5/4: Difference between revisions

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}}

{{Infobox Interval

| Name = classic~~/just~~ major third

| Name = just major third, classic(al) major third, ptolemaic major third

| Color name = y3, yo 3rd

| Sound = jid_5_4_pluck_adu_dr220.mp3

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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''' or '''~~classic~~ major third''' to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7 ¢ away from [[12edo]]'s major third of 400 ¢. 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 ¢. 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 [[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'''<ref>[https://marsbat.space/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>, '''classic(al) major third'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref>, or '''ptolemaic major third'''<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref> to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7 ¢ away from [[12edo]]'s major third of 400 ¢. 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 ¢. 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 here melodically in singing into a resonant [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4).

== Approximations by ~~EDOs~~ ==

== 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 [[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 (↓).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"

|-

! [[~~EDO~~]]

! [[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>

! 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 || ↓ ||

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| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || ↑ ||

|}

~~<references/>~~

== See also ==

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* [[List of superparticular intervals]]

* [[:File:5-4.mp3]] – sound sample that illustrates 5/4 as the interval between sung overtones

== Notes ==

[[Category:Third]]

[[Category:Major third]]

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 }}
 {{Infobox Interval
-| Name = classic/just major third
+| Name = just major third, classic(al) major third, ptolemaic major third
 | Color name = y3, yo 3rd
 | Sound = jid_5_4_pluck_adu_dr220.mp3
@@ Line 12: / Line 12: @@
 {{Wikipedia|Major third}}
-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''' or '''classic major third''' to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7 ¢ away from [[12edo]]'s major third of 400 ¢. 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 ¢. 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 [[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'''<ref>[https://marsbat.space/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>, '''classic(al) major third'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref>, or '''ptolemaic major third'''<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref> to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7 ¢ away from [[12edo]]'s major third of 400 ¢. 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 ¢. 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 here melodically in singing into a resonant [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4).
-== Approximations by EDOs ==
+== 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 (&uarr;) or flat (&darr;).
+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 (&uarr;) or flat (&darr;).
 {| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
 |-
-! [[EDO]]
+! [[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>
+! 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 || &darr; ||
@@ Line 62: / Line 62: @@
 | [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || &uarr; ||
 |}
-<references/>
 == See also ==
@@ Line 73: / Line 71: @@
 * [[List of superparticular intervals]]
 * [[:File:5-4.mp3]] – sound sample that illustrates 5/4 as the interval between sung overtones
+== Notes ==
+<references/>
 [[Category:Third]]
 [[Category:Major third]]