审阅范围 · Review scope
限定范围独立审阅。请保留正文中的出处与限制；不代表全库最终验收、实际演奏或学习效果验证。
Independent review within the stated scope. Keep the evidence and limitations stated in the text in view. This is not final corpus acceptance or validation of performance or learning outcomes.
发布状态。 本原创伴读的来源字段、计算与双语说明已通过独立审阅及最后的定点复核。下文的“候选稿／等待审阅”及数据文件中的候选状态保留为作者制作时点的记录。本次仅加此发布说明，未改原教材或补算缺失的中文对应物。实奏、动作适用性、声音、现代部件等价及教学效果仍未验证；相关工作译名继续保留限定。
Publication status. This original companion passed independent source-field, calculation and bilingual review, including the final focused recheck. Candidate/awaiting-review wording below and in the data retains its author-time status. Integration adds only this publication note; it neither edits the textbook nor fills missing source-Chinese counterparts. Performance, movement suitability, sound, modern-part equivalence and learning effects remain unvalidated; terminology qualifications remain in force.
比值、百分量与音分 / Ratios, percentages and cents
【本库原创补充材料】来源为§76表3.2，PDF实页207／页脚206。原表四个列头分别是音程、频率比、平均律值和“Error %”。这里的“理想”只指所选的小整数分数比，不等于所有音乐都应采用的最佳调律。四度、五度和八度在这些比值下分别是纯四度、纯五度和纯八度。
[Our original supplement] The source is §76 Table 3.2, physical page 207 / footer 206. Its four headers identify interval, frequency ratio, equal-tempered value and “Error %.” Here “ideal” denotes the selected small-integer ratio, not a universally best tuning. The fourth, fifth and octave represented here are perfect intervals.
设低音频率为同一个f，分数参考比为r，十二平均律比为e=2^(n/12)，n是半音数。高音分别为rf与ef。这个模型规定八度恰为2:1、每个半音同比例；没有加入实际琴弦非谐性、伸张调律或测量误差。
Let the shared lower frequency be f, the rational reference ratio r, and the twelve-tone equal-tempered ratio e=2^(n/12), where n counts semitones. The upper frequencies are rf and ef. This model fixes the octave at 2:1 with equal-ratio semitones; it includes no real-string inharmonicity, stretched tuning or measurement uncertainty.
三个不同的问题 / Three different questions
原表数字能由什么公式得到？
100(e−r)与五个原印“Error %”数值的两位小数舍入一致。它可解释为高音频率差相对于共同低音f的百分比；原表列头本身没有写出分母。相对分数参考高音偏差多少？
100(e/r−1)%才以rf为分母。本稿称为“相对差%”。相差多少音分？
1200 log₂(e/r)。正号表示平均律的高音高于分数参考，负号表示低于。Which formula reproduces the printed values?
100(e−r)rounds to all five printed “Error %” values. It can express the upper-frequency difference as a percentage of the shared lower frequency f; the source header itself does not state that denominator.What is the difference relative to the rational upper frequency?
100(e/r−1)%uses rf as denominator. We call this “relative difference %.”How many cents apart?
1200 log₂(e/r). Positive means the equal-tempered upper note is higher than the rational reference; negative means lower.
这不是把原表数字悄悄改掉，而是把它们的可复算含义与另一种常见百分量并列。 / The source values are retained while their reproducible meaning is distinguished from a different, common percentage measure.
| 源行 / Row | 音程 / Interval | r；半音n / r; semitones n | e（本稿计算）/ e calculated | 原印列值 / Printed value | 相对差% / Relative % | 音分差 / Cents |
|---|---|---|---|---|---|---|
| 3.2 R01 | 小三度 / Minor 3rd | 6/5; n=3 | 1.189207115 | -1.08 | -0.899407 | -15.641287 |
| 3.2 R02 | 大三度 / Major 3rd | 5/4; n=4 | 1.259921050 | 0.99 | +0.793684 | +13.686286 |
| 3.2 R03 | 纯四度 / Fourth | 4/3; n=5 | 1.334839854 | 0.15 | +0.112989 | +1.955001 |
| 3.2 R04 | 纯五度 / Fifth | 3/2; n=7 | 1.498307077 | -0.17 | -0.112862 | -1.955001 |
| 3.2 R05 | 纯八度 / Octave | 2/1; n=12 | 2.000000000 | 0 | +0.000000 | +0.000000 |
例如小三度：e≈1.189207，r=1.2。100(e−r)≈−1.079288，舍入为原印−1.08；100(e/r−1)≈−0.899407%。两数分母不同，不应互换。正负号也不表示音乐好坏、协和程度、可听阈值或演奏者能力。
For the minor third, e≈1.189207 and r=1.2. 100(e−r)≈−1.079288, rounding to the printed −1.08; 100(e/r−1)≈−0.899407%. Their denominators differ. The sign does not rate musical quality, consonance, audibility thresholds or a player's ability.
取一个纯计算例子f=100 Hz（Hz表示每秒周期数）：两个高音值分别是120 Hz和约118.920712 Hz，差约−1.079288 Hz。以100 Hz为分母得约−1.079288%；以120 Hz为分母得约−0.899407%。这两个分母回答的是不同问题。
For a purely calculated example, choose f=100 Hz (cycles per second). The two upper values are 120 Hz and about 118.920712 Hz, differing by about −1.079288 Hz. Dividing by 100 Hz gives about −1.079288%; dividing by 120 Hz gives about −0.899407%. The denominators answer different questions.
【我们的纸面练习】只选一行，把相同的假设低音频率代入rf与ef，核对两种百分量的分母。所得频率是模型例子，不是某台钢琴的实测值。本轮没有生成声音、听辨或证明哪种律制更有效。 / [Our paper exercise] Choose one row and substitute the same hypothetical lower frequency into rf and ef. Check the denominators of both percentages. The results are model examples, not measurements of a piano. No sound generation, listening comparison or efficacy ranking was performed.