Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。


Variation of major triad with temperament
音阶三和弦随音律变化

Harmonic Intervals in Cents

谐音间隔（以厘音为单位）

Expressing the musical intervals between successive harmonics in cents notation helps to show the relationship between the harmonics and the equal tempered musical scale. The equal tempered intervals are all in multiples of 100 cents, so the departure from such a multiple is an indication of how much the just interval is out of tune with equal temperament.

用音阶的音程表示连续泛音之间的音程关系，有助于展示泛音与等音阶音阶之间的关系。等音阶的音程都是100音分的倍数，因此偏离这样的倍数表明纯音程与等音阶的音程相差多少。

If any musical source produces a fundamental and a series of exact harmonices, then it is evident that the upper harmonics will be out of tune with the corresponding equal tempered notes. Examination of the illustration will make it evident that the 7th harmonic shows the most severe departure from any equal tempered interval. This is important in the design of brass instruments, since they use the upper resonances of the instrument as played notes in a harmonic sequence. Fortunately, the upper resonances of brass instruments can be tuned closer to equal temperament in the manufacturing process so that the problem is not so severe, but the seventh resonance is still troublesome.

如果任何音乐源产生基频和一系列精确的泛音，那么显然上泛音将与相应的等音律音阶不一致。查看图示可以明显看出，第七泛音与任何等音律音程的偏离最为严重。这一点在铜管乐器的设计中非常重要，因为它们使用乐器的上共振作为谐波序列中的音符。幸运的是，在制造过程中，铜管乐器的上共振可以调校得更接近等音律，因此这个问题并不那么严重，但第七共振仍是个问题。
 Index

Temperament and musical scales
索引 音乐的音阶与音调
 
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Major and Minor Triads

主三元组与次三元组

The major and minor triads are used widely in western music. In terms of just musical intervals, these triads can be expressed as whole number ratios of frequencies. The actual frequencies vary slightly in other temperaments, but the labeling of the intervals is the same.

在西方音乐中，大三和弦和小三和弦被广泛使用。从纯律的角度来看，这些三和弦可以表示为频率的整数比。虽然在其他调律系统中实际频率会略有变化，但音程的命名保持不变。

Note that the 4th, 5th and 6th harmonics of a harmonic sequence form a major triad: The pitch ratio 4:5:6 is the triad. The minor triad does not fit so neatly in the harmonic sequence, the smallest whole number ratios that produce a minor triad being 10:12:15.

注意，谐波序列中的第4、5和6个谐波形成一个大三和弦：谐波比4:5:6构成了这个三和弦。而小三和弦则不太符合谐波序列的结构，产生小三和弦的最小整数比为10:12:15。
Variation of major triad with temperament
音阶三和弦随音律变化
 Index

Temperament and musical scales
索引 音乐的音阶与音调
 
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Major Triad Differences

主要三和音差异
The differences between Just, Pythagorean , and Equal Tempered scales becomes evident in the tuning of a triad. The fifth is essentially the same for the three temperaments, but the major third differences are significant.
Just、Pythagorean 和 Equal Tempered 调音系统在三和弦的调谐中表现出差异。五度在三种调音系统中基本相同，但大三度的差异却相当显著。
 The frequencies for the major triad A-C#-E are shown at left. The three different major thirds are labeled in cents, and the difference can be seen to be larger than the just noticeable difference of 5¢.
主要三和弦A-C#-E的频率如左所示。这三个不同的主要三度在音分上标出，可以明显看出其差异大于5音分的可察觉差。

The difference between the Just and Phythagorean major thirds is 22¢, which is called the "syntonic comma". Nine commas make approximately a whole tone.

正音与毕达哥拉斯大三和弦的差异为22音分，这被称为“纯律音高”。九个这样的音高大致构成一个大二度。
Pythagorean intervals
毕达哥拉斯音程
 Index

Temperament and musical scales

Reference
Rossing
Science of Sound, 2nd ed.
Ch 7
索引、音色与音乐音阶 参考 Rossing《声音的科学》第二版 第7章
 
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Pitch Change with Temperature

温度变化对音调的影响

Since the speed of sound in air is given approximately by

由于空气中声速近似由

where TC is the celsius temperature,

其中 T C 是摄氏温度

then at temperature °C = °F,

然后在摄氏温度°C = 华氏温度°F，
中文译文中的待填/计算数值依次对应：1：t 2：tf。实际数值以上方原输入框为准。

the speed of sound is m/s = ft/s = mi/hr.

声音的速率是 m/s = ft/s = mi/hr。
中文译文中的待填/计算数值依次对应：1：v 2：vc 3：vm。实际数值以上方原输入框为准。

Since the pitch of air column instruments depends directly upon this speed of sound, then if the air column were at °C and were warmed by 1 °C, the pitch would increase (go sharp) by about cents.

由于空气柱乐器的音调直接取决于声速，因此如果空气柱在°C时，因升温1°C而使声速增加，其音调将升高（变尖）约 cents。
中文译文中的待填/计算数值依次对应：1：t2 2：dc。实际数值以上方原输入框为准。

If the column were warmed to °C, then it would be cents sharp.

如果该柱子被加热到°C，它就会变得锋利。
中文译文中的待填/计算数值依次对应：1：t3 2：dc2。实际数值以上方原输入框为准。

Note that you don't have to know the actual frequencies for these calculations since musical intervals are defined as pitch ratios.

请注意，这些计算不需要知道实际的频率，因为音乐音程是定义为音调比值的。
Pythagorean intervals
毕达哥拉斯音程
 Index

Temperament and musical scales

Reference
Rossing
Science of Sound, 2nd ed.
索引、音色与音乐音阶 参考 Rossing《声音的科学》第二版
 
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