Musical Scales and Temperament Musical scale – a succession of notes in ascending or distending order In Western music octave is divided in 12 semitones Chromatic scale - all 12 semitones Most music makes use of 7 selected notes (major or minor scales) There are many ways to construct musical scales Different scales are different ways of dividing octave (almost always) “Standard” scales: Pythagorean scale Mean-tone temperament Scale of just intonation Equal temperament Tuning – an adjustment of pitch in any instrument so that it corresponds to an accepted norm Temperament – a system of tuning in which the intervals deviate from acoustically pure (Pythagorean) intervals Intonation – the degree of accuracy with which pitches are produced

Scales and logarithms When we go from octave to octave up, each time we multiply frequency by 2. Examples: If we go 3 octaves up, frequency is 2x2x2 = 23 = 8 times higher If we go 10 octaves up, frequency is 210 = 1024 times higher On keyboards and on musical staff distance between notes is changed linearly If we go 3 octaves, it is 3 time as much as one octave If we go 10 octaves, it is 10 time as much as one octave This means that keyboard and musical staff have logarithmic scale: distance between keys and notes is proportional to the logarithm of the frequency

Pythagorean Scale Most consonant intervals: Pythagoras (born about 580 B.C.) Most consonant intervals: 1:1 – unison 2:1 – octave – 12 semitones 3:2 – perfect 5th – 7 semitones 4:3 – perfect 4th – 5 semitones The Pythagorean system is an attempt to build a complete chromatic scale from only two of the pure tones: the octave and the perfect fifth The goal was to close circle (Circle of fifths), i.e. to end up with the same note as started. Unfortunately, it is impossible. If octave has 12 semitones, then 7 octaves have 12x7 semitones. If perfect 5th has 7 semitones, 12 perfect 5th have 7x12 semitones.

Frequency ratios of notes in Pythagorean scale C F G C 1 4/3 3/2 2 9/8 C D E F G A B C 1 9/8 81/64 4/3 3/2 27/16 243/128 2 9/8 9/8 256/243 9/8 9/8 9/8 256/243 Circle of fifth C G D A E B F# C# G# D# A# E# Advantage: good for perfect 5th and 4th. Disadvantage: poor for 3d (E/C is 81/64~1.2656 instead 5/4=1.25) Syntonic comma: (81/64)/(5/4) = 1.0125

Mean-tone temperament Pythagorean 3d are out of tune Alterations of the Pythagorean scale have been developed C D E F G A B C C D-1/2δ E-δ F+1/4δ G-1/4δ A-3/4δ B-5/4δ C Syntonic comma: δ = (81/64)/(5/4) = 1.0125 Advantage: 3d and 6th sound much better Disadvantage: the 5th and 4th are no longer perfect intervals, deteriorates as more sharps and flats are added

Scale of just intonation Any tuning system that uses integers to represent the ratios for all intervals is called Just Intonation.  Just scales can include, but are not limited to, the use of the pure tones.  More than likely, a scale is derived from the use of one or more of the pure tone ratios (as in the Pythagorean Scale).  Just intonation systems are developed around one particular note, the root.  The other notes can be determined systematically (as in the Pythagorean Scale) or decided upon by choice.  Any scale created will be considered Just as long as the ratios are in integer form.  All the notes in the scale are individually determined from the root or a pre-established note in the scale.  Just tuning depends on the scale one is using.  Since all the notes in the scale are related to each other, and (more importantly) to the root of the scale, the notes will seem to be in tune as long as one stays in the same key.  However, if one modulates into another key in the same system, there will be some problems because the ratios between the notes and the new root will be different from those of the previous root. C D E F G A B C 1 9/8 5/4 4/3 3/2 5/3 15/8 2 9/8 10/9 16/15 9/8 10/9 9/8 16/15

(All semitones are the same) Equal temperament (All semitones are the same) Octave is divided into 12 equal semitone intervals A 440 B flat 466 B 494 C 523 C sharp 554 D 587 D sharp 622 E 659 F 698 F sharp 740 G 784 A flat 831 880 Semitone ratio: Whole tone: Advantage: 5th and 4th are reasonably good 3d and 6th are OK Modulation from key to key is easy