5/4: Difference between revisions
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== Approximations by edos == | == Approximations by edos == | ||
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. | 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}} | {{Interval_Edo_Approximation | interval = 5/4| max_edo=200}} | ||
Revision as of 07:13, 3 November 2025
| Interval information |
classic(al) major third,
ptolemaic major third
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, 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) contain good approximations[2] of the interval 5/4.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 3 | 1\3 | 400.00 | +13.69 | +3.42 |
| 6 | 2\6 | 400.00 | +13.69 | +6.84 |
| 22 | 7\22 | 381.82 | -4.50 | -8.24 |
| 25 | 8\25 | 384.00 | -2.31 | -4.82 |
| 28 | 9\28 | 385.71 | -0.60 | -1.40 |
| 31 | 10\31 | 387.10 | +0.78 | +2.02 |
| 34 | 11\34 | 388.24 | +1.92 | +5.44 |
| 37 | 12\37 | 389.19 | +2.88 | +8.87 |
| 50 | 16\50 | 384.00 | -2.31 | -9.64 |
| 53 | 17\53 | 384.91 | -1.41 | -6.22 |
| 56 | 18\56 | 385.71 | -0.60 | -2.80 |
| 59 | 19\59 | 386.44 | +0.13 | +0.62 |
| 62 | 20\62 | 387.10 | +0.78 | +4.05 |
| 65 | 21\65 | 387.69 | +1.38 | +7.47 |
| 81 | 26\81 | 385.19 | -1.13 | -7.62 |
| 84 | 27\84 | 385.71 | -0.60 | -4.20 |
| 87 | 28\87 | 386.21 | -0.11 | -0.77 |
| 90 | 29\90 | 386.67 | +0.35 | +2.65 |
| 93 | 30\93 | 387.10 | +0.78 | +6.07 |
| 96 | 31\96 | 387.50 | +1.19 | +9.49 |
| 109 | 35\109 | 385.32 | -0.99 | -9.02 |
| 112 | 36\112 | 385.71 | -0.60 | -5.59 |
| 115 | 37\115 | 386.09 | -0.23 | -2.17 |
| 118 | 38\118 | 386.44 | +0.13 | +1.25 |
| 121 | 39\121 | 386.78 | +0.46 | +4.67 |
| 124 | 40\124 | 387.10 | +0.78 | +8.09 |
| 140 | 45\140 | 385.71 | -0.60 | -6.99 |
| 143 | 46\143 | 386.01 | -0.30 | -3.57 |
| 146 | 47\146 | 386.30 | -0.01 | -0.15 |
| 149 | 48\149 | 386.58 | +0.26 | +3.27 |
| 152 | 49\152 | 386.84 | +0.53 | +6.69 |
| 168 | 54\168 | 385.71 | -0.60 | -8.39 |
| 171 | 55\171 | 385.96 | -0.35 | -4.97 |
| 174 | 56\174 | 386.21 | -0.11 | -1.55 |
| 177 | 57\177 | 386.44 | +0.13 | +1.87 |
| 180 | 58\180 | 386.67 | +0.35 | +5.29 |
| 183 | 59\183 | 386.89 | +0.57 | +8.72 |
| 196 | 63\196 | 385.71 | -0.60 | -9.79 |
| 199 | 64\199 | 385.93 | -0.38 | -6.37 |
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