1. Guerino Mazzola (Spring 2016 © ): Performance Theory II STRUCTURE THEORY II.1 (Tu Feb 03) Tuning, Intonation, and Dynamics

2. Guerino Mazzola (Spring 2016 © ): Performance Theory Tuning & Intonation define the performance of pitch global background of pitch calibration/gauging local pitch deformation for expressive purposes

3. Guerino Mazzola (Spring 2016 © ): Performance Theory Intonation/tuning curves Intonation/tuning deals with the performance of pitch. There is a symbolic pitch and a physical pitch, into which the symbolic pitch is transformed. physical pitch symbolic pitch HHHH

4. Guerino Mazzola (Spring 2016 © ): Performance Theory air waves 343 m/s The overall picture of sound waves instrument (sound source) musician auditory cortex ear room acoustics

5. Guerino Mazzola (Spring 2016 © ): Performance Theory Physical pitch Classical description of a (physical) sound event: periodic wave w(e) of frequency f [Hz], envelope H, amplitude A [dB], onset e [sec], and duration d [sec] wave w shift and squeeze support envelope H + w. H =

6. Guerino Mazzola (Spring 2016 © ): Performance Theory Physical pitch f = sound wave frequency pitch(f) = 1200/log 10 (2). log 10 ( f ) + const. [Ct] (Cent) Weber-Fechner Frequency ratio between octaves is 2: Sound with 2f is one octave higher than sound with f. frequencypitchdifference of frequenciesdifference of pitch f ~log(f) f1200/log(2).log(2) = 1200 [Ct] 2f~log(2f) = log(2) + log(f) 2f 4f~log(4f) = log(4) + log(f) = 2.log(2) + log(f)1200/log(2).log(2) = 1200 [Ct] f2f4f8f same!

7. Guerino Mazzola (Spring 2016 © ): Performance Theory Physical pitch Why 1200/log(2)? One octave ~ 1200 [Ct] 12 semitone steps 1 2 3 4 5 6 7 8 9 10 11 12 12-tempered tuning100 [Ct] frequency ratio f‘/f for 100 Ct? 1200/log(2).log(f‘/f) = 100 log(f‘/f) = log(2)/12 = log ( 12 √2) = log (2 1/12 ) f‘/f = 2 1/12 = 1.05946... tritone: 1200/log(2).log(f‘/f) = 600 log(f‘/f) = log(2)/2 = log ( 2 √2) = log (2 1/2 ) f‘/f = 2 1/2 = 1.41421...

8. Guerino Mazzola (Spring 2016 © ): Performance Theory Physical pitch Just tuning and 12-tempered tuning: choose basic frequency f 0 Just tuning: frequencies f = f 0.2 o.3 q.5 t, o, q, t = integers...-3,-2,-1,0,1,2,3,... Just tuning: frequencies f = f 0.2 o.3 q.5 t, o, q, t = integers...-3,-2,-1,0,1,2,3,... 12-tempered tuning: frequencies f = f 0.2 o/12 o = integer...-3,-2,-1,0,1,2,3,... 12-tempered tuning: frequencies f = f 0.2 o/12 o = integer...-3,-2,-1,0,1,2,3,... 100 0 200 300 400 500 600 700 800 900 1000 1100 (Ct) 45/32 = 2 -5.3 2.5 1

9. Guerino Mazzola (Spring 2016 © ): Performance Theory symbolic onset time E in q — physical onset time e in sec. — EEEE symbolic pitch H in semitone steps — physical pitch h in Cent — HHHH

10. Guerino Mazzola (Spring 2016 © ): Performance Theory — symbolic pitch H in semitone steps (scale) physical pitch h in Ct — HHHH MIDI = MusicalInstrumentDigitalInterface MIDI pitch has 128 (symbolic!) semitone steps 0127 c‘ ~ 60

11. Guerino Mazzola (Spring 2016 © ): Performance Theory What is intonation? pitch [Ct] scale [St] slope(H) = dh/dH [Ct/St] H h = h(H) Intonation(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] notated by S(H) S for German „Stimmung“ Intonation(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] notated by S(H) S for German „Stimmung“ how high (semitone = St) in the score? Ct tuning fork physical pitch h as a function of symbolic pitch H h = h(H)

12. Guerino Mazzola (Spring 2016 © ): Performance Theory What is intonation? h [Ct] H [St] Intonation(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] notated by S(H) Intonation(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] notated by S(H) 69 ~ a‘ h(H) slope =100 [Ct/St] S(H) = const. = 1/100 [St/Ct] pitch(440 Hz) 12-tempered tuning S(H) H H1H1H1H1 H0H0H0H0 S(H) = constant = S S

13. Guerino Mazzola (Spring 2016 © ): Performance Theory What is intonation? Intonation(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] notated by S(H) Intonation(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] notated by S(H) just tuning pitch(440 Hz) h [Ct] H [St] cd-de-eff+ga-ab-bc

14. Guerino Mazzola (Spring 2016 © ): Performance Theory f = f 0.2 o.3 q.5 t log(2) log(3) log(5) frequency for middle c frequency for middle c o, q, t integers, i.e. numbers in Ÿ = {...-2,-1,0,1,2,... } pitch(f) ~ log(f) = log(f 0 ) + o.log(2) + q.log(3) +t.log(5) ~ o.log(2) + q.log(3) +t.log(5) ~ o.log(2) + q.log(3) +t.log(5) o, q, t are unique for each f because of prime number factorization!

15. Guerino Mazzola (Spring 2016 © ): Performance Theory log(2) log(3) log(5) Euler space

16. Guerino Mazzola (Spring 2016 © ): Performance Theory (o/12).log(2), o = integer (3/12).log(2) fractions also ok for independence of directions!

17. Guerino Mazzola (Spring 2016 © ): Performance Theory Except for this fundamental problem, tuning/intonation works formally like tempo. However, the parallel is interesting, since similar questions concerning the relation of tuning to intonation, ie. the local and the global context, arise as for tempo/agogics: Does every piece have a tuning? Does every piece have a tuning? Several tunings (typically a problem for ensembles with different tuning approaches, piano, violin, reeds). Several tunings (typically a problem for ensembles with different tuning approaches, piano, violin, reeds). Tuning/intonation hierarchies (are tunings independent? is there intonation relative to other intonations), the example of the leader's tuning vs. the musician's one? For example, singers often have dramatically variable intonations (some go 1/2 tone from the given tuning in particular melodic situations). Tuning/intonation hierarchies (are tunings independent? is there intonation relative to other intonations), the example of the leader's tuning vs. the musician's one? For example, singers often have dramatically variable intonations (some go 1/2 tone from the given tuning in particular melodic situations). One should also investigate the non-European music traditions, eg. the Egyptian Maccam music, where intonation may vary quite often and according to not yet understood principles of gestural expressivity. One should also investigate the non-European music traditions, eg. the Egyptian Maccam music, where intonation may vary quite often and according to not yet understood principles of gestural expressivity. Discuss musical situations of tuning curves, such as change of tonality!

18. Guerino Mazzola (Spring 2016 © ): Performance Theory Examples: A. Look at a Schumann piece (op.15/1) and Webern‘s op.27/II in 4 different tunings: Pythagorean (no 5 component) Pythagorean (no 5 component) meantone (following Pietro Aron) meantone (following Pietro Aron) well-tempered following Bach well-tempered following Bach slendro (Balinesian) slendro (Balinesian) 12-tempered 12-tempered B. Egyptian Maqam music: Transcription by James Holden of original improv. recording C. Microtonal music Alois Hába (1893-1973): 1/4-tone, 1/5-tone compositions for strings

19. Guerino Mazzola (Spring 2016 © ): Performance Theory Dynamics A = sound wave pressure variantion amplitude in wave direction (longitudinal wave) loudness(A) = 20. log 10 (A/A 0 ) + const. [dB] (Dezibel) A 0 = 2.10 -5 N/m 2 (threshold) air waves 343 m/s instrument (sound source) musician auditory cortex ear room acoustics Weber-Fechner

20. Guerino Mazzola (Spring 2016 © ): Performance Theory

21. symbolic onset time E in ♩ — physical onset time e in sec. — EEEE symbolic loudness L in MIDI velocity steps (Vl, 0-127) — physical loudness l in dB — LLLL

22. Guerino Mazzola (Spring 2016 © ): Performance Theory

23. Tempo: T(E) = 1/slope(E) = 1/(de/dE) [ ♩ /sec] e 1 - e 0 = E 0 ∫ E 1 slope(E) dE = E 0 ∫ E 1 1/T(E) dE [sec] Intonation: S(H) = 1/slope(H) = 1/(dh/dH) [St/Ct] h 1 - h 0 = H 0 ∫ H 1 slope(H) dH = H 0 ∫ H 1 1/S(H) dH [Ct] Dynamics: I(L) = 1/slope(L) = 1/(dl/dL) [Vl/dB] l 1 - l 0 = L 0 ∫ L 1 slope(L) dL = L 0 ∫ L 1 1/I(L) dL [dB]