2. Set 6 Let there be music 1 Wow! We covered 50 slides last time! And you didn't shoot me!!

3. Stuff 2 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.

4. Last time 3 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?? Why these PARTICULAR frequencies??

5. Helmholtz’s Results Note from Middle CFrequency C264 D297 E330 F352 G396 A440 B496 4

6. Our immediate plan 5 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

7. 6 Tone Compare the results From these two sources. We can study tones with electronics

8. Oscilloscope 7 http://commons.wikimedia.org/wiki/Main_Page

9. One More Tool 8 Tone Signal Generator Electrical

10. In using these modern tools 9 1. 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.

11. One more thing These days, the tone generation and the oscilloscope can be “created” on a computer. This will often be our approach. 10

12. The Violin 11 L We will make some measurements based On these lengths.

13. Let’s Listen to the Violin 12 1) Let’s listen to the instrument, this time a real one.  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?

14. Guitar Tuning 13 StringScientific pitch Helmholtz pitch Interval Interval from middle Cmiddle C Frequency firstE4E4 e' major third major third above 329.63 Hz secondB3B3 b minor second minor second below 246.94 Hz thirdG3G3 g perfect fourth perfect fourth below 196.00 Hz fourthD3D3 d minor seventh minor seventh below 146.83 Hz fifthA2A2 A minor tenth below 110 Hz sixthE2E2 E minor thirteenth below 82.41 Hz

15. Consider Two Situations 14 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.

16. So… 15 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. k doubles m -> m/2 f doubles!

17. Octave 16 f 2f SUM Time 

18. The keyboard – a reference 17 The Octave Next Octave

19. The Octave 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

20. Properties of the octave 19 Two tones, one octave apart, sound well when played together. the same note 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. Pythagoras This allowed Pythagoras to create and understand the musical scale. This will be our next topic.

21. Pythagoras 20

22. 21 The ratios of these lengths Should be ratios of integers If the two strings, when struck At the same time, should sound “good” together.

23. Pythagoras 22 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.

24. Pythagoras Each number had its own personality - masculine 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. 23

25. 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

26. See you later …. 25