5/4: Difference between revisions
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{{interwiki | |||
| de = Naturterz | |||
| en = 5/4 | |||
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| ja = | |||
| ro = 5/4 (ro) | |||
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{{Infobox Interval | {{Infobox Interval | ||
| Name = just major third, classic(al) major third, ptolemaic major third | |||
| Name = classic major third | |||
| Color name = y3, yo 3rd | | Color name = y3, yo 3rd | ||
| Sound = jid_5_4_pluck_adu_dr220.mp3 | | Sound = jid_5_4_pluck_adu_dr220.mp3 | ||
}} | }} | ||
{{Wikipedia|Major third}} | |||
In [[5-limit]] [[ | In [[5-limit]] [[just intonation]], '''5/4''' is the [[frequency ratio]] between the 5th and 4th [[harmonic]]s. It has been called the '''just major third''', '''classic(al) major third''', or '''ptolemaic major third'''<ref>For reference, see [[5-limit]].</ref> to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7{{c}} away from [[12edo]]'s major third of 400{{c}}. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81/64]] by the syntonic comma of [[81/80]], which measures about 21.5{{c}}, and from the Pythagorean diminished fourth of [[8192/6561]] by the [[schisma]], which measures about 1.95{{c}}. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful". | ||
In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated | In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in [[:File: 5-4.mp3]] melodically in singing into a resonant [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4). | ||
== Approximations by | == Approximations by edos == | ||
Following [[edo]]s (up to 200, and also 643) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 5/4. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (↑) or flat (↓). | |||
Following [[ | |||
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5" | {| class="wikitable sortable right-1 center-2 right-3 right-4 center-5" | ||
|- | |- | ||
! [[ | ! [[Edo]] | ||
! class="unsortable" | deg\edo | ! class="unsortable" | deg\edo | ||
! Absolute <br> error ([[Cent|¢]]) | ! Absolute <br> error ([[Cent|¢]]) | ||
! Relative <br> error ([[Relative cent|r¢]]) | ! Relative <br> error ([[Relative cent|r¢]]) | ||
! ↕ | ! ↕ | ||
! class="unsortable" | Equally acceptable multiples <ref>Super | ! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref> | ||
|- | |- | ||
| [[25edo|25]] || 8\25 || 2.3137 || 4.8202 || ↓ || | | [[25edo|25]] || 8\25 || 2.3137 || 4.8202 || ↓ || | ||
|- | |- | ||
| [[28edo|28]] || 9\28 || 0.5994 || 1.3987 || ↓ || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]] | | [[28edo|28]] || 9\28 || 0.5994 || 1.3987 || ↓ || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]] | ||
|- | |- | ||
| [[31edo|31]] || 10\31 || 0.7831 || 2.0229 || ↑ || [[62edo|20\62]], [[93edo|30\93]] | | [[31edo|31]] || 10\31 || 0.7831 || 2.0229 || ↑ || [[62edo|20\62]], [[93edo|30\93]] | ||
|- | |- | ||
| [[34edo|34]] || 11\34 || 1.9216 || 5.4445 || ↑ || | | [[34edo|34]] || 11\34 || 1.9216 || 5.4445 || ↑ || | ||
| Line 37: | Line 39: | ||
| [[53edo|53]] || 17\53 || 1.4081 || 6.2189 || ↓ || | | [[53edo|53]] || 17\53 || 1.4081 || 6.2189 || ↓ || | ||
|- | |- | ||
| [[59edo|59]] || 19\59 || 0.1270 || 0.6242 || ↑ || [[118edo|38\118]], [[177edo|57\177]] | | [[59edo|59]] || 19\59 || 0.1270 || 0.6242 || ↑ || [[118edo|38\118]], [[177edo|57\177]] | ||
|- | |- | ||
| [[87edo|87]] || 28\87 || 0.1068 || 0.7744 || ↓ || [[174edo|56\174]] | | [[87edo|87]] || 28\87 || 0.1068 || 0.7744 || ↓ || [[174edo|56\174]] | ||
|- | |- | ||
| [[90edo|90]] || 29\90 || 0.3530 || 2.6471 || ↑ || [[180edo|58\180]] | | [[90edo|90]] || 29\90 || 0.3530 || 2.6471 || ↑ || [[180edo|58\180]] | ||
|- | |- | ||
| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || ↓ || | | [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || ↓ || | ||
| Line 58: | Line 60: | ||
|- | |- | ||
| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || ↓ || | | [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || ↓ || | ||
|- | |||
| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || ↑ || | |||
|} | |} | ||
== See also == | == See also == | ||
* [[8/5]] – its [[octave complement]] | * [[8/5]] – its [[octave complement]] | ||
* [[6/5]] – its [[fifth complement]] | * [[6/5]] – its [[fifth complement]] | ||
* [[5/2]] – the interval | * [[16/15]] – its [[fourth complement]] | ||
* [[ | * [[5/2]] – the interval up one [[octave]] which sounds even more [[consonant]] | ||
* [[ | * [[Ed5/4]] | ||
* [[ | * [[Gallery of just intervals]] | ||
* [[List of superparticular intervals]] | |||
== Notes == | |||
<references/> | |||
[[Category:Third]] | [[Category:Third]] | ||
[[Category:Major third]] | [[Category:Major third]] | ||