5/4
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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模板:C away from 12edo's major third of 400模板: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模板:C, and from the Pythagorean diminished fourth of 8192/6561 by the schisma, which measures about 1.95模板: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".
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
本表格只包含相对误差≤10%且平均律步数与5/4步数没有大于1的公因子的情况。
| 平均律步数 | 5/4的步数 | 绝对误差(ie) | 相对误差 |
|---|---|---|---|
| 3 | 1 | ||
| 22 | 7 | ||
| 25 | 8 | ||
| 28 | 9 | ||
| 31 | 10 | ||
| 34 | 11 | ||
| 37 | 12 | ||
| 53 | 17 | ||
| 59 | 19 | ||
| 65 | 21 | ||
| 81 | 26 | ||
| 87 | 28 | ||
| 90 | 29 | ||
| 96 | 31 | ||
| 109 | 35 | ||
| 115 | 37 | ||
| 121 | 39 | ||
| 143 | 46 | ||
| 146 | 47 | ||
| 149 | 48 | ||
| 152 | 49 | ||
| 171 | 55 | ||
| 183 | 59 | ||
| 199 | 64 | ||
| 202 | 65 | ||
| 205 | 66 | ||
| 208 | 67 | ||
| 211 | 68 | ||
| 227 | 73 | ||
| 233 | 75 | ||
| 239 | 77 | ||
| 255 | 82 | ||
| 258 | 83 | ||
| 264 | 85 | ||
| 267 | 86 |