WEEK 2 MUSIC AND MATHEMATICS MUSIC AND MATH WEEK 2
QUESTIONS ▸ How did we get from the Pythagorean tuning system we explored last time to the modern equal-temperament tuning system?
▸ Why do we associate major chords with “happy” and minor chords with “sad” - i.e. is it nature or nurture?
▸ Can we quantify the notions of consonance and dissonance in ways that aline with our experience? MUSIC AND MATHEMATICS
WHAT WE DISCOVERED LAST TIME ▸ Pythagorus developed a tuning system based on the ratios 2:1 and 3:2.
▸ Today we call the ratio 2:1 an octave and the ratio 3:2 a perfect fifth
▸ By stacking fifths and then adding or subtracting octaves to keep the notes within one octave, he developed a 7- note scale (now called the Pythagorean Diatonic Scale) that is the predecessor to the modern Western Major Scale MUSIC AND MATH WEEK 2
MODERN PERSPECTIVE: MUSIC THEORY ▸ 12 notes (the Chromatic or Dodecaphonic Scale)
▸ Important subsets of these notes ▸ Pentatonic Scale
▸ Major Scale MUSIC AND MATH WEEK 2
EQUAL TEMPERAMENT ▸ Online Pythagorean tuning (and other tuning systems that we’ll discuss), today we use an equal-tempered tuning system.
▸ Changes in pitch are still ratios (every note is a multiple of another note’s frequency)
▸ Just like Pythagorus, we end up with twelve notes MUSIC AND MATH WEEK 2
WHAT ARE THESE RATIOS? ▸ We want the ratio between successive notes to be the same.
▸ Call that ratio alpha.
▸ Then to get from any one note to any other we have… MUSIC AND MATH WEEK 2 MUSIC AND MATH WEEK 2
CONSTRUCT A FEW NOTES ▸ Start (as is standard today) with A = 440Hz.
▸ What’s the frequency of the first C above this?
▸ How about the E above that? MUSIC AND MATH WEEK 2
MAJOR AND MINOR CHORDS ▸ Major
▸ Start with any note N in the Chromatic Scale
▸ Play N, N+4, and N+7 (e.g. C-E-G, B♭-D-F)
▸ Minor
▸ Start with any note N in the Chromatic Scale
▸ Play N, N+3, and N+7 (e.g. C-E♭-G, D-F-A)
▸ Qualitatively, what’s different about how these sound? MUSIC AND MATH WEEK 2
NAMING CONVENTIONS Note Interval Name (Relative to the root, C) C Unison C# Minor Second D Major Second D# Minor Third E Major Third F Perfect Fourth F# Tritone/Augmented Fourth/Diminished Fifth G Perfect Fifth G# Minor Sixth A Major Sixth A# Minor Seventh B Major Seventh MUSIC AND MATH WEEK 2
BACK TO CHORDS ▸ Notice that a Major chord consists of a root note, its major third, and its perfect fifth.
▸ Notice that a Minor chord consists of a root note, its minor third, and its perfect fifth.
▸ More complex chords are formed by adding color in the form of other notes.
▸ E.g. Maj7: Root-Major Third-Perfect Fifth-Major Seventh MUSIC AND MATH WEEK 2
OVERTONES ▸ Just as colors can be shown to consist of mixtures of light at various frequencies, sounds can be shown to consist of mixtures of sinusoids at various frequencies and strengths.
▸ What kind of picture do we get when we add sin waves together? MUSIC AND MATH WEEK 2 EXAMPLES… MUSIC AND MATH WEEK 2
BACK TO ANCIENT GREECE ▸ Pythagorean Dodecaphonic Scale
▸ All constructed with ratios of 2:1 and 3:2 like before
▸ If we get everything back to within the octave and put them in order…. MUSIC AND MATH WEEK 2
WHAT ARE SOME OF THE PROBLEMS WITH THIS? ▸ Look at the intervals between the notes.
256 2187 256 2187 256 256 531441 256 256 2187 256 2187 256 243 2048 243 2048 243 243 524288 243 243 2048 243 2048 243 ▸ What problems do we foresee?
▸ Polyphony, Transposition
▸ The (horrific) ratio 531441:524288 is called the Pythagorean comma MUSIC AND MATH WEEK 2
AN EARLY “SOLUTION” FROM PTOLEMY ▸ Ptolemy
▸ Second Century scientist, mathematician, geographer
▸ Neat early model of the solar system
▸ Ptolemy’s Natural Chromatic Scale
▸ Look at the overtones: f, 2f, 3f, 4f, 5f, 6f, 7f,…
▸ There are lots of integer ratios in there. Which ones are closest to the Pythagorean thirds (minor and major)? MUSIC AND MATH WEEK 2
NATURAL (SYNTONIC) DIATONIC SCALE ▸ Ptolemy took the ratios in the Pythagorean Diatonic Scale (the major scale) and shifted the notes whose interval were “too far” from nice whole number ratios.
▸ Which one’s are those?
▸ What do we get?
▸ Listen and see if you notice any differences…
C D E F G A B ©
Syntonic Diatonic 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
PythagoreanDiatonic 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1
Difference 1/1 1/1 80/81 1/1 1/1 80/81 80/81 1/1 DISSONANCE… MUSIC AND MATH WEEK 2
CONSONANCE OF INTERVALS ▸ Terms:
▸ Latin consonare: sounding well together
▸ Foundations of Consonance
▸ A successful metric of consonance should…
▸ Decrease monotonically in proportion to increasing dissonance
▸ Self-evidently partition intervals into the relevant categories (e.g. perfect, imperfect, dissonant)
▸ Any ideas? MUSIC AND MATH WEEK 2
HOW SHOULD WE ORDER THE (NATURAL) INTERVALS IN TERMS OF DISSONANCE? Note Interval Name (Relative to the root, C) Ratio C Unison 1/1 C# Minor Second 16/15 D Major Second 9/8 D# Minor Third 6/5 E Major Third 5/4 F Perfect Fourth 4/3 F# Tritone 64/45 G Perfect Fifth 3/2 G# Minor Sixth 8/5 A Major Sixth 5/3 A# Minor Seventh 16/9 B Major Seventh 15/8 MUSIC AND MATH WEEK 2
Note Interval Name (Relative to the root, C) Ratio
C Unison 1/1 C# Minor Second 16/15 D Major Second 9/8 D# Minor Third 6/5 E Major Third 5/4 F Perfect Fourth 4/3 F# Tritone 64/45 G Perfect Fifth 3/2 G# Minor Sixth 8/5 A Major Sixth 5/3 A# Minor Seventh 16/9 B Major Seventh 15/8 MUSIC AND MATH WEEK 2
TEMPERING - AKA MAKING USEFUL SCALES ▸ Mean-tone
▸ Optimize the thirds and fifths in selected keys
▸ Well-tempered
▸ Make all keys usable, but allow some to be more purely intoned than others
▸ Equal-tempered
▸ Every key sounds the same MUSIC AND MATH WEEK 2
MEAN-TONE TEMPERED SCALE ▸ The natural major third is a ratio of 5/4 (again, this is said to be the natural one because it occurs in the harmonic series).
▸ The major scale has three major thirds in it.
▸ They are…
▸ What if we decide to preserve these?
C : D : E : F : G : A : B : C
5 5 5 4 4 4 MUSIC AND MATH WEEK 2
OTHER OPTIONS… ▸ Partch’s 43-tone scale
▸ Genesis of a Music (1947)
▸ “I am a composer seduced into carpentry.”
▸ Approximations to the pure small integer ratios were a travesty to the ear.
▸ Hindustani 22-Sruti Scale
▸ Split the octave into 22 degrees, called Sruti
▸ Based on just intervals without transposition Note degree Interval Value in cents Interval Name 1 1/1 0 unison 2 81/80 21.50629 syntonic comma 3 33/32 53.27296 undecimal comma 4 21/20 84.46723 minor semitone 5 16/15 111.7313 minor diatonic semitone 6 12/11 150.6371 3/4-tone, undecimal neutral second 7 11/10 165.0043 4/5-tone 8 10/9 182.4038 minor whole tone 9 9/8 203.9100 major whole tone 10 8/7 231.1741 septimal whole tone 11 7/6 266.8710 septimal minor third 12 32/27 294.1351 Pythagorean minor third 13 6/5 315.6414 minor third 14 11/9 347.4080 undecimal neutral third 15 5/4 386.3139 major third 16 14/11 417.5081 - 17 9/7 435.0843 septimal major third 18 21/16 470.7811 narrow fourth 19 4/3 498.0452 perfect fourth 20 27/20 519.5515 acute fourth 21 11/8 551.3181 harmonic augmented fourth 22 7/5 582.5125 septimal tritone 23 10/7 617.4880 Euler's tritone 24 16/11 648.6823 harmonic diminished fifth 25 40/27 680.4490 narrow fifth 26 3/2 701.9553 perfect fifth 27 32/21 729.2194 wide fifth 28 14/9 764.9162 septimal minor sixth 29 11/7 782.4924 - 30 8/5 813.6866 minor sixth 31 18/11 852.5924 undecimal neutral sixth 32 5/3 884.3591 major sixth 33 27/16 905.8654 Pythagorean major sixth 34 12/7 933.1295 septimal major sixth 35 7/4 968.8264 harmonic seventh 36 16/9 996.0905 Pythagorean minor seventh 37 9/5 1017.596 just minor seventh 38 20/11 1034.996 - 39 11/6 1049.363 21/4-tone, undecimal neutral seventh 40 15/8 1088.269 classic major seventh 41 40/21 1115.533 - 42 64/33 1146.727 - 43 160/81 1178.494 octave - syntonic comma