Mathematical Explanations of Consonance

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En provenance du cours de Duke University

Music as Biology: What We Like to Hear and Why

390 notes

Duke University

Music as Biology: What We Like to Hear and Why

390 notes

The course will explore the tone combinations that humans consider consonant or dissonant, the scales we use, and the emotions music elicits, all of which provide a rich set of data for exploring music and auditory aesthetics in a biological framework. Analyses of speech and musical databases are consistent with the idea that the chromatic scale (the set of tones used by humans to create music), consonance and dissonance, worldwide preferences for a few dozen scales from the billions that are possible, and the emotions elicited by music in different cultures all stem from the relative similarity of musical tonalities and the characteristics of voiced (tonal) speech. Like the phenomenology of visual perception, these aspects of auditory perception appear to have arisen from the need to contend with sensory stimuli that are inherently unable to specify their physical sources, leading to the evolution of a common strategy to deal with this fundamental challenge.

À partir de la leçon

Defining Music and Exploring Why We Like It

The tonal phenomena that need to be explained in any theory of music, and different approaches that have been take to provide answers.

Defining Music11:37

The Chromatic Scale and Some More Musical Terminology6:49

Consonance and Dissonance6:35

Tonality vs. Atonality: Frequency of Consonant vs. Dissonant Intervals4:27

Tension and Resolution5:08

Mathematical Explanations of Consonance5:01

A Physical Explanation of Consonance7:32

A Biological Explanation of Consonance16:01

Rencontrer les enseignants

Dale Purves
M.D.
Duke Institute for Brain Sciences

So in the remaining lessons in this module,

we're gonna be talking about the explanation of consonance.

What is it that makes tone combinations either pleasing,

consonant, or relatively displeasing?

And I emphasize relatively, because they're all relatively pleasing,

it's just a matter of degree.

So we're gonna be considering three explanations.

A mathematical explanation, a physical explanation, and a biological explanation.

So in this lesson let's begin and discuss mathematical explanation of consonance.

This goes back to Pythagoras.

And you'll remember that Pythagoras

was the ancient Greek philosopher whose major contributions I

think people would say were in mathematics and geometry.

But he was very interested in music as well,

as was his school of Pythagorean philosophy.

And he had a very simple idea of what made music consonant.

And his idea was that this was dependent on the ratios of strings,

the lengths of strings.

And he was half right and half wrong.

His framework for thinking about this was in terms of a philosophy that,

as those of you who know anything about ancient Greek philosophy will be aware,

had a very spiritual aesthetic context to it.

And he thought that the ratios that were musically

significant were these ratios of unison of the octave

of the perfect fifth and the perfect fourth.

And he limited his concept to these ratios because,

as we'll see and as all modern musicians know,

it gets very complicated after these simple whole number ratios.

Things get out of hand pretty quickly.

So Pythagoras was interested in the philosophy and

the spirituality of these ratios, and

didn't want to go beyond what he thought were these simple perfect ratios.

That he limited his concept of music too in fact he forbid ratios that

quickly in the post-Grecian era and he,

even in the era of ancient Greece, were being played all the time.

So this is a lyre.

The instrument of ancient Greece, or one of the instruments of ancient Greece.

And you can see that is has seven strings.

I mean some lyres had four strings, some had ten, this one has seven, but

you can appreciate that these Pythagorean

ratios were not what the popular music of the day was limited to.

They were playing music much as we

hear it today in the folk music and popular music of the time.

But Pythagoras has been sort of the dominant historical

figure in thinking about music in ancient Greece because of his philosophical and

spiritual concept of what musical tones were all about.

So the bottom line of Pythagorean ratios

was that they were based on string lengths.

And Pythagoras was aware that strings that were of the same length were

played together sounded good on two lyres for example.

That string lengths that were in the ratio of two to one,

an octave, sounded consonant, attractive, pleasing.

That ratios of three to two, the perfect fifth, were also pleasing.

And the perfect fourth, four to three, were pleasing as well.

So he limited His concept of music and

it's held over in Pythagorean tuning to these four ratios

that were expressed as the so-called tetrachord in ancient Greek music.

But be aware that the popular music of the day wasn't limited

to Pythagoras' rather

austere concept of what music could and should be based on mathematics.

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