Basic principles for producing temperaments

• Main musical commas (pythagorean - syntonic - enharmonic)
• Empirical way to obtain a comma (pythagorean-syntonic-enharmonic-schism)

Representational diagram of the tuning system

• Pythagorean Temperament
• Equal Temperament

            Since the creation of harmonic instruments in the ancient times, many expert mathematicians, when dealing with instrumental tuning, have aimed at satisfying the needs of human hearing.

            Whatever the tuning, the interval of octave must be acoustically pure. As is well known, every sound produces some harmonics that cannot be perceived by our hearing. If, however, we produce simultaneously two sounds our hearing recognises the harmonical differences. The interval is pure when no beats are perceived, that is, when the sound perfectly coincides with a harmonic.

            For reasons that we are going to explain, all intervals cannot be pure. Thus, we have a temperament, a compromise consisting in a certain variation of intervals, beginning with their acoustically pure value, aimed at satisfying the rigorous and inalterable condition of the pure octave. The simplest technique to obtain a temperament is to work on the fifths and fourths (complementary interval).

Basic principles for producing temperaments

            Temperaments originate from the fact that, in Western music, the octave is made up with fixed intervals. This fixed parameter, and the one hand, and on the other the limited number of the notes of the octave, raise problems of sound gamut. Their solution lies in the so-called musical commas, short intervals (of 1/10 of tone), defined by the nature of the different musical intervals.

Main musical commas

            The tuning practice concerns three types of comma regarding the octave, the fifth and major third, which lie at the basis of the musical temperament.

• Twelve consecutive and acoustically pure fifths do not produce the octave. We thus have the Pythagorean comma comma pitagorico (also called diatonic comma);

(Fig. 1 - Musical illustration of the Pythagorean comma)

• Four consecutive and acoustically pure fifths do not produce a pure major third. We thus have the Syntonic comma (also called major, natural or Didymus's comma);

(Fig. 2 - Musical illustration of the Syntonic comma)

• Three consecutive and acoustically pure thirds major do not produce the octave. They give birth to the Enharmonic comma.

(Fig. 3 - Musical illustration of the Enharmonic comma)

Empirical way to obtain a comma

            From C3 in the keyboard you must cover the 12 pure ascending fifths. In order to make this procedure as quick as possible, you should tune alternatively an ascending fifth and a descending fourth. This allows you to remain within the same octave (by convention in all tables the white notes are «tuned», the black notes need to be «tuned»). Once you have concluded the circle of the fifths, the last tuned note of the fifth, B#3, should coincide with C4. If we check the C3- C4 octave we notice that the B#3 is much more similar to C# than to C. This difference, that is, the one between the B# obtained from the «fifth» and the C obtained from the «octave», defines the Pythagorean comma.

            Starting from C3 let us tune the pure intervals as in Table 2, and let us take into account the last fifth. In this case E2 is almost an F. As a matter of fact, if we compare the E obtained from the pure major third C-E with the E obtained from pure fifth (or, if we prefer, from the pure fourth) A-E, the latter is clearly higher. The syntonic comma consists in the difference between two homophonous sounds.

Enharmonic comma

            From the same C3, let us tune three pure major thirds as in Table 3. In this case the B#2 = C3, obtained from the last third G# - B#, is higher (almost a C#) than the homophonous C3 obtained from the ocatve C2- C3. The difference between these sounds produces the enharmonic comma.

Algongside these three parameters we have a fourth one, the schism, that is, the difference between the Pythagorean and the syntonic comma.

Schism = Pythagorean comma - syntonic comma

            The most convenient representation of the cycle of the fifths is udoubtedly a circle divided as a clock. As a matter of fact, this scheme helps us evaluate the quality of the components of a given system. The pure fifth of our diagram will be marked as a "0", while the altered, or tempered, fifth as a whole number or as a fraction preceded by + or -.This fraction shows how much the comma must be modified.

Table 4 - Pythagorean System (with the wolf fifth on F [E#] - C)

The last fifth of the cycle is strongly reduced of one Pythagorean comma. In the Pythagorean system this fifth is also called "fifth of the wolf" or simply "wolf". The interval of fifth is a fundamental element in Western music. It represents a measure unity and allows us to understand the quality of the intervals at a glance. For example:

• The tone is made up with two consecutive fifths (C-G and G-D);

• The major third is made up with 4 consecutive fifths. The thirds cointaining exclusively pure fifths are called "Pythagorean". Therefore they are larger than the thirds containing tempered thirds.

• The semitone is made up with 7 consecutive fifths. The semitones containing pure fifths are larger than those containing the fifth "of the wolf";

• etc...

Table 5 -Tempered System

In the equal temperament the Pythagorean comma is divided into equal parts as regards all the fifths. In this way all tonalities will produce the same sound effect. The major thrid is more narrow than the Pythagorean third of 1/3 (4/12) of Pythagorean comma.

The major thrid is more narrow than the Pythagorean third of 1/3 (4/12) of Pythagorean comma. It follows that the Pythagorean third is larger of the pure third of 2/3 (8/12) of Pythagorean comma minus a schism.

There fore, tempering means to work upon the fourths and the fifths in order to change the intervals. When we deal exclusively with the fifths within a temperament we have a Pythagorean comma. On the contrary, when the thirds are taken into account, we have a syntonic comma.

            We will explain later on all the stages necessary for the practical realization of every single temperament. It is, however, difficult to put into words how to temper a fifth to 1/3, ¼ or even 2/7 of comma! The secret lies in the speed of the beat of the tempered fifth, a speed that varies from one temperament to another. My personal experience has taught me to regard continuous exercise and practice as the true key for reaching the adequate skill necessary for dealing with the different tunes.