1. The Power of Music II Things we Do Understand about Music ”All that exists in the universe is vibrating matter, pulsing energy, rhythmic movement” —Kay Gardner, 1990:74 www.mind-study.org/power-of-music.html ♫ 5:18

2. Revised Outline of the Course I.Music—Easy to Understand yet Inexplicable II.Things we Do Understand about Music III.Making Music IV.Music and Brain V.Music and Body, Music beyond Humans VI.Music Therapy 2

3. Topics for today Vibrating molecules Sound Waves Frequency, wave length Amplitude/volume Overtones/Harmonics The cycle of 5ths How we Hear 3

4. Topics for today Vibrating molecules Sound Waves Frequency, wave length Amplitude/volume Overtones/Harmonics The cycle of 5ths How we Hear 4

5. Physical properties of sound Vibrating air molecules air is mostly nitrogen and oxygen diameter of (small) 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 5

6. Some physical properties of sound Properties of sound waves Velocity: 1130 feet per second Frequency of vibration Wave length Amplitude (volume) 6

7. Sound Waves Mickey Hart (Grateful Dead) Campbell 1992: 41 7

8. More physical properties of sound Noise vs. musical tone Frequency of vibration Cycles per second (Hertz) Audible sound begins at 16 Hz Upper limit for humans is 18,000 Hz (for some, up to 20,000) Each doubling is next higher octave 8

9. 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 9

10. 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 10

11. 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 11

12. dB 12

13. Topics for today Vibrating molecules Sound Waves Frequency, wave length Amplitude/volume Overtones/Harmonics The cycle of 5ths Musical scales How we Hear 13

14. Pythagoras: Ratios and Harmonics Pythagoras (Πυθαγόρας) – Greek mathematician, philosopher, mystic – ca. 570 – ca. 495 BC Properties of vibrating strings 14

15. Ratios of vibrating strings — Examples (Early discovery attributed to Pythagoras) Ratio 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 15

16. The Fifth The interval between a full string vibrating and 2/3 The fifth has a frequency of 3/2 that of the base note Very basic for harmony – With the Third: major triad C — E — G Examples:  C — G  G — D  D — A  A — E  E — B 16

17. The circle of fifths 17

18. Frequencies around the circle The frequency of A – 440 Hz The frequency of “natural” E (a 5th higher) – 3/2 x 440 Hz – i.e. 660 Hz Up one more 5th, to B – 3/2 x 660 = 990 Up one more 5 th, to F# (G ♭ – 3/2 x 990 = 1485 Once again, to D ♭ – 3/2 x 1485 = 2227.5 – Divide by 8, same note 3 octaves lower: 278.4375 18

19. Continuing.. We have D ♭ at 278.4375 A ♭ : 417.65625 E ♭ : 626.484375 B ♭ : 939.726562 F : 1409.58984 C : 2114.38477 G : 3171.57715 D : 4757.36572 A : 7136.04858 — divide by 16: 446.003036 But the A we started with was 440.000000 We have a discrepancy of 6 Hz (about 1.3%) – After 12 steps (i.e., about 0.11 % per step) – About ½ cycle per step 19

20. Another way of looking at it The complete tour of the circle of 5ths is 12 steps So 12 times we multiply by 3:2 And we can divide by 2 as often as needed to keep the number from getting too high So, 3 12 ÷ 2 18 = 1.01364326 For A, we have 440 x 1.01364326 = 446.003036 (Since it is 1.01364326 rather than 1.00000000, and adjustment is needed) Adjustment: 20

21. Adjusting the frequencies The frequency of A – 440 Hz The frequency of “natural” E (a 5 th higher) – 3/2 x 440 Hz – i.e. 660 Hz 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 21

22. 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 22

23. 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) 23

24. 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 A# — 466.16 C above middle C — 523.251 Hz Highest note (88 th key): very high C — 4186.01 Hz 24

25. 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 25

26. Topics for today Vibrating molecules Sound Waves Frequency, wave length Amplitude/volume Overtones/Harmonics The cycle of 5ths How we Hear 26

27. A couple of technical terms Acoustics Psychoacoustics 27

28. Hearing Hearing was the last sense to develop Human brains can handle patterns of sound far more complex than the brains of any other animal (Jourdain 1997:4U) Playing a waltz to a goldfish; … to a dog or cat; … 28 “Hearing, it seems, is the difficult sense—slow to evolve, … reliant on the most intricate and fragile mechanical structures in the body. In was forged through millions of years of natural selection as countless lineages perished from detecting a predator too late, finding no mate, or overlooking a meal hidden nearby. Jourdain 1997: 2

29. Perception of Sound “When a brain isn’t up to the job, nothing occurs…” (ibid. 5T) “… a good ear for music …” ? – Really it is a good mind for music 29 “…every molecule in a concert hall sums every vibration from every instrument into one frenzied dance. Think of what a remarkable device it would take to watch that dance and infer from it every one of the original vibrations. Yet that—and more— is precisely what an ear does.—Jourdain 1997: 7

30. The ear (I) (1) On the outside: the pinna (pl. pinnae) – Just a funnel (a pretty fancy one) – Pinnae are a late evolutionary development Pinnae enhance certain frequency ranges – High frequency components – They also resonate to amplify certain high frequencies These frequencies are also important for speech perception The real ear is inside the head (2, 3, 4, 5) (2) the ear canal, about 1 inch long, resonates to boost higher frequencies 30

31. The Ear (II) (3) the ear drum – Converts the pressure wave of air into mechanical motion – For faintest sound, eardrum moves the width of a hydrogen atom (4) the middle ear: has three little bones (ossicles) – Hammer – Anvil – Stirrup (5) the inner ear 31

32. The Ear (III) (4) The middle ear (with its 3 occicles) – Concentrates the vibrations to 1/16 th as much area at the entrance to the inner ear as at the ear drum – The ossicles enhance middle frequencies—important for speech – Fish have no middle ear – Amphibians, reptiles, and birds have a middle ear with only one ossicle – The 2 additional ossicles of mammals boost the range of audible frequencies (5) the inner ear – Filled with fluid 32 http://www.the-scientist.com//?articles.view/articleNo/43884/title/Hearing-Explained/ Video:

33. The inner ear Converts vibrations into neurological signals – Information in a form the brain can use The inner ear functions as the retina does for vision – The outer and middle ears are like a lens Filled with fluid Contains three narrow chambers 1 ½ inches long – They are coiled 3 ½ times for compactness – Hence the name cochlea, from the Latin word for snail The middle chamber contains the organ of Corti 33

34. The cochlea 34

35. The organ of Corti Contains groups of tiny hair cells From which tiny hairs project Each group sensitive to a particular frequency The cells are connected to neurons About 14,000 receptor cells Connected to about 32,000 nerve fibers The nerve fibers extend from the cochlea, as the auditory nerve, to the brain Jourdain 12Bf 35

36. Frequency and amplitude of things we commonly hear 36

37. Hearing and Vision: a Comparison The interface: converts input to neural signals – Vision: the retina – Hearing: the organ of Corti (in the cochlea) Number of neurons involved – The retina has 100 million receptor cells Optic nerve has over 1,000,000 fibers – The organ of corti has 14,000 receptor cells Auditory nerve has 32.000 fibers Localizing the stimuli – Fairly obvious for vision – There is no way for one ear to know where the vibrations are coming from 37

38. Evolution of hearing First primitive hearing developed in fish – But fish don’t have a true inner ear Amphibians have primitive inner ears – With one ossicle Reptiles are sensitive to a broader frequency range Birds, broader yet – Sensitivity up to 10,000 Hz – But birds don’t hear bird calls as well as humans The whole evolutionary development up to human hearing.. – About 500 million years – Over 100 million generations 38

39. Hearing in the very young Fetuses can hear starting at 18 weeks (Mannes 80) They listen to mother’s heartbeat, voices and other sounds Some people are playing music to their unborn children – So far, no hard evidence that it has an effect – But, the case of Mozart.. 39

40. Means of Localizing sound Turning the head – For example, till both ears hear the sound at the same time More often, “calculation” by the brain, based on.. – Different time of arrival at left and right ears – Differing intensity of sound received by left and right ear – Differences in how sound is reflected in the pinnae We can also estimate the distance of a sound’s origin – Higher frequencies are more easily lost as distance increases – Our brains learn over time to estimate on basis of experience Humans can detect difference in horizontal position of about 1 degree for sounds of around 1000 Hz 40

41. How Good is Hi Fi? A stereo in your living room vs. being in a concert hall In a concert hall, vibrations from all over the orchestra plus reverberations from all over the walls and ceiling In your living room, two speakers present vibrations – From two microphones in a concert hall – All the vibrations come from the two speakers – They then re-reverberate around the walls and ceiling of the room The result is muddled You don’t get much improvement from using multiple mikes at different locations in the concert hall Live in a concert hall you have only two “mikes” — ears – Both receiving vibrations from all over the concert hall 41

42. How Good is Hi Fi? A stereo in your living room vs. being in a concert hall 42 “If music’s spaciousness is important to you, there’s no alternative to a good concert hall.” —Robert Jourdain (1997:24)

43. Music without Hearing (?) Evelyn Glennie – deaf percussionist –Mannes 2011: 7ff, 223:5 43

44. 44 T h a t ‘ s i t f o r t o d a y !