C	F

Scale tree

EDF scales can be approximated in EDOs by subdividing diatonic fifths. 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: 68.57143 cents (4\7/10 = 2\35) to 72 cents (3\5/10 = 3\50)

Fifth							Cents	Comments
4\7							68.5714
						27\47	68.9362
					23\40		69.000
						42\73	69.0411
				19\33			69.0909
						53\92	69.1304
					34\59		69.1525
						49\85	69.1765
			15\26				69.2308
						56\97	69.2784
					41\71		69.2958
						67\116	69.3103
				26\45			69.3333	Flattone is in this region
						63\109	69.3578
					37\64		69.3750
						48\83	69.3976
		11\19					69.4737
						51\88	69.5455
					40\69		69.5652
						69\119	69.5798
				29\50			69.6000
						66\131	69.6183	Golden meantone (696.2145¢)
					47\81		69.6296
						65\112	69.6429
			18\31				69.6774	Meantone is in this region
						61\105	69.7143
					43\74		69.7297
						68\117	69.7436
				25\43			69.7674
						57\98	69.7959
					32\55		69.8182
						39\67	69.8507
	7\12						70.000
						38\65	70.1539
					31\53		70.1887	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	70.2128	Garibaldi / Cassandra
				24\41			70.2409
						65\111	70.2703
					41\70		70.2857
						58\99	70.3030
			17\29				70.3448
						61\104	70.3846
					44\75		70.4000
						71\121	70.4132	Golden neogothic (704.0956¢)
				27\46			70.4348	Neogothic is in this region
						64\109	70.5487
					37\63		70.4762
						47\80	70.5000
		10\17					70.5882	The generator closest to a just 25/24 for EDOs less than 200
						43\73	70.6849
					33\56		70.7143
						56\95	70.7368
				23\39			70.7692
						59\100	70.8000
					36\61		70.8197
						49\83	70.8434
			13\22				70.9091	Archy is in this region
						42\71	70.9859
					29\49		71.0204
						45\76	71.0526
				16\27			71.1111
						35\59	71.1864
					19\32		71.2500
						22\37	71.3514
3\5							72.000

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.

Music

• http://www.archive.org/details/10Edf by Peter Kosmorsky