3/2: Difference between revisions

3/2, the just perfect fifth, is the largest superparticular interval, spanning the distance between the 2nd and 3rd harmonics. It is an interval with low harmonic entropy, and therefore high consonance.

Variations of the fifth (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third (see 5:4) as consonant. 3:2 is the simple JI interval best approximated by 12edo, after the octave.

Producing a chain of just perfect fifths yields Pythagorean tuning. Such a chain does not close at a circle, but continues infinitely. 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, meantone temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5:4 – or, in the case of quarter-comma meantone (see 31edo), identical.

In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".

Some better (compared to 12edo) approximations of the perfect fifth are 29edo, 41edo, and 53edo.

Approximations by EDOs

The following EDOs (up to 200) contain good approximations^[1] of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).

EDO	deg\edo	Absolute error (¢)	Relative error (r¢)	↕	Equally acceptable multiples ^[2]
12	7\12	1.9550	1.9550	↓	14\24, 21\36
17	10\17	3.9274	5.5637	↑
29	17\29	1.4933	3.6087	↑
41	24\41	0.4840	1.6537	↑	48\82, 72\123, 96\164
53	31\53	0.0682	0.3013	↓	62\106, 93\159
65	38\65	0.4165	2.2563	↓	76\130, 114\195
70	41\70	0.9021	5.2625	↑
77	45\77	0.6563	4.2113	↓
89	52\89	0.8314	6.1663	↓
94	55\94	0.1727	1.3525	↑	110\188
111	65\111	0.7477	6.9162	↑
118	69\118	0.2601	2.5575	↓
135	79\135	0.2672	3.0062	↑
142	83\142	0.5466	6.4675	↓
147	86\147	0.0858	1.0512	↑
171	100\171	0.2006	2.8588	↓
176	103\176	0.3177	4.6600	↑
183	107\183	0.3157	4.8138	↓
200	117\200	0.0450	0.7500	↑

↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
↑ Super EDOs up to 200 within the same error tolerance

@@ Line 5: / Line 5: @@
 | Cents = 701.95500
 | Name = just perfect fifth
-| Sound = jid_3_2_pluck_adu_dr220.mp3
 | Color name = w5, wa 5th
 | FJS name = P5
+| Sound = jid_3_2_pluck_adu_dr220.mp3
 }}
-'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[Gallery_of_Just_Intervals|interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance.
+'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance.
-Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third (see [[5/4|5:4]]) as consonant. 3:2 is the simple JI interval best approximated by [[12edo|12edo]], after the [[Octave|octave]].
+Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third (see [[5/4|5:4]]) as consonant. 3:2 is the simple JI interval best approximated by [[12edo]], after the [[octave]].
-Producing a chain of just perfect fifths yields Pythagorean tuning. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5:4 -- or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical.
+Producing a chain of just perfect fifths yields Pythagorean tuning. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5:4 – or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical.
 In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".
@@ Line 76: / Line 76: @@
 == See also ==
-* [[Wikipedia:Perfect fifth]]
-* [[Gallery of Just Intervals]]
 * [[4/3]] – its [[octave complement]]
-* {{OEIS|A060528}} – a list of EDOs with increasingly better approximations of 3:2 (and by extension 4:3)
+* [[Fifth complement]]
-* {{OEIS|A005664}} – denominators of the convergents to log<sub>2</sub>(3)
+* [[Gallery of just intervals]]
+* [[Wikipedia: Perfect fifth]]
+* {{OEIS| A060528 }} – a list of EDOs with increasingly better approximations of 3:2 (and by extension 4:3)
+* {{OEIS| A005664 }} – denominators of the convergents to log<sub>2</sub>(3)
 [[Category:3-limit]]

3/2: Difference between revisions

Revision as of 04:03, 13 March 2021

Approximations by EDOs

See also

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