Music in Math

1
Music in Math

• (or what many people probably dont think about
  when they think of math in music)

2
Factors of Composition Performance

• Rhythm
• Subdivision versus Addition
• Harmony
• Chord Construction
• Intonation
• Pythagoras and Vincenzo Galilei
• Form
• Serialism versus Minimalism

3
Rhythm

• Western Music is always based on 2 numbers 2 and
  3
  
• All Rhythmical denominations in Western Music are
  based on 2x (2/4, 4/1, 7/8, etc)
  
• African Music is based on polyrhythms
• Much Middle Eastern Rhythm is additive
• 20th Century art music combines all forms

4
Subdivision

• Rhythmic design (in composition) is based on
  finding a common pulse (beat) that repeats as
  major points of emphasis.
  
• The beat is divided into additional beats of more
  or less significance, and further divided after
  that.
  
• Performers will often divide the less significant
  beats even further.

5
Additive Rhythms

• Rhythmic design is based on finding a groove
  (thematic pattern) that repeats.
  
• The thematic pattern is constructed with notes of
  varied lengths, and tied together by bar lines
  for phrasing.
  
• 20th century music often does this with
  Arhythmicality, but usually uses Western Meters.
  

6
Harmony

• Western Music is based on the Septatonic scale in
  7 different modes and numerous derivations
  therein.
  
• The Ionian mode (Major scale) is based on the
  naturally occurring Pythagoras scale.
  
• Each scale represents a pattern of melodic
  construction from Tonic to Leading Tone

7
Circle of 5ths

• The 5th note of the scale becomes the first note
  of the next scale
  
• The pattern of interval relationships is
  maintained for each Scale
  
• The pattern loops back around at F Major.

8
Chord Construction and Palestrina Counterpoint

• Western Chords are based on the third note of the
  scale, ascending (first to third, third to fifth,
  fifth to seventh)
  
• Various rules are applied to notes moving against
  notes (counterpoint). For example, 5ths cannot
  be approached in the same direction, and may
  NEVER move in the same direction.

9
Chord Construction and Palestrina Counterpoint
(cont.)

• The Logic puzzle created here forms harmonies
  based on a limited amount of choices (if the
  soprano line cannot move up because it would
  resolve to a dissonance on a strong beat, it must
  move down).
• Limiting the amount of options available to the
  composer creates a more cohesive work.

10
Intonation

• A string plucked, a note sung, a pipe blown, a
  reed vibrated, or a block struck will oscillate
  at a given frequency (Hz).
  
• The relationship of the various pitches in a
  scale is based on the Overtone Series and
  Pythagoras's ratios (which are also based on the
  Overtone Series).

11
Overtones and Pythagoras

• Specific notes resonate at various pitches above
  the pitch produced. This can be simulated by
  blowing a bugle (or a trumpet without touching
  the valves) at a tighter velocity, resulting in
  different resonating frequencies.
• All objects resonate at a given frequency, and
  all overtones higher as well (hence broken
  glasses at opera houses).
  
• This is all due to the shape of waveforms that
  coincide with the original frequency.

12
The Overtone Series

• The initial pitches on the Naturally Occurring
  overtone series suggest broader, more consonant
  or perfect intervals.
  
• Further up, things get more imperfect more
  dissonant.
  
• The notes in parenthesis are not in tune with
  modern scales, possibly excepting the 6th
  overtone (the blue note).

13
Mean Tuning (Pythagoras)7-note scale

• 2/1 - the octave
• 3/2 - the perfect fifth
• 4/3 - the perfect fourth (the harmonic inverse of
  3/2)
  
• 5/4 - the major third
• 6/5 - the minor third
• 5/3 - the major sixth (the harmonic inverse of
  6/5)
  
• 8/5 - the minor sixth (the harmonic inverse of
  5/4)

14
Equal Temperament (Vincenzo Galilei, J.S. Bach,
the jerk)

• To play in all keys, an instrument needs to be
  chromatic (be able to play 12 notes per octave).
  This is problematic because the notes do not tune
  correctly to themselves.
• 100hz up 7 octaves 100100(2/1)7 12,900hz
• 100hz up 12 5ths 100100(3/2)12 13075hz
• 175hz (13075-12900) 12900 1.014 the jerk
  (the 5ths are about 25 cents sharp).

15
Equal Temperament (cont.)

• Equal Temperament solves this by making each half
  step equal to the 12th root of 2 larger.
  
• 100hz up 7 half steps (perfect 5th)
  1001002(1/12)7 249.8hz
  
• 100hz up a 5th 100(3/2)1 250.0hz
• Mean Tuning and Equal Temperament are within a
  few tenths of a percentage for most important
  intervals.

16
Form

• Called the most important element of
  composition, form is the means in which the
  piece is constructed, structured, planned, and
  where meaning is placed.
• Modern Compositions are mostly based on very old
  techniques of classic forms (e.g. Mozart) where
  sections are repeated in a Rondo or Sonata (ABA
  or Verse Chorus Solo relationships)

17
Serialism
0 6 8 5 7 E 4 3 9 T 1 2 6 0 2 E 1 5 T 9 3 2 7 8
4 T 0 9 E 3 8 7 1 2 5 6 7 1 3 0 2 6 E T 4 3 8 9
5 E 1 T 0 4 9 8 2 3 6 7 1 7 9 6 8 0 5 4 T E 2 3
8 2 4 1 3 7 0 E 5 4 9 T 9 3 5 2 4 8 1 0 6 5 T E
3 9 E 8 T 2 7 6 0 1 4 5 2 8 T 7 9 1 6 5 E 0 3 4
E 5 7 4 6 T 3 2 8 9 0 1 T 4 6 3 5 9 2 1 7 8 E 0

• Developed by Arnold Schönberg, Serialism is the
  practice of making a melody (series of notes)
  into a tone row where none of the pitches repeat
  until all are played.
• Using this technique, a matrix is constructed
  where 48 unique tone rows are created

18
Minimalism (Phase)

• Developed as a reaction against the impersonal
  nature of Serialism, many NY minimalists (Steve
  Reich, Phillip Glass, etc) formed a new school of
  thought.
• Using a minimum of materials over time that
  gradually changed, the form became designed based
  on the subtle differences. For example, playing
  a 10 minute recording just fast enough so that it
  would take 930 to finish, then playing it
  simultaneously with the original.
• This led to further developments in electronic
  music, such as phase distortion (flanges, warble
  effects), digital simulated reverb, and
  artificial acoustic modeling (e.g. Bose).

19
Purpose of Music

• Music is defined as the meaningful organization
  of sounds.
  
• Meaning is defined as a personal or expressive
  means in which the song was created
  (accessibility).
  
• Meaning is also defined as a greater intent that
  shows a deliberate, calculated, and evolving
  design (artistry).
  
• Mathematics fits well into both schools of
  thought, despite present conflicts between the
  two ideologies.