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### Music in Math

(or… what many people probably don’t think about when they think of math in music)

Factors of Composition & Performance

- Rhythm
- Subdivision versus Addition

- Harmony
- Chord Construction

- Intonation
- Pythagoras and Vincenzo Galilei

- Form
- Serialism versus Minimalism

Rhythm

- Western Music is always based on 2 numbers: 2 and 3
- All Rhythmical denominations in Western Music are based on 2^x (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

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.

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.

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

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.

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.

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.

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).

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.

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).

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)

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 = 100+100*(2/1)^7 = 12,900hz
- 100hz up 12 5ths = 100+100*(3/2)^12 ≈ 13075hz
- 175hz (13075-12900) ≈ 12900 * 1.014 ≈ the jerk (the 5ths are about 25 cents sharp).

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) = 100+100*2^(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.

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)

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

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 9:30 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).

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.

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