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Music, Language, and Cognition Study Group
Session 2.2, 9/28: Overview of tonality and syntax
Jonah Katz, IJN
1
A fact about particular pieces of music we’ll focus on: they doesn’t use
every note of the chromatic.
• a piece is composed of only a subset of 12-note chromatic
• this subset, which we’ll call a collection, defines the key of the
piece (3rd line up)
• the same collection is used in every octave
Lightning review of pitch space, continued
There exists a hierarchy of pitches in most kinds of Western music.
The vast majority of all Western music uses a very small number of
collections (in the sense of ordered sets of pitch intervals):
• <2, 2, 1, 2, 2, 2, 1>, the major scale we just saw
• <2, 1, 2, 2, 1, 2, 2>, the natural minor or aeolian mode
• <2, 2, 3, 2, 3>, the major pentatonic scale
• <3, 2, 2, 3, 2>, the minor pentatonic scale
The ‘shapes’ of these collections may be rooted in principles of
perceptual contrast or in physical properties of interval relationships.
But if so, then there exist various musical systems in other cultures that
violate one or more of the relative constraints on tonality.
1.4
1.1
The pitch continuum
A further fact about the music we’ve discussed: when more than one
note is sounded simultaneously, it is not the case that any note is
equally likely to be sounded with any other note.
• simultaneously sounded notes almost always form part of a
tertian triad (4th line up in the diagram)
Pitch is physically on a continuum (bottom line).
1.2
The chromatic
Out of the continuum, every genre G of music chooses a finite set of
discrete pitches which may occur in G.
• Central to our topic: the Western chromatic (2nd line up)
• breaks every octave into 12 semi-tones
1.3
The triad
Tertian triad:
• defined as three notes x, y, and z, where y is two steps (called a
third) above x in the relevant collection and z is two steps
above y.
• given that the diatonic family has 7 members, there are 7 triads
in the family.
The family (key, scale, collection)
1
I don’t know how typologically widespread tertian triads are.
• but my guess is ‘not very’
• Indian Carnatic music, which uses a chromatic derived from
the 12-tone one, and even uses versions of the diatonic
collection, does not use tertian chords
o vi
o vii diminished (notated vii0)
Confusingly, both scale degree numbers (for diatonic collections) and
these chord numbers have an alternate set of labels. I list them here for
reference, but I’ll try to use Roman numerals in general.
o I
‘tonic’
o ii
‘supertonic’
o iii
‘mediant’
o IV
‘subdominant’
o V
‘dominant’
o vi
‘submediant’
0
o vii
‘leading tone’ (more the note than the chord)
Given the makeup of the diatonic family, the thirds picked out by
chords will always span either 3 or 4 semi-tones.
• 4 semitones is a major third
• 3 is a minor third
• when y is a major third above x, and z is a minor third above y,
the chord quality is major
• when y-x is a minor third, z-y a major third, the chord is minor
• when both intervals are minor thirds, the chord is diminished
• chords with two major thirds don’t occur in the diatonic
family; they’re called augmented chords
• diatonic family contains 3 major, 3 minor, and 1 diminished
triad
Even more confusingly, some of the names given above are sometimes
used to refer to classes of chords that share a syntactic function, rather
than specific chords. I apologize on behalf of Western music theory,
and promise I’ll be clear when we eventually discuss these things.
We can refer to chords by their absolute pitch values, using x, which is
called the root, to label the chord, e.g. C major, D-sharp minor.
So the triad is an important entity in studying how simultaneouslysounded notes are organized. The word harmony could be plausibly
defined at this level: ‘how simultaneously sounded notes and sequences
of such collections of simultaneously-sounded notes are organized’. In
other words, understanding chords and sequences of chords is central
in the study of harmony.
We can also refer to chords by their relative position within the
diatonic collection.
• we do this by using Roman numerals, starting with the tonic
chord (chord with the tonic note as its root) and counting up
• using uppercase for major, lowercase for minor
• So the major collection contains
o I
o ii
o iii
o IV
o V
It turns out that the triad level shown in the diagram above is also
important for the stability hierarchies we discussed last year.
• centuries of intuition and Krumhansl’s results suggest that
pitches which belong to the tonic triad are more stable w/r/t a
key than pitches which do not
There may be a category error here, however:
2
•
•
•
•
•
•
it’s certain that pitches belonging to some chord are more
stable w/r/t that chord
and the tonic triad is the most stable chord w/r/t a key
it’s not actually clear that it’s meaningful to talk about a note’s
stability w/r/t a key
in Krumhansl’s experiments, all of the key-defining contexts
either sounded a tonic chord prior to the ‘probe tone’ or built
up strong expectations for a tonic chord to occur in the
position of the probe tone
so those judgments could be seen as judgments of stability
w/r/t a tonic chord, rather than a key
this won’t make much practical difference, because we’ll rarely
talk about the stability of a pitch w/r/t a key
The fifth also plays an important role in harmony.
• It is the most frequent relationship between the roots of
adjacent chords
• It is the most frequent relationship between the tonics of
adjacent regions of music in different keys
Some facts about harmony
2.1
Transitional probabilities
Given the summary of tertian triads (henceforth chords) above, we
know that (most) notes that co-occur in a piece of tonal music will be
members of the same chord. A further fact: combinations of
simultaneous notes that constitute chords do not occur freely in
sequences, that is, some sequences of chords are much more likely
than other sequences.
But it is worth noting that these stability judgments depend more
strongly on which chord is occurring in the music than they do on
which key a piece is in.
• for instance, if a piano is playing a piece in C major…
• and is sounding a D minor chord (ii)…
• and a singer sings the pitch A at the same time…
• it will be perceived as a much better ‘fit’ to the context than if
the singer sings a G
• despite the fact that G is arguably more stable in the key of C
than A is
1.5
2
In other words, there are some constraints on linear sequences of
chords in a piece of tonal music. Shown below is a table of common
two-chord sequences (called bigrams) in major-key Bach chorales,
taken from Rohrmeier (2007).
The fifth
Conventional wisdom in music theory, and to some extent
Krumhansl’s results, suggest that, amongst the pitches in the tonic
triad, the fifth scale degree (‘dominant’) is more stable than the third.
• This is the case in all of the tonalities (or collections) we’ve
discussed today.
• Also in Indian Carnatic music.
3
We’ll refer to this number, multiplied by the number of total
observations in the corpus, as the expected count for a bigram. The
observed count is the actual occurrence data shown above in table 1.
Expxy = N * p(x1) * p(y2)
Obs I
ii
iii IV V
vi
vii Total
I
132 36 474 668 191 43 1544
ii
116
35 11 100 59 5
326
iii
47
13
73 22
52 12 219
IV
351 63 31
138 29 45 657
V
1042 60 63 73
147 1
1386
vi
106 72 62 64 159
14 477
vii
92
1
4
3
2
4
106
Total 1754 341 231 698 1089 482 120 4715
Table 1. Bigram sequences in a corpus of Bach chorales. Rows
represent first chord, columns following chord. After Rohrmeier
(2007).
Where N is the total number of observed bigrams, and p(in) is the
probability of observing chord i in nth position.
Exp I
ii
iii IV V
vi
vii
I
112 76 229 357 158 39
ii
121
16 48 75 33 8
iii
81 16
32 51 22 6
IV
244 48 32
152 67 17
V
516 100 68 205
142 35
vi
177 34 23 71 110
12
vii
39
8
5 16 24 11
Table 2. Expected counts for the Bach corpus.
Obvious statement: some chord sequences occur much more often
than others.
• not particularly interesting in and of itself
• because some chords are more common than others
• and the sequences may just be reiterating those relations
Now we have a notion of how often sequences ought to occur if there
were no interesting constraints on them.
• Q: how closely does this model match reality?
• one answer: ratio of the values in table 1 to the values in table 2
o large numbers: sequence occurs more often than
expected
o small numbers (limit of 0): sequence occurs less often
than expected
• checking the Observed: Expected ratios (O/E values) will
give us some idea of whether the factors {preceding chord,
following chord} are independent.
If there are no interesting principles governing the combination of
chords into sequences:
• then every chord should be equally likely to follow every other
chord, all else being equal
• i.e., the likelihood of observing some chord shouldn’t depend
at all on the chords that precede and follow it
• so the probability of observing a sequence xy should be equal
to the product of the independent probabilities of observing x
in 1st position and observing y in 2nd position
4
I
ii
iii
IV V
I
1.18 0.48 2.07 1.87
ii
0.96
2.19 0.23 1.33
iii
0.58 0.82
2.25 0.43
IV
1.44 1.33 0.96
0.91
V
2.02 0.60 0.93 0.36
vi
0.60 2.09 2.65 0.91 1.44
vii
2.33 0.13 0.77 0.19 0.08
Table 3. O/E values for Bach corpus.
O/E
vi
1.21
1.77
2.32
0.43
1.04
In order to know whether something more than bigrams are at work
here, we’ll need to look at harmony in greater detail.
vii
1.09
0.60
2.15
2.69
0.03
1.15
2.2
Transitions and intuitions
Let’s start by checking whether these statistics are reflected in our
intuitions.
• pick some well-formed (= high likelihood) sequence
• take the same set of events and scramble them (preserving 1st
and last events as a control)
• see if they sound worse
0.37
There seem to be a number of fairly large and fairly small numbers in
the table.
• suggests that the two factors are not independent
• slightly more formal way of examining this: chi square test
o if we repeat this corpus analysis many, many times…
o with factors that really are independent…
o the sums of all O/E values in the table will
approximate a chi-square distribution
o with degrees of freedom (columns – 1) x (rows – 1)
• these particular values would be on the extreme end of such a
distribution: χ2 (30) = 49, p = 0.016
• so it’s pretty safe to conclude that these factors are not
independent in this sample
(1a)
F:
O/E:
OK, so what does that show?
• may be that Bach learned a grammar and the grammar results
in unequal probabilities for different sequences of chords
• may be that Bach just learned the bigram table shown above
(or some related statistical properties)
5
vi
ii
2.09
V
1.33
I
2.02
(1b)
(2b)
C: I
F:
vi
V
O/E:
1.44
ii
(2a)
2.3
O/E: 2.07
Mean: 1.44
I
1.44
V6 vi
I
1.87 1.04 0.60 2.07
IV
0.43
vi V
I
1.44
2.02
vi
I
V6
I
IV
I
0.60 1.87 2.02 2.07 1.44
So if we want to get a more detailed picture of what makes certain
transitions sound better than others, it will be useful to look outside a
particular diatonic collection…
The sense of typicality vs. atypicality may get sharper with longer
examples.
IV
IV
This still doesn’t sound that weird.
• I could mess around with diatonic sequences until I find one
that can be shuffled with a huge decrease in likelihood
• but it still won’t be so dramatic, intuitively
• because no diatonic sequence is really so ill-formed
• even in Bach
• but especially in more modern genres that you may be more
familiar with
0.96
Well, (1b) certainly doesn’t sound ill-formed.
• if you have a keen ear for harmony, it may sound ‘less typical’
or something like that
• but no sequence of chords from the same key really sound
‘ungrammatical’
• which is why there are no 0 values in table 1.
C: I
vi
O/E: 1.87 1.04 0.91 0.43
Mean: 1.36
I
0.60
V
Modulations and tonicizations
When the tonic center shifts during a piece of music, we call it
modulation or tonicization.
• difference between the two is one of scale:
o modulation tends to be for an extended period of time
o tonicization for shorter periods of time
• theory textbooks sometimes give rules for what ‘counts’ as a
modulation
6
•
but this needn’t interest us
(3b)
Modulation will be useful because it has a fairly dramatic perceptual
effect.
Of course, not all modulations are equally likely:
• modulation to a key that shares many notes with the original is
more likely than modulation to a key that shares fewer
• we say that keys with a lot of shared notes are closer than those
with fewer
• there is another dimension to key distance, but we ignore it for
the moment
vi
II
I
(4a)
Given that different keys contain different pitches, chords that are
licensed in one key are not necessarily licensed in another.
• so scrambling chords in examples with modulations can result
in outrageous-sounding variants
(3a)
vi
V7
F
G
a
G
C
Bortnjanskij’s “Tebe Pojom” in reduced form. Example after
Rohrmeier (2007). Brackets indicate distinct key areas in the excerpt.
II
V7
This piece has a number of modulations.
• when you put together elements that belong to different key
collections, it can sound totally wrong
• despite the fact that, in the original, there are elements from
different keys next to each other…
• and they don’t sound so weird
• so there must be something going on here beside likely and
unlikely transitional probabilities
I
Chord in block is dominant of following chord
7
(4b)
C7
F
G64-35
D7
a
F#0 G
E7
C
*
*
*
*
Scrambled Bortnjanskij. Asterisks represent unlikely bigrams.
Preview of the solution: the reason that some of the unlikely bigrams
here sound OK and others don’t is that the acceptable ones are
structurally licensed by one of their members. These structural
relations, in turn, relate to some of the diatonic dependencies we’ve
looked at.
More on that next week…
8