Pythagoras and the monochord

The Pythagorean School was the first to look for a relationship between the physics of sound and numbers and the symbolic implication of the latter. To obtain a theoretical "foundation" for the major or minor "pleasantness" (consonance) of sounds, Pythagoreans used the monochord; which was a very simple instrument made of a string held taut between two fixed ends beneath which a moveable bridge could freely slide in order to "break" the string into two segments of variable length. Listening to the sound produced by these two string segments, we realise that a consonant sound is only obtained when the ratio between the measurements of the two parts is a fraction composed of two small whole numbers. It was in this way, by calculating ratios of numbers from 1 to 4, that Pythagoras believed he could obtain all the consonances: the fourth (expressed as the ratio 4:3), the fifth (3:2), the octave (2:1) and the fifteenth (4:1). The numbers from 1 to 4 were the only ones allowed because their sums equal the perfect number 10, the tetraktys, which the Pythagoreans chose as the "foundation of immortality". The beauty of the relationships between sounds was this way "legitimised" by nature itself and a number was considered the supreme symbol of this miracle.

Pythagorean scale

The Pythagorean diatonic scale was generated by ascending and descending jumps of only those intervals indicated by Pythagoras (fourth, fifth and octave), as is evident by a simple mathematical calculation. It is completely satisfactory in the intonation of fourths, fifths and octaves; however, it presents several problems in the cases of other intervals, which become very serious if we adventure into changes of tonality and modulations. We fall into the error known as the "Pythagorean comma": if we begin from any note and ascend by 12 fifths, we should find the same note seven octaves higher; however, if we do the same the same calculation descending by 7 octaves the two results do not coincide because (3/2)¹² = 129.7 while 2⁷ = 128. This divergence creates highly perceivable intonation flaws.

For more details on the Pythagorean scale, see the page Pythagorean scale.

Ptolemaic (or natural) scale

Only in the 2^nd century BCE did Ptolemy attempt to retrieve the intervals of third and sixth considering them as the best consonances within a new scale, called natural, which reached its height of splendour in the 1500s through the work of the music theorist Gioseffo Zarlino. This scale is constructed on major and minor triads, which would build the foundation of occidental harmony centuries later. In this scale, ratios are calculated of whole numbers from 1 to 6, however, here too some intervals work better and others worse (above all sharp fourths and flat fifths). It is impossible to ascend by a sixth and then a fourth and then descend by two perfect fifths without falling into the "fatal" error, called in this case, the "Zarlinian comma", which makes the scale highly unstable.

For more details on the natural scale, see the page Natural scale.

Temperate scale and the invention of equal temperament

It became necessary to have a type of "intervallic compromise" that would distribute the (inevitable) error so slightly over all the intervals as making it almost inaudible. This is how "equal temperament" took hold between the 1500s and the 1600s. The name means "division of scales into n equal intervals"; in the case of the western chromatic octave, this equals 12 semitones. All interval ratios of semitone, tone, third, fourth, fifth, sixth, seventh and eighth would be absolutely equal in every register of an instrument. While this may be to the detriment of perfect intonation, it was certainly to the benefit of a harmonic stability allowing for the instantaneous transposition in any tonality, as well as any type of modulation (change from one tonality to another). It is very interesting to compare temperate scales of different civilizations: the division of the octave into n parts is constant, however, the number of these parts radically changes the colour and harmonic rate of each scale.

For more details on the temperate scale, see the page Equal temperament.

In-depth study and links

For a modern scientist, the discovery of a link between "simple ratios" of the lengths of the parts of a monochord and the consonance of a sound, does not lead to a Pythagorean "metaphysical" (let's say almost numerological) explanation. Once it has been established that sound is a physical entity, the presence of these ratios must have an origin connected to the production modality of the sounds (in this case, the length of the vibrating string). If you are interested in understanding how physics explains the "miracle" of consonance of simple ratios, visit the page that deals with these aspects.

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