3/2: Difference between revisions

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Cerdded41 (talk | contribs)
autopatrolled
192 edits
Added table with 3/2 approximations in different EDOs (5-EDO to 31-EDO). Also added a overview of the fifths classes (and their resulting EDOs).
Cerdded41 (talk | contribs)
autopatrolled
192 edits
Added the links to the EDOs in the approximation table.
Line 83: Line 83:
!error
!error
|-
|-
|5
|[[5edo]]
|3/5
|3/5
|720
|720
Line 89: Line 89:
| +18.045¢
| +18.045¢
|-
|-
|7
|[[7edo]]
|4/7
|4/7
|685.714
|685.714
Line 95: Line 95:
| -16.241¢
| -16.241¢
|-
|-
|8
|[[8edo]]
|5/8
|5/8
|750
|750
Line 101: Line 101:
| +48.045¢
| +48.045¢
|-
|-
|9
|[[9edo]]
|5/9
|5/9
|666.667
|666.667
Line 107: Line 107:
| -35.288¢
| -35.288¢
|-
|-
|10
|[[10edo]]
|6/10
|6/10
|720
|720
Line 113: Line 113:
| +18.045¢
| +18.045¢
|-
|-
|11
|[[11edo]]
|6/11
|6/11
|654.545
|654.545
Line 119: Line 119:
| -47.41¢
| -47.41¢
|-
|-
|12
|[[12edo]]
|7/12
|7/12
|700
|700
Line 125: Line 125:
| -1.955¢
| -1.955¢
|-
|-
|13
|[[13edo]]
|8/13
|8/13
|738.462
|738.462
Line 131: Line 131:
| +36.507¢
| +36.507¢
|-
|-
|14
|[[14edo]]
|8/14
|8/14
|685.714
|685.714
Line 137: Line 137:
| -16.241¢
| -16.241¢
|-
|-
|15
|[[15edo]]
|9/15
|9/15
|720
|720
Line 143: Line 143:
| +18.045¢
| +18.045¢
|-
|-
|16
|[[16edo]]
|9/16
|9/16
|675
|675
Line 149: Line 149:
| -26.955¢
| -26.955¢
|-
|-
|17
|[[17edo]]
|10/17
|10/17
|705.882
|705.882
Line 155: Line 155:
| +3.927¢
| +3.927¢
|-
|-
|18
|[[18edo]]
|11/18
|11/18
|733.333
|733.333
Line 161: Line 161:
| +31.378¢
| +31.378¢
|-
|-
|19
|[[19edo]]
|11/19
|11/19
|694.737
|694.737
Line 167: Line 167:
| -7.218¢
| -7.218¢
|-
|-
|20
|[[20edo]]
|12/20
|12/20
|720
|720
Line 173: Line 173:
| +18.045¢
| +18.045¢
|-
|-
|21
|[[21edo]]
|12/21
|12/21
|685.714
|685.714
Line 179: Line 179:
| -16.241¢
| -16.241¢
|-
|-
|22
|[[22edo]]
|13/22
|13/22
|709.091
|709.091
Line 185: Line 185:
| +7.136¢
| +7.136¢
|-
|-
|23
|[[23edo]]
|13/23
|13/23
|678.261
|678.261
Line 191: Line 191:
| -23.694¢
| -23.694¢
|-
|-
|24
|[[24edo]]
|14/24
|14/24
|700
|700
Line 197: Line 197:
| -1.955¢
| -1.955¢
|-
|-
|25
|[[25edo]]
|15/25
|15/25
|720
|720
Line 203: Line 203:
| +18.045¢
| +18.045¢
|-
|-
|26
|[[26edo]]
|15/26
|15/26
|692.308
|692.308
Line 209: Line 209:
| -9.647¢
| -9.647¢
|-
|-
|27
|[[27edo]]
|16/27
|16/27
|711.111
|711.111
Line 215: Line 215:
| +9.156¢
| +9.156¢
|-
|-
|28
|[[28edo]]
|16/28
|16/28
|685.714
|685.714
Line 221: Line 221:
| -16.241¢
| -16.241¢
|-
|-
|29
|[[29edo]]
|17/29
|17/29
|703.448
|703.448
Line 227: Line 227:
| +1.493¢
| +1.493¢
|-
|-
|30
|[[30edo]]
|17/30
|17/30
|720
|720
Line 233: Line 233:
| +18.045¢
| +18.045¢
|-
|-
|31
|[[31edo]]
|18/31
|18/31
|696.774
|696.774

Revision as of 23:06, 20 June 2021

Interval information
Ratio 3/2
Factorization 2-1 × 3
Monzo [-1 1⟩
Size in cents 701.955¢
Name just perfect fifth
Color name w5, wa 5th
FJS name [math]\displaystyle{ \text{P5} }[/math]
Special properties superparticular,
reduced,
reduced harmonic
Tenney height (log2 nd) 2.58496
Weil height (log2 max(n, d)) 3.16993
Wilson height (sopfr(nd)) 5

[sound info]
Open this interval in xen-calc

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 ↑
  1. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  2. ↑ Super EDOs up to 200 within the same error tolerance
Comparison of 3/2 approximations and "fifth classes", with 3/2 = 701.955 cents. (from 5-EDO to 31-EDO, no subsets of 12-EDO.)
EDO degree cents fifth category error
5edo 3/5 720 pentatonic EDO +18.045¢
7edo 4/7 685.714 perfect EDO -16.241¢
8edo 5/8 750 supersharp EDO +48.045¢
9edo 5/9 666.667 superflat EDO -35.288¢
10edo 6/10 720 pentatonic EDO +18.045¢
11edo 6/11 654.545 superflat EDO -47.41¢
12edo 7/12 700 diatonic EDO -1.955¢
13edo 8/13 738.462 supersharp EDO +36.507¢
14edo 8/14 685.714 perfect EDO -16.241¢
15edo 9/15 720 pentatonic EDO +18.045¢
16edo 9/16 675 superflat EDO -26.955¢
17edo 10/17 705.882 diatonic EDO +3.927¢
18edo 11/18 733.333 supersharp EDO +31.378¢
19edo 11/19 694.737 diatonic EDO -7.218¢
20edo 12/20 720 pentatonic EDO +18.045¢
21edo 12/21 685.714 perfect EDO -16.241¢
22edo 13/22 709.091 diatonic EDO +7.136¢
23edo 13/23 678.261 superflat EDO -23.694¢
24edo 14/24 700 diatonic EDO -1.955¢
25edo 15/25 720 pentatonic EDO +18.045¢
26edo 15/26 692.308 diatonic EDO -9.647¢
27edo 16/27 711.111 diatonic EDO +9.156¢
28edo 16/28 685.714 perfect EDO -16.241¢
29edo 17/29 703.448 diatonic EDO +1.493¢
30edo 17/30 720 pentatonic EDO +18.045¢
31edo 18/31 696.774 diatonic EDO -5.181¢
  • The many and various 3/2 approximations in different EDOs can be classified as (after Kite Giedraitis):
    • superflat EDO - fifth is narrower than 686 cents.
    • perfect EDO - fifth is 686 cents wide (and 4/7 steps).
    • diatonic EDO - fifth is between 686.1 - 719.9 cents wide.
    • pentatonic EDO - fifth is exactly 720 cents wide.
    • supersharp EDO - fifth is wider than 720 cents.

See also

Retrieved from "https://en.xen.wiki/index.php?title=3/2&oldid=72859"