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

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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

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Coursework paper by Will Stromolo at NYU

Renat Ek: spectrum

QuickTime™ and aTIFF (Uncompressed) decompressor

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equal amp

weak f3 f4

Renat Ek: dissonance curve

Renat Ek: dissonance vs tuning