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Transcript

Set 6 Let there be music Wow! We covered 50 slides last time! And you didn't shoot me!! 1 Stuff As of Saturday morning, the grades have yet to be posted on myUCF site. Today we continue with our musical interlude. We still have to cover two basic Physics concepts: Energy Momentum We will return to these topics later. 2 Last time We looked at strings, how they vibrate and mentioned the factors that determine the vibrational frequency of a string. We also remembered the Helmholtz result (next slide) that each note on the musical scale had a specific frequency. But were those specific frequencies selected? Why not different ones?? Why these PARTICULAR frequencies?? 3 Helmholtz’s Results Note from Middle C C D E Frequency 264 297 330 F G A B 352 396 440 496 4 Our immediate plan Apply modern electronic methods, with the assistance of a guest violinist, to answer these questions. Apply these methods to the understanding of 1) The scale progression 2) The development of chords 5 We can study tones with electronics Tone Compare the results From these two sources. 6 Oscilloscope http://commons.wikimedia.org/wiki/Main_Page 7 One More Tool Tone Signal Generator Electrical 8 In using these modern tools We postpone understanding how some of these tools work until later in the semester. 2. We must develop some kind of strategy to convince us that this approach is appropriate. 1. 9 One more thing • These days, the tone generation and the oscilloscope can be “created” on a computer. • This will often be our approach. 10 The Violin L We will make some measurements based On these lengths. 11 Let’s Listen to the Violin 1) Let’s listen to the instrument, this time a real one. 12 The parts One tone alone .. E on A string E on the E string Both together (the same?) A Fifth A+E open strings Consecutive pairs of fifths – open strings. A second? Third? Fourth? Seventh? Guitar Tuning 13 Interval from middle Frequency C major third 329.63 Hz above String Helmholtz Scientific pitch pitch first E4 e' second B3 b minor second below third G3 g perfect fourth 196.00 Hz below fourth D3 d minor seventh 146.83 Hz below fifth A2 A sixth E2 E minor tenth below minor thirteenth below 246.94 Hz 110 Hz 82.41 Hz Consider Two Situations For the same “x” the restoring force is double because the angle is double. The “mass” is about half because we only have half of the string vibrating. 14 For the same “x” the restoring force is double because the angle is double. So… F kx 1 f 2 k m The “mass” is about half because we only have half of the string vibrating. k doubles m -> m/2 f doubles! 15 1 f 2 1 2 2f k 1 m 2 4k 1 m 2 2k m/2 k 4 m Octave 1 0.5 f 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005 -0.5 -1 1 0.5 2f -0.5 -1 1.5 1 0.5 SUM -0.5 -1 -1.5 Time 16 The keyboard – a reference The Octave 17 Next Octave The Octave 12 tones per octave. Why 12? … soon. Played sequentially, one hears the “chromatic” scale. Each tone is separated by a “semitione” Also “half tone” or “half step”. Whole Tone = 2 semitones 18 Properties of the octave Two tones, one octave apart, sound well when played together. In fact, they almost sound like the same note! A tone one octave higher than another tone, has double its frequency. Other combinations of tones that sound well have frequency ratios that are ratios of whole numbers (integers). It was believed olden times, that this last property makes music “perfect” and was therefore a gift from the gods, not to be screwed with. This allowed Pythagoras to create and understand the musical scale. This will be our next topic. 19 Pythagoras 20 The ratios of these lengths Should be ratios of integers If the two strings, when struck At the same time, should sound “good” together. 21 Pythagoras Born in Samos, Ionia Remembered as a mathematician. Well educated; learned to play the lyre, read poetry, and could recite Homer. Believer that ALL relations could be reduced to number. All things are numbers; the whole cosmos is a scale and a number. He developed the Pythagorean Theorm. 22 Pythagoras Each number had its own personality - masculine 23 or feminine, perfect or incomplete, beautiful or ugly. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. He was a fine musician, and he used music as a means to help those who were ill. Pythagoras The beliefs that Pythagoras held were [2]: (1) that at its deepest level, reality is mathematical in nature, (2) that philosophy can be used for spiritual purification, (3) that the soul can rise to union with the divine, (4) that certain symbols have a mystical significance, and (5) that all brothers of the order should observe strict loyalty and secrecy. So it is no surprise that he looked at the lengths of strings that sounded well together as a religious issue as well as a scientific issue. Luckily, in this case, it worked .. sort of. 24 See you later …. 25