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Music in Math PowerPoint PPT Presentation

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

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Music in Math

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Music in math l.jpg

Music in Math

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


Factors of composition performance l.jpg

Factors of Composition & Performance

  • Rhythm

    • Subdivision versus Addition

  • Harmony

    • Chord Construction

  • Intonation

    • Pythagoras and Vincenzo Galilei

  • Form

    • Serialism versus Minimalism


Rhythm l.jpg

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


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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.


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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.


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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


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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.


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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.


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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.


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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).


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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.


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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).


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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)


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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).


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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.


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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)


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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


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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).


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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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