Computerized real time tuning Ladislav Šipeky Ladislav Šipeky31.1.2012 FSTA1/18

Computerized real time tuning

Ladislav Šipeky

Ladislav Šipeky 31.1.2012 FSTA 1/18

Mathematics-Music Tuning Fuzzy logic in music Dissonance curve Algorithms

Future work

Computerized real time tuning

Ladislav Šipeky 31.1.2012 FSTA 2/18

Symbolic relationship

h + Q = h.

?4

1

2

1

Computerized real time tuningMathematics - music

Ladislav Šipeky 31.1.2012 FSTA 3/18

Tuning- terminology

Tuning: Tuning is an exact definition of the pitch of one or more tones.

Octave: Given two sounds with frequencies f1 and f2, we say that f2 is one octave higher than f1 if f2 is twice f1.

Interval:2

1

f

f

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 4/18

Pythagorean tuning

2/1 - Octave / 12 semitones

3/2 - Fifth (perfect) /7 semitones

4/3 - Fourth (perfect) /5 semitones

2p3q

p, q ϵ Z

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 5/18

Pythagorean comma

FG = 1024 / 729

GI = 729 / 512

524288 / 531441

712

22

3

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 6/18

Just intonation

major third / 4 semitones

2p3q5r

p, q, r ϵ Z

4

5

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 7/18

Equal temperament

Cents

12 2

2

1221 log1200,

f

fffd

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 8/18

Pytagorean tuning [cent]

Just intonation[cent]

12-tones equal temperamen[cent]

C 0 0 0

D 203,91 203,91 200

E 407,82 386,13 400

F 498,04 498,04 500

G 701,96 701,96 700

A 905,86 905,86 900

H / B 1109,78 1088,26 1100

C 1200 1200 1200

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 9/18

BeatsAcoustical effect - interference of close frequences

21 fffbeats

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 10/18

INTERVAL DEVIATION

prime / unisono (-12, 12) major second (160, 230)major third (372, 430)perfect fourth (472, 530)perfect fifth (672, 730)major sixth (866, 930)major seventh (1066, 1136)

Garbuzov zones

Computerized real time tuningTuning

Ladislav Šipeky 31.1.2012 FSTA 11/18

Computerized real time tuningFuzzy logic in music

Fuzzy logic in music

0

2222

2221

2222

2

(x)

else

fxfff

xffxf

fxfff

fx

2ff2f 2f 2f

Ladislav Šipeky 31.1.2012 FSTA 12/18

Computerized real time tuningDissonance curve

Spectral analysis of real toneResult of Fourier transform

Threshold

Taken from: http://www.phys.unsw.edu.au/jw/sound.spectrum.html

Ladislav Šipeky 31.1.2012 FSTA 13/18

Dissonance curve

Taken from: http://eceserv0.ece.wisc.edu/~sethares/consemi.html

1212

212121 ),,,( ffbsffas eellllffd

Computerized real time tuningDissonance curve

Ladislav Šipeky 31.1.2012 FSTA 14/18

Dissonance curve of „real“ tones

Taken from: http://eceserv0.ece.wisc.edu/~sethares/consemi.html

Computerized real time tuningDissonance curve

Ladislav Šipeky 31.1.2012 FSTA 15/18

Real time tuning algorithms

Hermode Tuning system (Werner Mohrlok)

Adaptive tuning (William A. Sethares)

Computerized real time tuningAlgorithms

Ladislav Šipeky 31.1.2012 FSTA 16/18

Usage Interpretation Evaluation of musicians ...

Future work “More” correct formula of dissonance curves Implementation of musical analysis to existing algorithms Else

Computerized real time tuningFuture work

Ladislav Šipeky 31.1.2012 FSTA 17/18

Thank you for your attention

Mgr. art., Mgr. Ladislav Šipeky

ladislav.sipeky@stuba.sk

Department of Mathematics

Faculty of Civil Engineering

Slovak University of Technology Bratislava

Radlinského 11

813 68 Bratislava

Slovakia

http://www.math.sk

Ps. For lunch will be: borshch and something for vegetarians

Computerized real time tuning

Ladislav Šipeky 31.1.2012 FSTA 18/18