Fundermentals of Temperament

I compiled a Glossary which includes brief definitions of the following terms used in the following discussion.

• [->] Just tone
• [->] Just Temperament and Pythagorean Temperament
  • [->] Just Temperament
  • [->] Pythagorean Temperament
• [->] Explanation of technical term
  • [->] Pythagorean Comma (PC)
  • [->] Syntonic Comma (SC)
  • [->] Cent

1.Just tone

``The interval which frequency ratio can be expressed by simple integer ratio.'' is called Just Tone. Using these tones at a time, there is no vibration and the sound is very good. Generally this condition is called ``harmonize''.

Usually discussing on temperament, the following intervals are treated as consonance tones on Just temperament.

1:1 Unison
1:2 Octave
2:3 fifth
3:4 fourth
4:5 major third
5:6 minor third

Major triad consists of Perfect fifth and major third.

Frequency ratio = 4:5:6

Similarly, minor triad consists of Perfect fifth and minor third.

Frequency ratio = 10:12:15

However, it is impossible to make every major third and minor third as Just tone on a temperament. The following temperaments are derived from the difference in handling of tones in the dissonance and consonance point of view.

2. Just Temperament and Pythagorean Temperament

Every temperament is based on Just temperament and Pythagorean Temperment. That is why it is good for you to start learning these 2 temperament to understand the general technical term of temperament.

1) Just Temperament

Just Temperament is basically derived from Just Tone of main triad (C,F,G in C dur). Followings are the calculation of the tone of Just Temperament.

G = 3/2 x C  =  3/2
E = 5/4 x C  =  5/4
F = 4/3 x C  =  4/3
A = 5/4 x F  =  5/3
D = G / 4/3  =  9/8
H = 5/4 x G  = 15/8

Arrangement in pitch order

C     D      E      F      G      A      H
1    9/8    5/4    4/3    3/2    5/3   15/8
 \  /   \  /   \  /   \  /   \  /   \  /
  9/8   10/9  16/15    9/8   10/9    9/8

The result reveals that whole tone has 2 types, ie. 9/8 and 10/9. The former is called major whole tone and the latter is called minor whole tone.

->[Characteristic of Just Temperament]

2) Pythagorean Temperament

The tone is closely same as the octave tone when the tone 3/2 to power 12 of the original tone. (In fact, the tone is little bit higher than octave.) By using this theory, Pytagorean Temperament divides the scale between octave by 12. To adjust octave, the last 5th become narrower than 3/2 of the tone. (Usually the narrow 5th is set between Gis and Es.)

C   G   D   A   E   H   Fis   Cis   Gis   Es   B   F   C   ...
 \ / \ / \ / \ / \ / \ /   \ /   \ /   \ /  \ / \ / \ /
 3/2 3/2 3/2 3/2 3/2 3/2   3/2   3/2   3/2  PC  3/2 3/2 3/2

->[Characteristic Pythagorean Temperament]

3. Explanation of technical term

1) Pythagorean Comma (PC is an abbreviation)

The difference between original tone and the tone which 3/2 to power 12 of original tone. 5th steps between Es and Gis becomes narrow as Pythagorean Comma on Pythagorean Temperament.

2) Syntonic Comma (SC is an abbreviation)

The difference between major third which made by 3/2 to power four and major third on Just Temperament. (24 cents) The scale is same as the difference between E on Pythagorean Temperament and E on Just Temperament.

Cent

A unit of a natural logarithm of frequency ratio. The unit is defined to adjust a semitone on Equal Temperament as 100.

If we uses R as frequency ratio and C as cent, the relation between R and C is the following.

R = 2^( C/1200 )  <->  C = 1200 x log_{2} R

(Miyauchi)