Musical Scales

The octave, fifth, and fourth are the intervals which have been considered to be consonant throughout history by essentially all cultures, so they form a logical base for the building up of musical scales. A typical strategy for using these univerally consonant intervals is the circle of fifths. 八度、五度和四度是历史上几乎所有文化都认为是和谐的音程，因此它们构成了构建音乐音阶的逻辑基础。利用这些普遍和谐的音程的一种典型策略是五度圈。

Musical Intervals

The term musical interval refers to a step up or down in pitch which is specified by the ratio of the frequencies involved. For example, an octave is a music interval defined by the ratio 2:1 regardless of the starting frequency. From 100 Hz to 200 Hz is an octave, as is the interval from 2000 Hz to 4000 Hz. The intervals which are generally the most consonant to the human ear are intervals represented by small integer ratios. Intervals represented by exact integer ratios are said to be Just intervals, and the temperament which keeps all intervals at exact whole number ratios is Just temperament.

Consonance and Dissonance

Two tones are said to be consonant if their combination is pleasing to the ear, and dissonant if displeasing. The simplest approach to quantifying consonance is to say that two tones are consonant if their frequencies are related by a small integer ratio. The ratio determines the musical interval. For example, the octave 2:1, fifth 3:2, and fourth 4:3 are presumed to be universally consonant musical intervals because most persons in any culture or period of history have considered them to be pleasing tone combinations and have built musical compositions around them.

For example, in the buildup of a pentatonic scale by a circle of fifths, a natural whole tone of ratio 9/8 emerges, satisfying the condition for consonance. A semitone like E-F also emerges, and the ratio 256/243 suggests dissonance. 例如，在通过五度圈构建五声音阶时，自然全音的比值为9/8，满足和声的条件。一个半音如E-F也出现，比值256/243则暗示不和谐。

When you define "consonance" as "pleasing to the ear", then of course you have to ask "whose ear?". You can get into such intense debate about what is "pleasing" that some have come to define music as "sounds organized by human beings" to accede the endless variety. The use of consonance here is limited to giving a suggestion of a simple rule that yields musical intervals that are pleasing to most people, i.e., "consonant".

Circle of Fifths

A full chromatic scale can be created by using just the perfect fourth and fifth musical intervals. This is characteristic of the Pythagorean temperament. This process can be pictured on the circle of fifths. The outer circle visits all twelve notes on the chromatic scale by going up by fifths (or down by fourths) . The inner circle goes down by fifths (or up by fourths). To create all these notes in the same octave, you could drop down an octave when necessary to stay in the original octave. 通过使用纯四度和纯五度的音乐音程，可以构建出完整的十二平均律。这是毕达哥拉斯调律的特征。这个过程可以在五度圈上形象地表现出来。外圈通过每次上五度（或下四度）来访问十二平均律中的所有音。内圈则通过每次下五度（或上四度）来访问。为了在同一个八度内生成所有这些音，必要时可以降一个八度以保持在原八度内。 Why fifths? 为什么五度？