Pythagorean Scale (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 (three) of the pure tones: the octave and the perfect fifth/forth The goal was to close circle (Circle of fifths), i.e. to end up with the same note as started. It is impossible to achieve exactly, but it is possible to achieve approximately If octave has 12 semitones, then 7 octaves have 12x7 semitones. If perfect 5th has 7 semitones, 12 perfect 5th have 7x12 semitones.

Questions: How to construct a convenient scale? How to set up an interval between notes? What is known: Perfect fifth plus perfect forth is an octave Perfect fourth up is the same as octave up and then perfect fifth down Perfect fifth up is the same as octave up and then perfect fourth down So we can use octave and perfect fifth to construct perfect fourth 12 perfect 5th are approximately equal to 7 octaves 12 semitones/octave! 7 semitones/ perfect 5th!

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# (F)

More about the circle of fifth

Pythagorean scale (continues) 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 Games with numbers Tone: Diatonic semitone (C/B, F/E): Chromatic semitone (F#/F): Chromatic semitone Diatonic semitone Pythagorean coma =

Pythagorean temperament Comparing scales The Pythagorean fifth is 2 cents greater leading to 24 cents overlap The circle of fifth: Equal temperament Pythagorean temperament

Mean-tone temperament Pythagorean 3d are out of tune (E/C is 81/64~1.2656 instead 5/4=1.25) Alterations of the Pythagorean scale have been developed An attempt to alter the Pythagorean scale by flattening the third so that the major and minor third correspond to the just intervals It, however, leads to the sharps and flats being more out of tune 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

Pythagorean temperament Comparing scales The meantone temperament, the fifth is 3.5 cents short of the equal temperament, which leads to a short of 3.5x12=42 cents when the circle of fifths is completed The Pythagorean fifth is 2 cents greater leading to 24 cents overlap The circle of fifth: Equal temperament Mean tone temperament Pythagorean temperament

Scale of just intonation The just scale is based on the major triad, a group of three notes that sound particularly harmonious, where the first two notes are a major 3rd apart and the last two are a minor 3rd apart, with frequency ratios of 4:5:6 λf:λi ff:fi example # of half steps 4:5 5:4 major third (C,E) or (Ab,C) 4 5:6 6:5 minor third (C,Eb) or (A,C) 3 All seven notes of the just scale can be obtained by letting these three triads consist of notes with the frequency ratios 4:5:6 If C:E:G is 4:5:6, it also means 1:5/4:3/2 G:B:D => B = 5/4 * 3/2 = 15/8 and D = 3/2 * 3/2 = 9/4. To drop the D to the same octave it becomes 9/8 F:A:C => F = 2 ÷3/2 = 4/3 and A = 2 ÷6/5 = 5/3 (this is in opposite direction) 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

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 We have three intervals: 9/8, 10/9 and 16/15 called the major whole tone, the minor whole tone, and the semitone respectively Disadvantage: Five fifths are perfect but the other are not perfect. D:A is imperfect. And the same is true of the fourths. When you do the sharps and flats, enharmonic notes (G# and Ab) are not equal as they can be derived by various triad combinations

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.

Modulation of one tone by another (nonlinear effect) One frequency is considerably smaller then the other 1) Amplitude modulation 2) Frequency modulation