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| | es = | | | es = |
| | ja = | | | ja = |
| | | ro = 5/4 (ro) |
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| {{Infobox Interval | | {{Infobox Interval |
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| {{Wikipedia|Major third}} | | {{Wikipedia|Major third}} |
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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'''<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 [[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". |
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| 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). | | 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). |
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| == 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) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 5/4. |
| | | {{Interval edo approximation|interval = 5/4| max_edo=200}} |
| {| class="wikitable sortable right-1 center-2 right-3 right-4 center-5" | |
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| ! [[Edo]]
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| ! class="unsortable" | deg\edo
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| ! Absolute <br> error ([[Cent|¢]])
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| ! Relative <br> error ([[Relative cent|r¢]])
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| ! ↕
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| ! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>
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| | [[25edo|25]] || 8\25 || 2.3137 || 4.8202 || ↓ ||
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| | [[28edo|28]] || 9\28 || 0.5994 || 1.3987 || ↓ || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]]
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| | [[31edo|31]] || 10\31 || 0.7831 || 2.0229 || ↑ || [[62edo|20\62]], [[93edo|30\93]]
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| | [[34edo|34]] || 11\34 || 1.9216 || 5.4445 || ↑ ||
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| | [[53edo|53]] || 17\53 || 1.4081 || 6.2189 || ↓ ||
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| | [[59edo|59]] || 19\59 || 0.1270 || 0.6242 || ↑ || [[118edo|38\118]], [[177edo|57\177]]
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| | [[87edo|87]] || 28\87 || 0.1068 || 0.7744 || ↓ || [[174edo|56\174]]
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| | [[90edo|90]] || 29\90 || 0.3530 || 2.6471 || ↑ || [[180edo|58\180]]
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| | [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || ↓ ||
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| | [[121edo|121]] || 39\121 || 0.4631 || 4.6701 || ↑ ||
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| | [[143edo|143]] || 46\143 || 0.2997 || 3.5718 || ↓ ||
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| | [[146edo|146]] || 47\146 || 0.0123 || 0.1502 || ↓ ||
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| | [[149edo|149]] || 48\149 || 0.2635 || 3.2714 || ↑ ||
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| | [[152edo|152]] || 49\152 || 0.5284 || 6.6930 || ↑ ||
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| | [[171edo|171]] || 55\171 || 0.3488 || 4.9704 || ↓ ||
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| | [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || ↓ ||
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| | [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || ↑ ||
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| |}
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| == See also == | | == See also == |
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| * [[6/5]] – its [[fifth complement]] | | * [[6/5]] – its [[fifth complement]] |
| * [[16/15]] – its [[fourth complement]] | | * [[16/15]] – its [[fourth complement]] |
| * [[5/2]] – the interval plus one [[octave]] sounds even more [[consonant]] | | * [[5/2]] – the interval up one [[octave]] which sounds even more [[consonant]] |
| | * [[Ed5/4]] |
| * [[Gallery of just intervals]] | | * [[Gallery of just intervals]] |
| * [[List of superparticular intervals]] | | * [[List of superparticular intervals]] |