5/4：修订间差异

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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''', '''classic(al) major third''', or '''ptolemaic major thir'''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".  
在[[5限纯律]]中，'''5/4''' 是第5泛音与第4泛音之间的频率比，其大小约为386.3¢.


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).
它比[[12ed2]]的4步低约13.7¢；如果用[[3限纯律]]的[[81/64]]逼近，误差则是[[81/80]]≈21.5¢；如果用[[3限纯律]]的[[8192/6561]]逼近，误差则是[[32805/32768]]≈1.95¢.


== Approximations by edos ==
== Approximations by edos ==

2025年12月2日 (二) 11:03的版本

在5限纯律中，5/4 是第5泛音与第4泛音之间的频率比，其大小约为386.3¢.

它比12ed2的4步低约13.7¢；如果用3限纯律的81/64逼近，误差则是81/80≈21.5¢；如果用3限纯律的8192/6561逼近，误差则是32805/32768≈1.95¢.

Approximations by edos

本表格只包含相对误差的绝对值≤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 +126.4941 +3.42%
22 7 -385.101 -8.24%
25 8 -748.249 -4.82%
28 9 -2888.1427 -1.4%
31 10 +2210.8565 +2.02%
34 11 +900.9429 +5.44%
37 12 +602.0688 +8.87%
53 17 -1229.5229 -6.22%
59 19 +13635.6185 +0.62%
65 21 +1255.797 +7.47%
81 26 -1534.063 -7.62%
87 28 -16207.4293 -0.77%
90 29 +4905.0016 +2.65%
96 31 +1459.3731 +9.49%
109 35 -1744.1179 -9.02%
115 37 -7634.743 -2.17%
121 39 +3737.9906 +4.67%
143 46 -5776.02 -3.57%
146 47 -140249.0112 -0.15%
149 48 +6570.9626 +3.27%
152 49 +3276.4236 +6.69%
171 55 -4963.3778 -4.97%
183 59 +3029.1111 +8.72%
199 64 -4507.6517 -6.37%
202 65 -9887.116 -2.95%
205 66 +62387.8247 +0.47%
208 67 +7703.0121 +3.9%
211 68 +4160.1807 +7.32%
227 73 -4216.0421 -7.77%
233 75 -36355.6122 -0.92%
239 77 +5825.8383 +5.92%
255 82 -4013.4235 -9.17%
258 83 -6479.1151 -5.74%
264 85 +34678.4334 +1.1%
267 86 +8522.3662 +4.52%

另见

检索自“https://tun.wiki/index.php?title=5/4&oldid=33”