Temperament History, Physics, and Psychophysics of Harmony Martin - - PowerPoint PPT Presentation

Temperament

History, Physics, and Psychophysics of Harmony Martin Caspe – Phys 536 – March 12, 2019

Contents

• Introduction – Who am I, and why did I choose this topic?
• History of Tunings in the West
• Pythagoreans – Middle Ages – Baroque Transition – Even Temperament as Standard
• Physics
• Characteristics of Tone – Beats and Roughness – Pure Tones versus Real Music
• Psychophysics / Psychoacoustics
• The Magic Octave – Other Intervals – Consonance and Dissonance – Building Scales
• Conclusion

History of Tunings – Pythagoras!

Pythagoras discovers: Octaves (f2=2 f1) Perfect fifths, (2f2=3 f1) Perfect fourths (3f2=4 f1) Apocryphal tale of anvils Invention and experimentation with monochord

Circle of Fifths and Pythagorean Comma But nobody cared for centuries!

History of Tunings – Pythagoras’ Legacy

Boethius – 500 CE Music of the Spheres

Organum – 9th – 13th Centuries Plainchant with harmonies, mostly fifths and fourths

Harpsichord – 14th – 16th Centuries Dedicated string for each key

Physics – Comparison of Pure Tones: Pythagorian to Even Tempered

Experimental Setup:

• Dual output signal generator
• CRATE CR-280 amplifier
• Oscilloscope
• Ibanez acoustic/electric guitar

Physics – Comparison of Pure Tones: Pythagorean to Even Tempered Fifths

• Pythagorean tuning has perfect fifths (and octaves, and fourths)

as expected

• Beats are evident in the even tempered fifths intervals
• Comparing only two notes (one interval) of pure tones it is

difficult to hear the dissonance in even tuning, although it is there. ➔ Why are such subtle fluctuations such a big deal?

Physics – Comparison of Pure Tones: Pythagorean to Even Tempered Fifths

Physics – Pure Sines Versus Vibrating String

Physics – Pure Sines Versus Vibrating String

Psychophysics – Perception of Phenomena

Psychophysics: “a branch of psychology concerned with the effect of physical processes (such as intensity of stimulation) on the mental processes of an organism” Psychoacoustics: “a branch of science dealing with the perception of sound, the sensations produced by sounds, and the problems of communication”

2019 Merriam-Webster: https://www.merriam-webster.com/

Psychophysics – Characteristics of Tone

As discussed in class: Pitch Frequency Loudness Amplitude Timbre Complexity (quality, overtones, attack decay) Add: Consonance Special frequency intervals – Octave, Fifth, Fourth Dissonance Frequencies interfere – Beats, Roughness, Wolftones Harmonies and chords are build from consonances, and avoid dissonances

Psychophysics – Consonance to Dissonance

Frequency Ratio Interval “Perfect” consonances 1/1 Unison 2/1 Octave 3/2 Fifth 4/3 Fourth “Imperfect” consonances 5/3 Major sixth 5/4 Major third 6/5 Minor third 8/5 Minor sixth

Psychophysics – The Magical Octave

Tonic &

• vertones

(ascending pitch) Octave &

• vertones

f1 2*f1 f2 = 2*f1 3*f1 4*f1 2*(2*f1) = 4*f1 5*f1 6*f1 3*(2*f1) = 6*f1

• In real instruments, harmonic

“overtones” exist

• Consider two strings vibrating on

the interval of the fifth: f2 = 2 f1

• Each string is also producing
• vertone harmonics in whole-

number multiples of it’s dominant frequency

• All overtones overlap – very pure,

ringing tone

• All cultures have Octave concept,

but may divide into 5, 17, or 22 parts, not 12

Psychophysics – Harmonic Interference

Tonic &

• vertones

(ascending pitch) Perfect fifth &

• vertones

f1 f2 = 3/2 f1 2*f1 3*f1 2*(3/2 f1) = 3*f1 4*f1

Very close – may beat

3*(3/2 f1) = 9/2*f1 5*f1 6*f1 4*(3/2 f1) = 6*f1

• In real instruments, harmonic

“overtones” exist

• Consider two strings vibrating on

the interval of the fifth: f2 = 3/2 f1

• Each string is also producing
• vertone harmonics in whole-

number multiples of it’s dominant frequency

• Even tuned in perfect fifths,

beats may occur in the upper harmonics

Psychophysics – Is the Major Third Dissonant?

Beat spans ~3.5 scope divisions of 4.0ms ➔ ~14ms = ~70 Hz

Conclusions

• The perception of music and how intervals of notes sounds is a

complex interaction between the physical phenomena and the psychophysical way we interpret them

• The behaviors of pure sine wave tones gives clues to

understanding how we perceive real music, but it is too simplistic

• This presentation just scratched the surface of a very large field
• f study

Bibliography

Roederer, J., “Introduction to the Physics and Psychophysics of Music”, 2nd. Ed., Springer-Verlag, 1975 Isacoff, S., “T emperament”, Vintage Books Division of Random House, 2001 Johnston, I., “Measured T

• nes”, Adam Hilger impring of IOP Publishing Ltd, 1989

Hamilton, C., “Sound and its Relation to Music”, Oliver Ditson Co, 1912 Wood, A., “The Physical Basis of Music”, Cambridge University Press, Reprinted 1925