4/3: Difference between revisions
Wikispaces>k9assassin **Imported revision 545204998 - Original comment: ** |
→Temperaments: - duplicate from 3/2. Bixby not worth the space |
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{{interwiki | |||
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| ro = 4/3 (ro) | |||
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{{Infobox Interval | |||
| Name = just perfect fourth | |||
| Color name = w4, wa 4th | |||
| Sound = jid_4_3_pluck_adu_dr220.mp3 | |||
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{{Wikipedia|Perfect fourth}} | |||
'''4/3''' is the [[frequency ratio]] of the '''just perfect fourth'''. As its inversion is the perfect fifth, [[3/2]], 4/3 is the [[octave reduced]] form of the third [[subharmonic]]. 4/3 is one of the most common intervals one finds in the world's [[Approaches to Musical Tuning|musical traditions]], past and present. | |||
[[ | |||
4 | Among many other uses, 4/3 forms the basis of [[tetrachord]]s in many musical traditions, such as [[Ancient Greek music]], as well as in modern [[just intonation]] and [[xenharmonic|xenharmony]]. | ||
[[ | == History == | ||
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. | |||
& | == Chord construction == | ||
Much like 3/2, 4/3 is valuable as a framework for constructing [[chord]]s. However, while 3/2 provides the framework for [[5-limit]] triads involving intervals like 5/4 and [[6/5]], 4/3 provides a possible framework for [[7-limit]] triads involving intervals like [[7/6]] and [[8/7]], though such triads are [[Condissonance|ambisonances]] (that is, they're both consonant and dissonant at the same time) at best. | |||
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Because up to two instances of 4/3 can fit within the span of an [[octave]], it is very easy to create xenharmonic chords using 4/3 as a framework. Regardless, the usage of 4/3 as a framework for chords is intimately connected with the use of [[tritave]]s in the same capacity- at least in [[Octave #Octave equivalence|octave-equivalent]] systems- due to the same pitch classes being involved in both 6:7:8 and 4:7:12 where 7 is kept as the same note, thus rendering the two chords as different voicings of the same underlying harmonic unit. | |||
== Approximations by EDOs == | |||
4 | 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 4/3. 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 || ↓ || | |||
|- | |||
|} | |||
<references/> | |||
== Temperaments == | |||
4/3 can be used as an alternative generator for temperaments generated by an octave and a fifth of 3/2, such as [[meantone]], [[superpyth]], and [[schismic]]. See [[3/2 #In regular temperament theory]] for details. | |||
== See also == | |||
* [[3/2]] – its [[octave complement]] | |||
* [[9/8]] – its [[fifth complement]] | |||
* [[Fourth complement]] | |||
* [[Ed4/3]] | |||
* [[Gallery of just intervals]] | |||
[[Category:Fourth]] | |||
[[Category:Over-3 intervals]] | |||
[[Category:Tritave-reduced harmonics]] | |||