5/4: Difference between revisions

In 5-limit just intonation, 5/4 is the frequency ratio between the 5th and 4th harmonics. It has been called the just major third, classic(al) major third, or ptolemaic major third^[1] to distinguish it from other intervals in that neighborhood. Measuring about 386.3 ¢, 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 ¢, and from the Pythagorean diminished fourth of 8192/6561 by the schisma, which measures about 1.95 ¢. 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 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 edos

Following edos (up to 200, and also 643) contain good approximations^[2] of the interval 5/4. 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 ^[3]
25	8\25	2.3137	4.8202	↓
28	9\28	0.5994	1.3987	↓	18\56, 27\84, 36\112, 45\140
31	10\31	0.7831	2.0229	↑	20\62, 30\93
34	11\34	1.9216	5.4445	↑
53	17\53	1.4081	6.2189	↓
59	19\59	0.1270	0.6242	↑	38\118, 57\177
87	28\87	0.1068	0.7744	↓	56\174
90	29\90	0.3530	2.6471	↑	58\180
115	37\115	0.2268	2.1731	↓
121	39\121	0.4631	4.6701	↑
143	46\143	0.2997	3.5718	↓
146	47\146	0.0123	0.1502	↓
149	48\149	0.2635	3.2714	↑
152	49\152	0.5284	6.6930	↑
171	55\171	0.3488	4.9704	↓
199	64\199	0.3841	6.3691	↓
643	207\643	0.0004	0.0235	↑

Notes

↑ For reference, see 5-limit.
↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
↑ Super-edos up to 200 within the same error tolerance

[1] For reference, see 5-limit.

[2] rror magnitude below 7, both, absolute (in ¢) and relative (in r¢)

[3] Super-edos up to 200 within the same error tolerance

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-{| class="wikitable"
+{{interwiki
-|-
+| de = Naturterz
-| [[File:5_4_glyph.png|alt=5 4 glyph.png|113x123px|5 4 glyph.png]]
+| en = 5/4
-|-
+| es =
-| JI Glyph for 5/4
+| ja =
-|}
+| ro = 5/4 (ro)
+}}
+{{Infobox Interval
+| Name = just major third, classic(al) major third, ptolemaic major third
+| Color name = y3, yo 3rd
+| Sound = jid_5_4_pluck_adu_dr220.mp3
+}}
+{{Wikipedia|Major third}}
-'''5/4'''
+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".
-{{Monzo| -2 0 1 }}
+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).
-.31371 cents
+== 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;).
-[[File:jid_5_4_pluck_adu_dr220.mp3]] <br/> [[:File:jid_5_4_pluck_adu_dr220.mp3|jid_5_4_pluck_adu_dr220.mp3]]
+{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
+|-
-In [[Just Intonation]], '''5/4''' is the frequency ratio between the 5th and 4th harmonics. 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".
+! [[Edo]]
+! class="unsortable" | deg\edo
-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 [http://udderbot.wikispaces.com/home udderbot] (from the fundamental up to 5 and then noodling between 5 and 4).
+! Absolute <br> error ([[Cent|¢]])
+! Relative <br> error ([[Relative cent|r¢]])
-[[File:5-4.mp3]] <br/> [[:File:5-4.mp3|5-4.mp3]]
+! &#8597;
+! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>
-/4 converted to cents (¢): 1200 * log (5/4) / log (2) = 386.314...
+|-
+|  [[25edo|25]]  ||  8\25  || 2.3137 || 4.8202 || &darr; ||
-/4 the interval, like all ''intervals'', refers to a ''relation'' between two pitches. It has been called the ''just major third'' to distinguish it from the other intervals in that neighborhood.
+|-
+|  [[28edo|28]]  ||  9\28  || 0.5994 || 1.3987 || &darr; || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]]
+|-
+|  [[31edo|31]]  || 10\31  || 0.7831 || 2.0229 || &uarr; || [[62edo|20\62]], [[93edo|30\93]]
+|-
+|  [[34edo|34]]  || 11\34  || 1.9216 || 5.4445 || &uarr; ||
+|-
+|  [[53edo|53]]  || 17\53  || 1.4081 || 6.2189 || &darr; ||
+|-
+|  [[59edo|59]]  || 19\59  || 0.1270 || 0.6242 || &uarr; || [[118edo|38\118]], [[177edo|57\177]]
+|-
+|  [[87edo|87]]  || 28\87  || 0.1068 || 0.7744 || &darr; || [[174edo|56\174]]
+|-
+|  [[90edo|90]]  || 29\90  || 0.3530 || 2.6471 || &uarr; || [[180edo|58\180]]
+|-
+| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || &darr; ||
+|-
+| [[121edo|121]] || 39\121 || 0.4631 || 4.6701 || &uarr; ||
+|-
+| [[143edo|143]] || 46\143 || 0.2997 || 3.5718 || &darr; ||
+|-
+| [[146edo|146]] || 47\146 || 0.0123 || 0.1502 || &darr; ||
+|-
+| [[149edo|149]] || 48\149 || 0.2635 || 3.2714 || &uarr; ||
+|-
+| [[152edo|152]] || 49\152 || 0.5284 || 6.6930 || &uarr; ||
+|-
+| [[171edo|171]] || 55\171 || 0.3488 || 4.9704 || &darr; ||
+|-
+| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || &darr; ||
+|-
+| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || &uarr; ||
+|}
-:'' See also [[Gallery of Just Intervals]]''
+== See also ==
+* [[8/5]] – its [[octave complement]]
+* [[6/5]] – its [[fifth complement]]
+* [[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]]
-[[Category:5-limit]]
+== Notes ==
-[[Category:just_interval]]
+<references/>
-[[Category:major_third]]
-[[Category:ratio]]
-[[Category:superparticular]]
-<!-- interwiki -->
+[[Category:Third]]
-[[de:Naturterz]]
+[[Category:Major third]]

5/4: Difference between revisions

Latest revision as of 13:37, 16 April 2025

Approximations by edos

See also

Notes

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