Tone interval theory
Laura Dilley, Ph.D.
Speech Communication Group
Massachusetts Institute of Technology
and
Departments of Psychology and Linguistics
The Ohio State University
Chicago Linguistics Society
Annual Meeting
April 9, 2005
Overview
• What’s the problem?
– Failure of descriptive apparatus for
some tonal systems
• Why concepts from music theory can
help resolve the problems
• Introduction to tone interval theory
Prior assumptions
• Early autosegmental theory made several
strong claims regarding tones
– Tones, segments represented on different tiers
– Tones are exactly like segments

• The claim that tones are exactlyxlike
segments leads to a failure of descriptive
adequacy for some tonal systems
Exactly like segments?
• Idea: Tones, segments are defined without
reference to one another in series
• No inherent relativity of tones to other tones
• Relative heights of tones are not part of the
phonology
Relative height
must be part of
phonetics
– But cf. Jakobson, Fant and Halle (1952)
Strong phonetic view (Pierrehumbert 1980)
• Extended autosegmental theory to
English
• Treated relative tone height as part of
phonetic component of grammar
– Phonological primitives based on H, L
tones plus phonetic tone scaling rules
• Insufficent constraints on relative tone
height in phonetic rules lead to problems
with descriptive adequacy, testability
Defining descriptive adequacy
• Q: What should a theory of the phonology and
phonetics of tone and intonation do?
• A: Define a clear and consistent relation
between phonology and aspects of F0 shape.
• A: Support descriptive linguistic intuitions
– E.g., LHL should correspond to a rising-falling
pattern
A phonology-phonetics test case
H
• Q: If we assume that LHL corresponds to L
then what are the critical restrictions on H, L?
• A: H must be higher than adjacent L, and L
must be lower than adjacent H.
– Permits a sequence of H, L tones to give rise to a
predictable F0 shape
• What would happen if these restrictions are
not in place?
L
Some dire consequences
• If critical restrictions on adjacent H, L are not in place:
– Cannot predict F0 shape from phonology (overgeneration)
– Cannot describe an F0 contour in terms of a unique
phonological specification (indeterminacy)
– Cannot test a theory
Phonetic rules (Pierrehumbert 1980)
1. In Hi (+T) (T+)Hj: f(Hj) = f(Hi) ·[p(H*j)/p(H*i)]
2. In H+L:
f(L) = k·f(H), 0 < k < 1
3. In H (+T) L+: f(L) = n·f(H)· [p(H)/p(L)], 0 < n < k
4. In H(+T) L-:
f(L-) = p0·f(H), 0 < p0 < k
5. In H+L Hi and H L+Hi:
6. In H- T:
f(Hi) = k·f(Hi), 0 < k < 1
f(T) = f(H-) + f(T)
7. f(L%) = 0
8. f(Li+1) = f(L*i)·[p(L*i)/p(Li+1)]
Pierrehumbert (1980)
Example: H* L+H*. Rewrite as: H1 L H2
f(T) = F0 level of tone T
p(T) = tone scaling value of tone T (“prominence”)
f(L) = n • f(H1) • [p(H1)/p(L)], for 0 < n < 1
[f(L)/f(H1)] = n • [p(H1)/p(L)]
• Therefore, the F0 of L, f(L), is higher than the F0 of
H1, f(H1) when [p(H1)/p(L)] > 1/n.
• The F0 of L can also be higher than F0 of H2 (Dilley
2005)
• No restrictions are in place to prevent this.
Pierrehumbert and Beckman (1988)
• Example: H* L+H*. Rewrite as: H1 L H2
• Each tone is independently assigned a value
for a parameter p (for prominence), where p
determines F0
H1 L2 H3 → p(H1) p(L2) p(H3)
1
h
p(H)
0
H1
H3
L2
L2
H1
H2
H3
L1
l
• Critical restrictions are not in place
0
L3
p(L)
1
Summary and implications
• Treating tones as exactly like segments
relegated relative tone height to phonetics
– Phonetic rules, mechanisms were proposed
to control relative tone height
• In no version of the phonetic theory do
the rules specify sufficient constraints
• This leads to a failure of descriptive
adequacy and testability
What to do?
Q: Is the problem adequately addressed
simply by adding constraints to
phonetic rules?
A: No.
There is evidence that relative tone height
is part of phonology, not the phonetics.
The problems run deeper: phonological
categories are not fully supported by
data.
Relative height is phonological
Contrastive downstep: Igbo (Williamson 1972)
ámá ‘street’
ám!á ‘distinguishing mark’
Contrastive upstep: Acatlán Mixtec (Pike and
Wistrand 1974)
?íkúmídá ‘we (incl.) have’
?íkúmíd^á ‘you (pl. fam.) have’
!
= downstep, ^ = upstep
Music as inspiration
• Claim: Music theoretic concepts provide
a way of addressing problems in
intonational and tonal phonology
– Describing relative tone height as part of the
phonological representation
– Achieving descriptive adequacy, testability
– Pitch range normalization
– Typological differences among tonal
systems
– Others
Frequency
(Hz)
233
277 311
370 415 466
554 622
220 247 262 294 330 349 392 440 494 523 587 659
A#
Notes
A
C#
F#
D#
B C D
E
F
G#
A#
G A
C#
B C
D#
D E
One semitone = 122  1.05946
Key of C
G G A
392 392 440
Frequency
Ratios
1.0
Key of F
C
G
392 523
0.95
0.89
1.33
1.12
B
494
C C
C
D
262 262 294 262
1
F
349
E
330
0.95
0.89
1.33
1.12
• Musical scales and melodies are represented in
terms of frequency ratios (Burns, 1999)
More on melodic representation
• Nature of frequency ratios differs for distinct
musical cultures
– e.g., Number and size of scale steps
• Layers of representation for musical melody
(Handel 1989):
– Up-down pattern: Whether successive notes are
e.g., higher, lower than other notes ALL melodies
– Interval: Distance between notes, cf. a specific
frequency ratio
SOME melodies
– Scale: Relation between a note and a tonic referent
note in a particular key
SOME melodies
Scales and frequency ratios
• Scales correspond to a set of ratios defined with
respect to a tonic (referent) note
I
II
Ratio
1
1.12 1.26 1.33 1.50 1.68 1.89
Tonic
C (Key)
C
D
E
F
G
A
B
262
294
330
349
392
440
494
F
G
A
Bb
C
D
E
349
392
440
466
523
598
659
F (Key)
III
IV
V
VI
VII
Layers of representation
G
392
G
392
A
440
C
523
B
494
r<1
r>1
r<1
G
392
3
8
Up-down
pattern
r=1 r>1
Interval
Scale
V4
1.0
1.12
0.89
1.33
0.95
V4
VI4
V4
I5
VII4
•Each successive layer of representation encodes
more information than the previous layer
Tone interval theory
• Tone intervals, I, are abstractions of frequency
ratios
• Tones, T, are timing markers that are coordinated
with segments via metrical structure (cf.  onsets)
• Tone intervals relate a tone to one of two kinds of
referent:
1) Referent is another tone (up-down pattern, interval)
T1 T2 → I1,2 = T2/T1
2) Referent is the tonic,  (cf. scale)
Iμ,2 = T2/μ
Tone interval theory, cont’d.
• Every pair of adjacent tones in sequence is
joined into a tone interval in ALL languages
T1 T2 T3 … Tn → I1,2 I2,3 … In-1,n
(I1,2 = T2/T1)
• Each tone interval is then assigned a relational
feature (cf. up-down pattern)
higher implies that T2 > T1 or I1,2 > 1
lower implies that T2 < T1 or I1,2 < 1
same implies that T2 = T1 or I1,2 = 1
I1,2=1 I2,3>1 I3,4<1
I4,5>1 etc.
Tone interval theory, cont’d.
• SOME languages further restrict these ratio
values (cf. Interval)
I1,2=1 I2,3=1.12 I3,4=0.89 I4,5=1.33 etc.
• SOME languages define tones with respect to
a tonic (cf. Scale)
• Tones, tone intervals occupy different tiers and
are coindexed (cf. tonal stability)
x
x
x
x.
T 1 T2 T3 … Tn
I1,2 I2,3 … In-1,n
Advantages of this approach
 Defining the phonology in this way:
 Achieves descriptive adequacy and generates
testable predictions
 Proposes explicit connection with music
 Builds on earlier work
T1 T2 T3
I1,2 >1 I2,3<1

TH2
I1,2 >1
TL1
I2,3<1
TL3
Summary and Conclusions
• Autosegmental theory was based on the strong
claim that tones are exactly like segments
– Relative tone height was relegated to phonetics
• Theories attempting to extend this approach
intonation languages have led to problems
– E.g., inability to generate testable predictions
• Relative tone height is almost certainly part of
phonology, not phonetics
Summary, cont’d.
• Musical melodies are represented in terms of:
– Frequency ratios between notes in sequence and
between a note and the tonic
– Up-down pattern, interval, and scale
• Tone interval theory
– The representation is based on tone intervals
(abstractions of frequency ratios)
– Notion of up-down pattern permits a clear definition
between phonology, phonetics
– Builds on earlier work


Thank you.
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