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Music and Mind II The Sound of Music ”All that exists in the universe is vibrating matter, pulsing energy, rhythmic movement” —Kay Gardner, 1990:74 www.mind-study.org ♫

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Where we are in the course I.The Appeal of Music II.The Sound of Music III.The Hearing of Music IV. The Structure of Music V.The Making of Music VI.The Power of Music 2 www.mind-study.org

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Topics for today Vibrating molecules Waves Frequency, wave length Amplitude/volume Overtones/Harmonics The cycle of 5ths (& the cycle of 4ths) Effects of sound on physical objects 3

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Topics for today Vibrating molecules Waves Frequency, wave length Amplitude/volume Overtones/Harmonics The cycle of 5ths (& the cycle of 4ths) Effects of sound on physical objects 4

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Some physical properties of sound Vibrating air molecules Sound waves traveling through the air from molecule to molecule Properties of sound waves Velocity: 1130 feet per second Frequency of vibration Wave length Amplitude (volume) 5

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Some physical properties of sound Vibrating air molecules air is mostly nitrogen and oxygen diameter of molecule about 0.3 nm Sound waves traveling through the air from molecule to molecule The sound wave is not an actual thing It is an abstraction The reality is the vibrating molecules 6

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Frequency and wave length Frequency of vibration Measured in cycles per second: Hertz (Hz) Wave length Inversely proportional to frequency Since the velocity does not vary So, high frequency – short wave length Perceived as “high pitch” low frequency – long wave length 7

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Frequency and wave length (II) Higher frequency — shorter wave length Double the frequency — half the wave length perceived as the next higher octave Examples Middle C Frequency 261.63 Hz Wave length 131.87 cm (51.92 inches) A Frequency 440 Hz Wave length 78.41 (30.87 inches) Low bass notes from Mormon Tabernacle organ have waves more than 60 feet long 8

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Amplitude/Volume Range from loudest to softest music: 1 million to one Measured in decibels – Zero decibels is the faintest sound a human ear can hear – The decibel (dB) scale is logarithmic Ten decibels is 10 times louder than 0 decibels So 20 decibels is 100 times as loud as 0 decibels – 30 decibels is one thousand times as loud – Some examples Whisper 30 dB Conversation 60 dB Jet takeoff120 dB 9

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dB 10

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Topics for today Vibrating molecules Waves Frequency, wave length Amplitude/volume Pythagorean ratios Overtones/Harmonics The cycle of 5ths (& the cycle of 4ths) Musical scales Effects of sound on physical objects 11

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Ratios of vibrating strings — Examples (Early discovery attributed to Pythagoras) Example Frequency 1C 1/2C’ x2 1/4C” x4 1/3G’ x3 2/3G x3/2 3/4 F x4/3 4/5E x5/4 1/5E” x5 1/6G” x6 1/7B ♭ ” x7 12

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More on vibrating strings Wave length and frequency depend on 1)Length of string For each next fret on a guitar, ½ tone higher pitch Q: Why are the frets not spaced equally? Each next fret makes the string 5.95% shorter 2)Weight of string 3)Tightness of string 13

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The circle of fifths 14

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The circle of fifths (and fourths) vis-à-vis Pythagorean ratios 15 C F G B ♭ D E ♭ A A ♭ E D ♭ B G ♭ /F ♯ x2/3 x3/4 x2/3 X3/4 (i.e., 4/9) (8/27) (32/243) (64/729) (16/81) (i.e., 9/16) (27/64) (81/256) (243/1024) (729/4096)

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The circle of fifths (and fourths) vis-à-vis frequencies 16 C F G B ♭ D E ♭ A A ♭ E D ♭ B G ♭ /F ♯ x3/2 x4/3 x3/2 x4/3 x4x3 x4/3 x3/3 x4/3 (i.e., 9/4) (27/8) (243/32) (729/64) (81/16) (i.e., 16/9) (64/27) (256/81) (1024/243) (4096/729)

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The circle of fifths 17

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The circle of fifths (and fourths) vis-à-vis frequencies F ♯ and G ♭ : same note (?) – Has to be the same for the circle of fifths to work The frequency of F ♯ is 729/64 x the frequency of C – 729/64 = 11.391 – Divide by 8 for same note three octaves lower: 1.424 The wave-length of G ♭ is 729/4096 x the wave-length of C – 4096/729 = 5.619 – Divide by 4 for same note two octaves lower: 1.405 So frequency of F ♯ is 1.424 x frequency of C And frequency of G ♭ is 1.405 x frequency of C Not the same — but pretty close 18

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The circle of fifths (and fourths) vis-à-vis frequencies F ♯ and G ♭ : NOT exactly the same note – But if they aren’t the same the circle of 5ths won’t work Frequency of F ♯ is 1.424 x frequency of C Frequency of G ♭ is 1.405 x frequency of C Not the same — but Close enough for the circle of fifths to work If we make a little adjustment – Compromise between 1.424 and 1.405 1.415 We spread out the adjustment over the six steps along the circle from C to F#/G ♭ The discrepancy from “natural” is too slight to be noticed by most people 19

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Adjusting the frequencies The frequency of A – 440 Hz Hence, the frequency of “natural” E is 660 Hz – 3/2 x 440 With minor adjustment to make a well-tempered scale, – The frequency of E is 659.255 Less than 1 cycle per second of difference Too little to bother most people – The frequency of D#/E ♭ is 622.25 – 622.45 ÷ 440 = 1.4142 20

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J.S. Bach and the well-tempered clavier The Well-Tempered Clavier, by J. S. Bach, is one of the world's great intellectual treasures. Each of its two volumes contains a prelude and fugue in every major and minor key of the chromatic scale. Book I, which was completed in 1722, was the first cycle of compositions in this conception. Book I begins with a prelude in C Major, followed by a fugue in the same key. These are followed by a prelude and fugue in C minor, C#/Db major/minor, D major/minor, etc. Each pair moves up the chromatic scale until every key has been represented. In Book II, which was completed in 1744, Bach effects another complete transversal of the chromatic cycle. One of Bach's primary purposes in composing these cycles was to demonstrate the feasibility of the "well tempered" tuning system that would allow for composition in every key. http://www2.nau.edu/tas3/wtc.html 21

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J.S. Bach and the well-tempered clavier (das Wohltemperierte Klavier) A monument in the history of Western music, The Well- Tempered Clavier represents not only the culmination of J. S. Bach's own maturation process but also the impetus for the emerging style and structure of modern keyboard music. Mozart, Beethoven, and Chopin were influenced by its polyphonic richness and depth of harmony, and Schumann counseled young musicians to "make The Well-Tempered Clavier your daily bread.” (Amazon blurb on a 2013 edition) 22

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The modern piano(forte): 88 keys Lowest note: very low A — 27.5 Hz Next higher A — 55 Hz Next higher A — 110 Hz C below middle C — 130.813 Hz A below middle C— 220 Hz Middle C — 261.626 Hz A above middle C — 440 Hz C above middle C — 523.251 Hz Highest note (88 th key): very high C — 4186.01 Hz 23

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Topics for today Vibration Overtones/Harmonics Effects of sound on physical objects 24

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Cymatics https://www.youtube.com/watch?v=9R4Bkwh9h9c 0 – 4:30 8:50 -12:20 Monks chanting Om —Steven Halpern https://www.youtube.com/watch?v=alT1KfE8_sk 13:20 – 16:20 25

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26 T h a t ‘ s i t f o r t o d a y !

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