# perfect 5th, perfect fifth, p5

[Joseph Monzo]

The interval which encompasses 5 degrees of a diatonic scale. It is usually close in size to the frequency ratio 3:2, which is approximately 702 cents.

In the usual 12-edo tuning, the perfect 5th encompasses 7 semitones, and is thus exactly 700 cents. This is the precise dividing point between tunings of the historically important meantone and pythagorean families, and 12-edo can function as a tuning of both families. The primary difference between those two families, in terms of the use of the diatonic-scale in "common-practice", is that in the historically precedent pythagorean tuning, the diatonic-semitone (the minor-2nd, called limma in ancient terminology) is the smaller of the two different-sized semitones, and the chromatic-semitone (the augmented-prime or augmented-unison, called apotome in ancient terminology) is the larger of the two, whereas in the later meantone tunings exactly the reverse is true: the meantone diatonic-semitone is larger and the meantone chromatic-semitone is smaller. In 12edo, both types of semitones are exactly the same size, 100 cents: therefore, no 5th smaller than 12-edo's 5th of 700 cents can function as a pythagorean 5th, and no 5th larger than 700 cents can function as a meantone. (See the 12-edo article for a detailed explanation of why 12-edo belongs to both families).

Below is a graph of the size of the best approximation to the 3:2 "5th" in cents, for all the EDOs from 10 to 72:

Below is a table showing the sizes of "perfect-5th" for various tunings of the historically important pythagorean and meantone families, in descending order of size in cents:

```  tuning       ~cents of generator

(243:160 ratio 723.4612904621024   5-limit JI "super-5th": 3:2 * 81:80 [syntonic-comma])

pythagorean:
3\5-edo     720.0               also: 6\10, 9\15, 12\20, 15\25, 18\30, 21\35, etc. (approximate upper limit of what is recognizably a "perfect-5th")
22\37-edo    713.5135135135135
16\27-edo    711.1_              also: 32\54, 48\81, ... 320\540
90\152-edo   710.5263157894736
13\22-edo    709.0909090909091
319\540-edo   708.8888888888889
85\144-edo   708.3_
53\90-edo    706.6_              also: ... 159\270, ... 318\540
183\311-edo   706.1093247588425
10\17-edo    705.8823529411766   also: 20\34, 30\51
317\540-edo   704.4444444444445
27\46-edo    704.3478260869565
17\29-edo    703.448275862069    also: 34\58, 51\87
89\152-edo   702.6315789473684
24\41-edo    702.439024390244    also: 120\205
127\217-edo   702.3041474654378
182\311-edo   702.250803858521
79\135-edo   702.2_              also: 158\270, 316\540
55\94-edo    702.1276595744681

3:2 ratio    701.9550008653874   pythagorean "perfect-5th"

31\53-edo    701.8867924528302
38\65-edo    701.5384615384614

----------------------
12-edo       700.0               also: 14\24, 21\36, 28\48, 35\60, 42\70, 49\84, 56\96, ... 84\144, ... 315\540
----------------------

meantones:
1/11-comma    699.9998836293224
60\103-edo   699.029126
53\91-edo    698.901099
46\79-edo    698.734177
181\311-edo   698.3922829581994
39\67-edo    698.507463
1/6-comma     698.3706193
32\55-edo    698.181818
57\98-edo    697.959184
157\270-edo   697.7_              also: 314\540
25\43-edo    697.674419
1/5-comma     697.6537429
3/14-comma    697.3465102
43\74-edo    697.297297
2/9-comma     697.1758254
61\105-edo   697.142857
18\31-edo    696.7741935483871   also: 36\62, ... 126\217-edo
1/4-comma     696.5784285
65\112-edo   696.428571
47\81-edo    696.296296
7/26-comma    696.164846
29\50-edo    696.0
5/18-comma    695.9810315
2/7-comma     695.8103467
40\69-edo    695.652174
313\540-edo   695.5_
51\88-edo    695.454545
62\107-edo   695.327103
1/3-comma     694.7862377
11\19-edo    694.7368421052631   also: 22\38, 33\57, ... 88\152-edo
180\311-edo   694.5337620578779
15\26-edo    692.3076923076923
19\33-edo    690.909090909091
23\40-edo    690.0
4\7-edo     685.7142857142857   also: 8\14, 12\21, 16\28, 20\35 ... etc. (approximate lower limit of what is recognizably a meantone perfect-5th)

(40:27 ratio   680.4487112686725   5-limit JI "wolf-5th": 3:2 / 81:80 [syntonic-comma])

```
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[John Chalmers, Divisions of the Tetrachord]

The interval 3/2 in Just Intonation or the closest approximation to 702 cents (Â¢) in an equal temperament.

The Diapente in Greek.

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