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Consonance & Scales
Chris Darwin
Perception of Musical Sounds: 2007
Plomp & Levelt: pure tones
max dissonance at ~ CB/4
1 2 3 4 5 6 7 8 91 kHz +1000 + i*10 Hz
bandwidth at 1kHzis ~136 Hz
1 semitone up is1060 Hz
Beating between harmonics
Harmonic structure & scales
Consonant intervals determined by interactions between similar-frequency harmonics
To minimise dissonant harmonic interactions
For integer harmonics (fn = n*f1) consonant intervals will be low-integer ratios
octave = 2:1fifth = 3:2fourth = 4:3maj 3rd= 5:4
Plomp & Levelt: complex tones
model data
Tuning of scales
Two incompatible requirements:
• Make intervals maximally consonant
• Keep step sizes equal so as to allow transposition
Comparison of tuning systems
2n/12 3n/2m ratios ofsmallest integers
Pythagorean tuning
Pythagorean comma
Pro: only uses 2 and 3Con: 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
Just Intonation
Pro: Perfect intervalsCon: Not all tones same size - so can’t transpose
C
C#
http://www.physics.mcgill.ca/~guymoore/ph224/different_scales.html
Just Intonationhttp://www.justintonation.net/
Equal temperament
Make each semitone equal: 21/12 = 1.059...
Con: No perfect intervals apart from octave
Pro: Equally good/bad in every key - can transpose
Comparison of tuning systems
examples in different tunings (Frère Jacques)http://www.physics.mcgill.ca/~guymoore/ph224/different_scales.html
Stretched scales
Sethares, W. A. (1993). "Local consonance and the relationship between timbre and scale," J. Acoust. Soc. Am. 94, 1218-1228.
QuickTime™ and aCinepak decompressor
are needed to see this picture.
http://eceserv0.ece.wisc.edu/~sethares/consemi.html
Amadeus
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.
Renat Ek: spectrum & tuning
http://homepages.nyu.edu/~wws217/AMA/RelatingSpectrumAndTuningOfTheClassicalThaiRenatEk.pdf
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Coursework paper by Will Stromolo at NYU
Renat Ek: spectrum
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
equal amp
weak f3 f4
Renat Ek: dissonance curve
Renat Ek: dissonance vs tuning