4/3

(Redirected from Perfect fourth)
 Ratio 4/3 Factorization 22 × 3-1 Monzo [2 -1⟩ Size in cents 498.045¢ Name just perfect fourth Color name w4, wa 4th FJS name $\text{P4}$ Special properties square superparticular,reduced Tenney height (log2 nd) 3.58496 Weil height (log2 max(n, d)) 4 Wilson height (sopfr (nd)) 7 Harmonic entropy(Shannon, $\sqrt{nd}$) ~3.81657 bits https://en.xen.wiki/w/File:Jid_4_3_pluck_adu_dr220.mp3[sound info] open this interval in xen-calc
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4/3 is the frequency ratio of the just perfect fourth, which is easily one of the more heavily discussed intervals outside of xenharmony- in fact, some of these usages have gone on to inspire other music theories within xenharmonic contexts, such as certain ideas about tetrachords. As its inversion is the perfect fifth, 3/2, 4/3 is the octave reduced form of the third subharmonic. In the florid organum of Medieval music, 4/3 was reliably considered a consonance, and indeed was frequently emphasized. Once major thirds with a tuning approximating 5/4 began to be treated as consonances, however, the perception of 4/3 was altered to where it was at times considered a dissonance. However, as of late, the perfect fourth is once again being reevaluated as a consonance.

Approximations by EDOs

The following EDOs (up to 200) contain good approximations[1] of the interval 4/3. 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 ()
Equally acceptable multiples [2]
12 5\12 1.9550 1.9550 10\24, 15\36
17 7\17 3.9274 5.5637
29 12\29 1.4933 3.6087
41 17\41 0.4840 1.6537 34\82, 51\123, 68\164
53 22\53 0.0682 0.3013 44\106, 66\159
65 27\65 0.4165 2.2563 54\130, 81\195
70 29\70 0.9021 5.2625
77 32\77 0.6563 4.2113
89 37\89 0.8314 6.1663
94 39\94 0.1727 1.3525 78\188
111 46\111 0.7477 6.9162
118 49\118 0.2601 2.5575
135 56\135 0.2672 3.0062
142 59\142 0.5466 6.4675
147 61\147 0.0858 1.0512
171 71\171 0.2006 2.8588
176 73\176 0.3177 4.6600
183 76\183 0.3157 4.8138
200 83\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