2. Consonance & Scales Chris Darwin Perception of Musical Sounds: 2007

3. Plomp & Levelt: pure tones max dissonance at ~ CB/4 1 2 3 4 5 6 7 8 9 1 kHz + 1000 + i*10 Hz bandwidth at 1kHz is ~136 Hz 1 semitone up is 1060 Hz

4. Beating between harmonics

5. Harmonic structure & scales Consonant intervals determined by interactions between similar-frequency harmonics To minimise dissonant harmonic interactions For integer harmonics (f n = n*f 1 ) consonant intervals will be low-integer ratios octave = 2:1 fifth = 3:2 fourth = 4:3 maj 3rd= 5:4

6. Plomp & Levelt: complex tones model data

7. Tuning of scales Two incompatible requirements: Make intervals maximally consonant Keep step sizes equal so as to allow transposition

8. Comparison of tuning systems 2 n/12 3 n /2 m ratios of smallest integers

9. Pythagorean tuning Pythagorean comma Pro: only uses 2 and 3 Con: Only octave and fifth perfect. Wolf fifth to avoid comma, so can’t transpose C C# Frère Jac http://www.physics.mcgill.ca/~guymoore/ph224/different_scales.html

10. Just Intonation Pro: Perfect intervals Con: Not all tones same size - so can’t transpose C C# http://www.physics.mcgill.ca/~guymoore/ph224/different_scales.html

11. Just Intonation http://www.justintonation.net/

12. Equal temperament Make each semitone equal: 2 1/12 = 1.059... Con: No perfect intervals apart from octave Pro: Equally good/bad in every key - can transpose

13. Comparison of tuning systems examples in different tunings (Frère Jacques) http://www.physics.mcgill.ca/~guymoore/ph224/different_scales.html

14. Stretched scales Sethares, W. A. (1993). "Local consonance and the relationship between timbre and scale," J. Acoust. Soc. Am. 94, 1218-1228. http://eceserv0.ece.wisc.edu/~sethares/consemi.html Amadeus

15. Bill Sethares In the same way that instrumental sounds with harmonic partials (for instance, those caused by vibrating strings and air columns) are closely related to the scales of the West, so the scales of the gamelans are related to the spectrum, or tonal quality, of the instruments used in the gamelan. Similarly, the unusual scales of Thai classical music are related to the spectrum of the xylophone-like renat. Sethares, W. A. (1998). Tuning, Timbre, Spectrum, Scale Springer-Verlag.

16. Renat Ek: spectrum & tuning http://homepages.nyu.edu/~wws217/AMA/RelatingSpectrumAndTuningOf TheClassicalThaiRenatEk.pdf Coursework paper by Will Stromolo at NYU

17. Renat Ek: spectrum equal amp weak f3 f4

18. Renat Ek: dissonance curve

19. Renat Ek: dissonance vs tuning