4/3: Difference between revisions

Line 12:

'''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 (↑) or flat (↓).

{| 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¢]])

! ↕

! 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 || ↑ || [[24edo|10\24]], [[36edo|15\36]]

|-

| [[17edo|17]] || 7\17 || 3.9274 || 5.5637 || ↓ ||

|-

| [[29edo|29]] || 12\29 || 1.4933 || 3.6087 || ↓ ||

|-

| [[41edo|41]] || 17\41 || 0.4840 || 1.6537 || ↓ || [[82edo|34\82]], [[123edo|51\123]], [[164edo|68\164]]

|-

| [[53edo|53]] || 22\53 || 0.0682 || 0.3013 || ↑ || [[106edo|44\106]], [[159edo|66\159]]

|-

| [[65edo|65]] || 27\65 || 0.4165 || 2.2563 || ↑ || [[130edo|54\130]], [[195edo|81\195]]

|-

| [[70edo|70]] || 29\70 || 0.9021 || 5.2625 || ↓ ||

|-

| [[77edo|77]] || 32\77 || 0.6563 || 4.2113 || ↑ ||

|-

| [[89edo|89]] || 37\89 || 0.8314 || 6.1663 || ↑ ||

|-

| [[94edo|94]] || 39\94 || 0.1727 || 1.3525 || ↓ || [[188edo|78\188]]

|-

| [[111edo|111]] || 46\111 || 0.7477 || 6.9162 || ↓ ||

|-

| [[118edo|118]] || 49\118 || 0.2601 || 2.5575 || ↑ ||

|-

| [[135edo|135]] || 56\135 || 0.2672 || 3.0062 || ↓ ||

|-

| [[142edo|142]] || 59\142 || 0.5466 || 6.4675 || ↑ ||

|-

| [[147edo|147]] || 61\147 || 0.0858 || 1.0512 || ↓ ||

|-

| [[171edo|171]] || 71\171 || 0.2006 || 2.8588 || ↑ ||

|-

| [[176edo|176]] || 73\176 || 0.3177 || 4.6600 || ↓ ||

|-

| [[183edo|183]] || 76\183 || 0.3157 || 4.8138 || ↑ ||

|-

| [[200edo|200]] || 83\200 || 0.0450 || 0.7500 || ↓ ||

|-

|}

== See also ==

* [[3/2]] – its [[octave complement]]

@@ Line 12: / Line 12: @@
 '''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 ==
 * [[3/2]] – its [[octave complement]]