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 exactly like segments leads to a failure of descriptive adequacy for some tonal systems x

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 –But cf. Jakobson, Fant and Halle (1952) Relative height must be part of phonetics

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 Q: If we assume that LHL corresponds to 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 H L

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 Some dire consequences

Phonetic rules (Pierrehumbert 1980) 1. In H i (+T) (T+)H j : f(H j ) = f(H i ) ·[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-) = p 0 ·f(H), 0 < p 0 < k 5. In H+L H i and H L+H i : f(H i ) = k·f(H i ), 0 < k < 1 6. In H- T: f(T) = f(H-) + f(T) 7. f(L%) = 0 8. f(L i+1 ) = f(L* i )·[p(L* i )/p(L i+1 )]

Pierrehumbert (1980) Example: H* L+H*. Rewrite as: H 1 L H 2 f(T) = F0 level of tone T p(T) = tone scaling value of tone T (“prominence”) f(L) = n f(H 1 ) [p(H 1 )/p(L)], for 0 < n < 1 [f(L)/f(H 1 )] = n [p(H 1 )/p(L)] Therefore, the F0 of L, f(L), is higher than the F0 of H 1, f(H 1 ) when [p(H 1 )/p(L)] > 1/n. The F0 of L can also be higher than F0 of H 2 (Dilley 2005) No restrictions are in place to prevent this.

Pierrehumbert and Beckman (1988) Example: H* L+H*. Rewrite as: H 1 L H 2 Each tone is independently assigned a value for a parameter p (for prominence), where p determines F0 H 1 L 2 H 3 → p(H 1 ) p(L 2 ) p(H 3 ) Critical restrictions are not in place H1H1 L2L2 H3H3 H1H1 L2L2 H3H3 h l 1 p(H) 0 p(L) 1 L1L1 H2H2 L3L3

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

Musical scales and melodies are represented in terms of frequency ratios (Burns, 1999) Frequency (Hz) ABCDEFGABCDE A#A# C#C# D#D# F#F# G#G# A#A# C#C# D#D# Notes G 392 A 440 G 392 C 523 B 494 C 262 C 262 D 294 C 262 F 349 E Frequency Ratios One semitone = 12  2  Key of C Key of F

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 –Interval: Distance between notes, cf. a specific frequency ratio –Scale: Relation between a note and a tonic referent note in a particular key ALL melodies SOME melodies

Scales and frequency ratios Scales correspond to a set of ratios defined with respect to a tonic (referent) note I II III IV V VI VII C (Key) C D E FGAB F(Key) FGAB b C D E Tonic Ratio

G 392 A 440 G 392 C 523 B Up-down pattern Interval Scale 3838 r = 1 r > 1 r < 1 r > 1 r < 1 V4V4 V4V4 VI 4 V4V4 I5I5 VII 4 Layers of representation 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) 2) Referent is the tonic,  (cf. scale) T 1 T 2 → I 1,2 = T 2 /T 1 I μ,2 = T 2 / μ

Tone interval theory, cont’d. Every pair of adjacent tones in sequence is joined into a tone interval in ALL languages Each tone interval is then assigned a relational feature (cf. up-down pattern) higher implies that T 2 > T 1 or I 1,2 > 1 lower implies that T 2 < T 1 or I 1,2 < 1 same implies that T 2 = T 1 or I 1,2 = 1 I 1,2 =1 I 2,3 >1 I 3,4 1 etc. T 1 T 2 T 3 … T n → I 1,2 I 2,3 … I n-1,n (I 1,2 = T 2 /T 1 )

Tone interval theory, cont’d. SOME languages further restrict these ratio values (cf. Interval) I 1,2 =1 I 2,3 =1.12 I 3,4 =0.89 I 4,5 =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 T 2 T 3 … T n I 1,2 I 2,3 … I n-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 T 1 T 2 T 3 I 1,2 >1 I 2,3 <1  T1T1 T2T2 T3T3 I 1,2 >1 I 2,3 <1 L H L

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.    