4/3: Difference between revisions

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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 [[tetrachord]]s. As its inversion is the perfect fifth, [[3/2]], 4/3 is the [[octave reduced]] form of the third [[subharmonic]]. In the [[Wikipedia: Medieval music #Early polyphony: organum|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.
'''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 [[tetrachord]]s. As its inversion is the perfect fifth, [[3/2]], 4/3 is the [[octave reduced]] form of the third [[subharmonic]]. In the [[Wikipedia: Medieval music #Early polyphony: organum|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 [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute <br> error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! &#8597;
! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
|-
|  [[12edo|12]]  ||  5\12  || 1.9550 || 1.9550 || &uarr; || [[24edo|10\24]], [[36edo|15\36]]
|-
|  [[17edo|17]]  ||  7\17  || 3.9274 || 5.5637 || &darr; ||
|-
|  [[29edo|29]]  ||  12\29  || 1.4933 || 3.6087 || &darr; ||
|-
|  [[41edo|41]]  ||  17\41  || 0.4840 || 1.6537 || &darr; || [[82edo|34\82]], [[123edo|51\123]], [[164edo|68\164]]
|-
|  [[53edo|53]]  ||  22\53  || 0.0682 || 0.3013 || &uarr; || [[106edo|44\106]], [[159edo|66\159]]
|-
|  [[65edo|65]]  ||  27\65  || 0.4165 || 2.2563 || &uarr; || [[130edo|54\130]], [[195edo|81\195]]
|-
|  [[70edo|70]]  ||  29\70  || 0.9021 || 5.2625 || &darr; ||
|-
|  [[77edo|77]]  ||  32\77  || 0.6563 || 4.2113 || &uarr; ||
|-
|  [[89edo|89]]  ||  37\89  || 0.8314 || 6.1663 || &uarr; ||
|-
|  [[94edo|94]]  ||  39\94  || 0.1727 || 1.3525 || &darr; || [[188edo|78\188]]
|-
| [[111edo|111]] ||  46\111 || 0.7477 || 6.9162 || &darr; ||
|-
| [[118edo|118]] ||  49\118 || 0.2601 || 2.5575 || &uarr; ||
|-
| [[135edo|135]] ||  56\135 || 0.2672 || 3.0062 || &darr; ||
|-
| [[142edo|142]] ||  59\142 || 0.5466 || 6.4675 || &uarr; ||
|-
| [[147edo|147]] ||  61\147 || 0.0858 || 1.0512 || &darr; ||
|-
| [[171edo|171]] || 71\171 || 0.2006 || 2.8588 || &uarr; ||
|-
| [[176edo|176]] || 73\176 || 0.3177 || 4.6600 || &darr; ||
|-
| [[183edo|183]] || 76\183 || 0.3157 || 4.8138 || &uarr; ||
|-
| [[200edo|200]] || 83\200 || 0.0450 || 0.7500 || &darr; ||
|-
|}
<references/>
== See also ==
== See also ==
* [[3/2]] – its [[octave complement]]
* [[3/2]] – its [[octave complement]]

Revision as of 13:09, 4 December 2021

Interval information
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 [math]\displaystyle{ \text{P4} }[/math]
Special properties square superparticular,
reduced,
reduced subharmonic
Tenney norm (log2 nd) 3.58496
Weil norm (log2 max(n, d)) 4
Wilson norm (sopfr(nd)) 7

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

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 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 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

See also

Retrieved from "https://en.xen.wiki/index.php?title=4/3&oldid=81464"