Study of temperament Temperament is an interdisciplinary subject of acoustics, music acoustics, mathematics and musicology. All studies of pitch in music involve temperament. For example: the structure and intonation of melody intervals; the principle of harmony in mode and harmony theory; the various interval relationships when multiple voices are combined vertically; the theory of modulation; the determination of intonation and phoneme in instrument manufacturing and tuning; duet singing , Pitch adjustment in chorus ensemble;...etc. Since temperament is closely related to the existence of music itself, although the study of temperament must use the methods of physics and mathematics, it must also involve the scales and modes actually used in the music of various ethnic groups around the world. Rhythm studies in practice In terms of application and development, it ultimately serves the improvement of music performance, the development of music creation, and the overall improvement of music culture. Regardless of whether the length ratio, frequency ratio or period ratio is used, there are inconveniences: when comparing the size of two approximate musical intervals, multiplication or division must be used. Without some calculations, one cannot know which one is larger and how much larger. To add or subtract two intervals, multiplication and division must be performed. To how many times an interval is expanded and how many equal parts it is divided into, we need to perform exponentiation and square root calculations. With the development of mathematics, the concept of logarithm began to be introduced into the measurement of musical intervals in the 19th century, and the concept of "interval value" was established. The method of calculating the interval value is to convert the frequency ratio of a certain interval into a logarithm and formulate a certain unit name according to a certain purpose. With the interval value, the size of the interval can be seen at a glance. The addition and subtraction of intervals can be calculated by adding and subtracting the interval value. How many times the interval is expanded or divided into equal parts can also be calculated by simple multiplication and division. Many countries now use "cent" as the unit of interval value. This was created by the British mathematician, comparative musicologist, and linguist A.J. Ellis (1814-1890). The interval value of the octave in twelve equal temperaments (the most commonly used temperament in equal temperament, as well as Indian equal temperament, eight equal temperaments, six equal temperaments, etc.) is 1200 cents, and the relationship between each semitone is 100 cents.

The number of notes of any interval in any temperament can be calculated by the common logarithm based on the frequency ratio: first find the proportionality constant, and then multiply the common logarithm of the frequency ratio of each interval by the proportionality constant, that is have to. The proportionality constant is: the number of notes in the octave ÷ the common logarithm of the octave frequency ratio 2 = 1200 ÷ 0.30103 = 3986.313. If you want to find the number of notes in a pure fifth, multiply the logarithm (0.17609) of the frequency ratio of the pure fifth (3:2=1.5) by the proportionality constant: 0.17609×3986.313=701.950. The decimal is often rounded to 702 cents. In addition to cents, other unit systems for interval values ??include the "Savar", which was created by the Frenchman F. Savard (1791 ~ 1841). Just multiply the common logarithm of the frequency ratio of a certain interval by 1000 to get the Savard number, so 1 Savard ≈ 4 cents. "Miyou" (μ) means "thousandth of an octave". German musicologist H. Lieman and music educator and linguist C. Aitz (1848~1924) both used this interval value. An octave is measured as 1,000 "miyus", so 1 miyu = 1.2 cents. "Full tone" means "six octaves", which was coined by Japanese musicologist Naozu Tanabe. The octave is measured as 6 whole tones, 1 whole tone = 200 cents.