2edf

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2edf

3edf →

Prime factorization	2 (prime)
Step size	350.978¢
Octave	3\2edf (1052.93¢) (semiconvergent)
Twelfth	5\2edf (1754.89¢) (semiconvergent)
Consistency limit	2
Distinct consistency limit	1
Special properties	Highly composite

2edf, if the attempt is made to use it as an actual scale, would divide the just perfect fifth into two equal parts, each of size 350.9775 cents, which is to say sqrt (3/2) as a frequency ratio. It corresponds to 3.4190 edo. If we want to consider it to be a temperament, it tempers out 6/5, 9/7, 32/27, and 81/80 in the patent val.

Factoids about 2edf

60/49 and 49/40 are good rational representations of the square root of 3/2.

2edf's step size is close to the generator of the hemififths temperament, which tempers out 2401/2400 and 5120/5103 in the 7-limit.

Intervals

#	Cents
1	350.98
2	701.96

Scale tree

Todo: correct maths, review The text and the table incoherently mix up EDO and EDF calculations. This section should also be moved to a more appropriate page.

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: 342.8571 cents (4\7/2 = 2\7) to 360 cents (3\5/2 = 3/10)

Fifth							Cents	Comments
4\7							342.857
						27\47	344.681
					23\40		345.000
						42\73	345.2055
				19\33			345.45
						53\92	345.652
					34\59		345.763
						49\85	345.882
			15\26				346.153
						56\97	346.392
					41\71		346.479
						67\116	346.551
				26\45			346.6	Flattone is in this region
						63\109	346.789
					37\64		346.875
						48\83	346.988
		11\19					347.368	The generator closest to a just 11/9 for EDOs less than 200
						51\88	347.72
					40\69		347.826
						69\119	347.899
				29\50			348.000
						76\131	348.092	Golden meantone (696.2145¢)
					47\81		348.148
						65\112	348.214
			18\31				348.387	Meantone is in this region
						61\105	348.571
					43\74		348.648
						68\117	348.718
				25\43			348.837
						57\98	348.980
					32\55		349.09
						39\67	349.254
	7\12						350.000
						38\65	350.769
					31\53		350.843	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	351.064	Garibaldi / Cassandra
				24\41			351.2195
						65\111	351.351
					41\70		351.429
						58\99	351.51
			17\29				351.724
						61\104	351.923
					44\75		352.000
						71\121	352.066	Golden neogothic (704.0956¢)
				27\46			352.174	Neogothic is in this region
						64\109	352.294
					37\63		352.381
						47\80	352.500
		10\17					352.941
						43\73	353.425
					33\56		353.571
						56\95	353.684
				23\39			353.846
						59\100	354.000
					36\61		354.098
						49\83	354.217
			13\22				354.54	Archy is in this region
						42\71	354.930
					29\49		355.102
						45\76	355.263
				16\27			355.5
						35\59	355.932
					19\32		356.250
						22\37	356.756
3\5							360.000

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.