5/4: Difference between revisions
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| Monzo = -2 0 1 | | Monzo = -2 0 1 | ||
| Cents = 386.31371 | | Cents = 386.31371 | ||
| Name = classic major third | | Name = classic/just major third | ||
| Color name = y3, yo 3rd | | Color name = y3, yo 3rd | ||
| FJS name = M3<sup>5</sup> | | FJS name = M3<sup>5</sup> | ||
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In [[5-limit]] [[ | 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''' 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 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). | 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 (↓). | ||
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== See also == | == See also == | ||
* [[8/5]] – its [[octave complement]] | * [[8/5]] – its [[octave complement]] | ||
* [[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 plus one [[octave]] sounds even more [[consonant]] | ||
* [[Gallery of | * [[Gallery of just intervals]] | ||
* [[Wikipedia:Major third | * [[List of superparticular intervals]] | ||
* [[Wikipedia: Major third]] | |||
* [[:File:5-4.mp3]] – sound sample that illustrates 5/4 as the interval between sung overtones | * [[:File:5-4.mp3]] – sound sample that illustrates 5/4 as the interval between sung overtones | ||
Revision as of 13:43, 9 September 2021
| Interval information |
reduced,
reduced harmonic
[sound info]
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 or classic major third 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¢. 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
Following EDOs (up to 200, and also 643) contain good approximations[1] 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 [2] |
|---|---|---|---|---|---|
| 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 | ↑ |
See also
- 8/5 – its octave complement
- 6/5 – its fifth complement
- 16/15 – its fourth complement
- 5/2 – the interval plus one octave sounds even more consonant
- Gallery of just intervals
- List of superparticular intervals
- Wikipedia: Major third
- File:5-4.mp3 – sound sample that illustrates 5/4 as the interval between sung overtones