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.
Defining Scales and Modes
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Music as Biology: What We Like to Hear and Why
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Musical Scales
Why a small number of basic scales are used in music worldwide, and how a biological framework explains this and related puzzles.
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- Dale PurvesM.D.
Duke Institute for Brain Sciences
Welcome back to the fifth of these modules.
And today we're gonna be talking about scales.
And you remember in sort of outlining the menu of problems,
there are a bunch of issues that pertain to scales.
Why is music organized at scales?
What's the chromatic scale?
Why are the number of scales that we use relatively small, and on and on.
There are a whole series of questions that have to do with the nature of scales.
And let's begin by trying to define scales, and
another word that's sometimes used in, what I find,
a rather difficult to understand way in relation to scales which is modes.
So let's go back and look at a slide I showed you before, the chromatic scale.
Which is, in a very real sense, the superset from which the popular
scales that we use all the time not only in Western music, but
in a variety of musical cultures are drawn.
So let's just review what I told you before, because it's complicated.
So you remember that in the chromatic scale, there are 13 notes,
12 intervals, that each of these intervals,
notes, has a name, unison for two notes played together.
And the notes are always in relation to the toning note, we'll come back and
say more about that in a bit.
Unison minor, second major, second major, third and so on down the list of names
that you see here, with these abbreviations that are commonly used and
that are indicated here on the piano keyboard.
So this is a scale that's an easy way to
think about it as the piano keyboard, and remember that the piano
keyboard is not something that defines the scale, it's the other way around.
Piano keyboard is an empirical organization of notes to facilitate
playing that has come from the organization that we like to hear,
as you'll remember when I talked about consonance.
How these intervals are exactly determined, these 12 intervals of
the chromatic scale superset, are exactly determined, refers to tuning systems.
And there are many tuning systems over musical history, but
there are three that have been much debated,
much commented on, and used in a variety of contexts.
These three are the Pythagorean tuning system, the just intonation system,
and equal temperament.
And again, these are complicated issues, but let me see if I can make them
as simple, at least in the simplistic way I tend to think about them.
So Pythagorean tuning systems, as you'll recall from what we said before,
is a tuning system that is based on the lengths of strings that
Pythagoras determined to be consonant according to their ratios.
So if two strings are the same length, they are played together and
they make a pleasing tone combination.
When their ratios
are in the relationship of 2:1, an octave, they're also consonance.
That's the second most consonant combination ratio.
The fifth, the ratio of the string lengths 3:2 is also pleasing, and
finally the ratio of the perfect fourth.
The 4:3 ratio string length is also pleasing, and Pythagoras,
based on his philosophy and his mathematical inclination, limited his
concept of the scales and tuning to those ratios.
That doesn't mean that in ancient Greece all the music was played
in Pythagorean tuning.
Far from it.
The popular music in ancient Greece was all over the place and
used tuning systems that are much more in use today.
So the Pythagorean tuning system is really sort of a touchstone in music theory
and philosophy, but has no great practical significance, and didn't even in the time
that it was being enunciated by the Pythagorean school in ancient Greece.
Just intonation is also a ratiometric tuning system that is indicated here.
The ratio as being between the fundamental notes of the two tones.
You remember the fundamental note of a tone, each tone is a harmonic series.
The initial fundamental, called usually F0,
which again is a confusing issue, that's the fundamental frequency of that series.
And the ratios that you see here are between the fundamentals
of the two tones in question.
So again, unison tones are in the relationship of 1:1.
For the octave, they're in the relationship of 2:1, for
a perfect fifth 3:2, for a perfect fourth, I mentioned in Pythagorean tuning,
4:3, and so on for the ratios that you see here.
Both Pythagorean tuning and just intonation tuning have a problem.
And that problem is that when you try to divide, parse,
the octave into components that are in ratios of string lengths for
Pythagoras, fundamental frequencies, or
just intonation tuning, which was widely used for
many centuries and continues to be used today in some special circumstances,
you can't fit the intervals neatly into an octave.
You can fit them into a single octave, but they don't form equal parts.
And then, as a result of that inequality,
when you try to extend to a second or third or fourth octave,
and you remember that the piano has about seven octaves,
the guitar about three or four,
then you get into trouble because you get mismatches
between the compilation of intervals going across octaves.
That's just a reality that's evident, and
has always been evident in Pythagorean tuning or just intonation tuning.
Beginning in the Renaissance, this problem and it's a fundamental
problem in tuning musical instruments and having them play together in ensembles.
If you wanna beyond music that extends over just one octave,
then you need to come up with a system that divides the octave
into equal components, not ratiometric components,
but equal components that then are the same over any octave.
So again, over the 7 octaves of the piano, equal temperament, which is this way of
dividing the octave into 12 exactly equal intervals solves the problem.
But the debate for centuries has been whether that compromise
takes away from the quality of what many people
consider sort of the brighter quality of just intonation tuning, and
that is a debate that's never really been settled.
In equal temperament, that's the methodology here in this column,
that equal temperament uses, and it simply divides the octave,
as I said, into 12 equal components, and each component is designated as 100 cents.
That's just for ease of comparison, ease of describing this in music theory,
and indeed, for thinking about instrumental tuning.
So there are 1,200 cents in an octave, each semitone, each unit,
and we'll come back to talk about the semitone a lot as we go on here.
Each unit is 100 cents.
So equal temperament again, since the Renaissance,
has increasingly been used because, of course, instrument makers
want to have their instruments accord with the ensemble playing.
That is, of course, everywhere in music, so they are economically and
mechanically interested in equal temperament.
And again, the compromise that exists when
you divide the octave into 12 equal parts compared to
its ratiometric division, that's taken by many as being a little bit brighter than
a compromise that's accepted and frankly not noticed by most of us.
So all of the music you hear today, virtually, and
the music that you download in midi format, it's all equal temperament.
And nobody really complains about this except
perhaps musical [FOREIGN] and we deal with it fine, but
it's still conceptually an over system, where does that come from?
It's just a practical compromise and
it's not clear why that should be.
So I refer to the chromatic scale as a superset.
And the reason for that is that the scales that are in popular use are much smaller.
They don't use all of the 12 intervals, 13 notes,
with a chromatic scale, as we'll come to later.
That's actually not really a scale in
the sense in which we're going to be discussing it.
It's an organizational set, and the scales that are used
are generally five interval, six note scales, or seven interval,
eight note scales, and we'll talk about that more as we go along.
But let me use as an example of a scale that you probably are all familiar with
certainly in Western music, but again as I say this is not limited to Western music.
This is the Major Scale,
just as an example of the popular scales that are commonly used in music.
This scale is the do re mi fa sol la ti do scale that you all first
heard about probably in kindergarten, and here it is on the piano keyboard.
And you musicians will recognize this as a C.
And let me emphasize that any scale
begins on a note, and you can begin a scale on any note on the piano.
But those scales, the notes based on their frequencies, are called A, B, C,
D, E, F, G, and
those alphabetical designations indicate specific frequencies.
So this happens to be C, you keyboard
players will recognize this as C on the piano, and if this were
the fourth octave in the middle range of the piano, this would be C4 and
it would have a particular frequency.
Middle C on the piano is 262 hertz, so A, B, C, D ,E,
F, G are designators that indicate actual frequencies.
And you need that because ensemble playing of course
means that you have to tune to the same frequencies.
So the concert tuning a frequency as A, concert A, which is 440 hertz.
So again this is another complication, but it should be obvious why you need that.
But the major scale on the piano, as any scale, begins with a tonic.
We'll say more about the tonic.
And has, as in this scale,
seven intervals, eight notes going from the tonic to the major 2nd,
to the major 3rd, to the perfect 4th, perfect 5th,
major 6th, major 7th, and back to the octave.
So any scale defines the intervals
over an octave that are used by a composer,
by a player, to generate music in a way that he or
she wants to generate it, wants to play it, likes to hear it.
And all scales are different and we'll come to this momentarily.
All scales are different in that the intervals that they use are not the same.
These are not the 12 equal intervals of the chromatic scale,
the semitone intervals each being 100 cents.
These are intervals that vary, so these intervals are from here to here,
two semitones, two more semitones, but
here to here, from the major 3rd to the perfect 4th, is just one semitone.
From the perfect 4th to the perfect 5th, another whole tone,
another whole tone, another whole tone, and another semitone interval.
So depending on the arrangement of those intervals, whole tone,
semitone, they don't have to be just whole tones or semitones.
But the way you arrange the steps between those intervals has everything to do,
as we'll see, with the nature of that scale, the quality that it generates,
and this leads me to the complicated idea of modes.
So scales are referred to as modes when
the arrangement of the scale, and this is music history, it's not really logic.
But scales are referred to as modes when they are used, or
when the scale that's being used has a religious or
a ritualistic context of some sort, and we'll talk about this
more when we come to talk about the emotions that different scales generate.
But in general, modes, complicated term,
if you look it up on Wikipedia, you'll find sort of a difficult
music history explanation of what this word means and
how it's used and it's by no means simple, but in general modes,
refer to the context of music being religious or ritualistic in some way.
And the emphasis is not strictly on major and minor,
which we're going to talk about in the last of these modules.
So again, all this may seem hopelessly complex.
Well that's the nature of the beast, it is complex, the history is complex.
But this is what we have to talk about when we're talking about scales and
how they're used, why they're used, and
what the explanation is in mathematical terms,
in physical terms, or in biological terms.
So let's listen now to a demonstration of the major scale and
some of its embellishments.
Again, scales, even though they are simplistic or
relatively simplistic in the way that I've just described them,
these intervals are almost always embellished in some way.
So you can have intervals that are used in music that are specifically notated,
as in Persian music and some other musical traditions,
we'll talk about those in the last module, that use formally
notated intervals between the semitone interval and the equal temperament tuning.
But in Western music or
any form of music, these embellishments are obvious as glissandos,
vibrato, changes in the tonality that go between the so
called semitone intervals, not so called, the semitone.
They go between the semitone interval that give the music a more
specific characterization.
And probably the most obvious example of this is in blues music and
Western music, where the late B.B. King,
who just passed away, was a master at bending the strings of
the guitar to generate these shifts, these slides, really,
between the notes of the scale that give blues music, in part,
the particular characteristics that it has when a master like King plays it.
So let's hear some of this from Ruby.
>> Before the 1600s, the natural minor scale was referred to as the Aeolian mode.
Here's how it sounds sung and played.
[MUSIC]
Post 1600, the Aeolian and D'ionian modes rose in prominence.
The Lonian mode is also known as the major scale.
Here's how it sounds.
[MUSIC]
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