David Lewin
Encyclopedia
David Lewin was an American music theorist, music critic
and composer
. Called "the most original and far-ranging theorist of his generation" (Cohn 2001), he did his most influential theoretical work on the development of transformational theory
, which involves the application of mathematical group theory
to music.
and studied piano from a young age. He graduated from Harvard
in 1954 with a degree in mathematics. Lewin then studied theory and composition with Roger Sessions
, Milton Babbitt
, Edward Cone, and Earl Kim
at Princeton University
, earning an M.F.A. in 1958. He returned to Harvard as a Junior Fellow in the Harvard Society of Fellows from 1958–61. Following teaching positions at the University of California, Berkeley
(1961–1967), the State University of New York at Stony Brook
(1967–1979), and Yale University
(1979–1985), he returned to Harvard as the Walter W. Naumberg Professor of Music in 1985. Lewin was a Guggenheim Foundation Fellow in 1983–1984, served as the president of the Society for Music Theory
from 1985–1988 and was a member of the American Academy of Arts and Sciences
. He received honorary doctoral degrees from the University of Chicago in 1995, from the New England Conservatory of Music in 2000, and posthumously from Université Marc Bloch de Strasbourg, France, in 2006.
to full orchestra
. In 1961, he became the first professional musician to compose a computer-generated piece at Bell Laboratories (Cohn 2001).
, Edward T. Cone
, and Milton Babbitt
wrote music criticism. Starting during the late 1960s (with articles on Schoenberg, crystallized in his debate with Cone), Lewin began work with text/music relations. During the late 1970s, Lewin's work in this area became more explicitly concerned with issues in literary theory
, publishing articles in 19th-Century Music
. "Studies in Music with Text," published posthumously, demonstrates Lewin's concerns in this area, while also demonstrating a synthesis of critical/theoretical insights/methods/etc. His most far-reaching essay in this area is "Music Theory, Phenomenology, Modes of Perception."
The first group includes his innovations in transformational theory
, as expressed in numerous articles and in his treatise Generalized Musical Intervals and Transformations. In this work, Lewin applied group theory
to music, investigating the basic concepts, interval
and transposition
, and extending them beyond their traditional application to pitch. Based on a powerful metaphor of musical space, this theory can be applied to pitch, rhythm and metre, or even timbre. Moreover, it can be applied to both tonal and atonal repertories.
Lewin's writing on the relationship between text and music in song and opera involves composers from Mozart to Wagner to Schoenberg. In one interesting example, "Music Analysis as Stage Direction," he discusses how structural aspects of the music can suggest dramatic interpretations.
Important writings for the discipline of music theory include "Behind the Beyond" (1968–9), a response to Edward Cone, and "Music Theory, Phenomenology, and Modes of Perception" (1986).
While Lewin's rigorous formal theory may seem forbidding, his writing is marked by a sense of poetry and a critical awareness of disciplinary issues and cultural biases. He often makes clear which dense sections can be skipped by readers unfamiliar with mathematics, and connects his abstract theory to practical musical considerations, such as performance and music perception. For example, in Musical Form and Transformation: Four Analytic Essays, Lewin provides ear-training exercises to develop an ability to hear more difficult musical relationships. His work has influenced later theorists, such as Richard Cohn
, Robert Morris, Henry Klumpenhouwer
, John Clough, Brian Hyer, and Norman Carey and David Clampitt. Posthumously, in 2003, a symposium on David Lewin's theories was conducted at the Mannes Institute for Advanced Studies in Music Theory.
Critic
A critic is anyone who expresses a value judgement. Informally, criticism is a common aspect of all human expression and need not necessarily imply skilled or accurate expressions of judgement. Critical judgements, good or bad, may be positive , negative , or balanced...
and composer
Composer
A composer is a person who creates music, either by musical notation or oral tradition, for interpretation and performance, or through direct manipulation of sonic material through electronic media...
. Called "the most original and far-ranging theorist of his generation" (Cohn 2001), he did his most influential theoretical work on the development of transformational theory
Transformational theory
Transformational theory is a branch of music theory developed by David Lewin in the 1980s, and formally introduced in his most influential work, Generalized Musical Intervals and Transformations...
, which involves the application of mathematical group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
to music.
Biography
Lewin was born in New York CityNew York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...
and studied piano from a young age. He graduated from Harvard
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
in 1954 with a degree in mathematics. Lewin then studied theory and composition with Roger Sessions
Roger Sessions
Roger Huntington Sessions was an American composer, critic, and teacher of music.-Life:Sessions was born in Brooklyn, New York, to a family that could trace its roots back to the American revolution. His mother, Ruth Huntington Sessions, was a direct descendent of Samuel Huntington, a signer of...
, Milton Babbitt
Milton Babbitt
Milton Byron Babbitt was an American composer, music theorist, and teacher. He is particularly noted for his serial and electronic music.-Biography:...
, Edward Cone, and Earl Kim
Earl Kim
Earl Kim was a Korean-American composer.Kim was born in Dinuba, California, to immigrant Korean parents. He began piano studies at age ten and soon developed an interest in composition, studying in Los Angeles and Berkeley with, among others, Arnold Schoenberg, Ernest Bloch, and Roger Sessions...
at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, earning an M.F.A. in 1958. He returned to Harvard as a Junior Fellow in the Harvard Society of Fellows from 1958–61. Following teaching positions at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
(1961–1967), the State University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....
(1967–1979), and Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
(1979–1985), he returned to Harvard as the Walter W. Naumberg Professor of Music in 1985. Lewin was a Guggenheim Foundation Fellow in 1983–1984, served as the president of the Society for Music Theory
Society for Music Theory
The Society for Music Theory is an American organisation devoted to the promotion of music theory as a scholarly and pedagogical discipline...
from 1985–1988 and was a member of the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
. He received honorary doctoral degrees from the University of Chicago in 1995, from the New England Conservatory of Music in 2000, and posthumously from Université Marc Bloch de Strasbourg, France, in 2006.
Composition
While Lewin is primarily known as a theorist, he was also an active composer who wrote works for a wide range of forces, from solo voiceVoice
Voice may refer to:* Human voice* Voice control or voice activation* Writer's voice* Voice acting* Voice vote* Voice message-In film:* Voice , a 2005 South Korean film* The Voice , a 2010 Turkish horror film directed by Ümit Ünal...
to full orchestra
Orchestra
An orchestra is a sizable instrumental ensemble that contains sections of string, brass, woodwind, and percussion instruments. The term orchestra derives from the Greek ορχήστρα, the name for the area in front of an ancient Greek stage reserved for the Greek chorus...
. In 1961, he became the first professional musician to compose a computer-generated piece at Bell Laboratories (Cohn 2001).
Criticism
Lewin's theoretical work may best be understood against his background in 1950/60s avant-garde compositional circles on the North American East Coast. Most of these composers were also music critics. Benjamin BoretzBenjamin Boretz
Benjamin Boretz is an American composer and music theorist.-Life and work:Boretz was born in Brooklyn, New York and graduated with a degree in music from Brooklyn College...
, Edward T. Cone
Edward T. Cone
Edward Toner Cone was an American composer, music theorist, pianist, and philanthropist.Cone studied composition under Roger Sessions at Princeton University, receiving his bachelor's in 1939...
, and Milton Babbitt
Milton Babbitt
Milton Byron Babbitt was an American composer, music theorist, and teacher. He is particularly noted for his serial and electronic music.-Biography:...
wrote music criticism. Starting during the late 1960s (with articles on Schoenberg, crystallized in his debate with Cone), Lewin began work with text/music relations. During the late 1970s, Lewin's work in this area became more explicitly concerned with issues in literary theory
Literary theory
Literary theory in a strict sense is the systematic study of the nature of literature and of the methods for analyzing literature. However, literary scholarship since the 19th century often includes—in addition to, or even instead of literary theory in the strict sense—considerations of...
, publishing articles in 19th-Century Music
19th-Century Music
19th-Century Music is a U.S. triannual music journal published by University of California Press, in Berkeley, California, and established in 1977. Dealing with musical life in Europe and the Americas during the era of the "long century" 19th-Century Music is a U.S. triannual music journal...
. "Studies in Music with Text," published posthumously, demonstrates Lewin's concerns in this area, while also demonstrating a synthesis of critical/theoretical insights/methods/etc. His most far-reaching essay in this area is "Music Theory, Phenomenology, Modes of Perception."
Theory
David Lewin's work in music theory was both influential and eclectic. Broadly, his writings can be divided into three overlapping groups: formal or mathematically-based theory, more interpretive writing on the interaction of music and text, and metatheoretical discussions on the methodology and purpose of contemporary music theory.The first group includes his innovations in transformational theory
Transformational theory
Transformational theory is a branch of music theory developed by David Lewin in the 1980s, and formally introduced in his most influential work, Generalized Musical Intervals and Transformations...
, as expressed in numerous articles and in his treatise Generalized Musical Intervals and Transformations. In this work, Lewin applied group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
to music, investigating the basic concepts, interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
and transposition
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
, and extending them beyond their traditional application to pitch. Based on a powerful metaphor of musical space, this theory can be applied to pitch, rhythm and metre, or even timbre. Moreover, it can be applied to both tonal and atonal repertories.
Lewin's writing on the relationship between text and music in song and opera involves composers from Mozart to Wagner to Schoenberg. In one interesting example, "Music Analysis as Stage Direction," he discusses how structural aspects of the music can suggest dramatic interpretations.
Important writings for the discipline of music theory include "Behind the Beyond" (1968–9), a response to Edward Cone, and "Music Theory, Phenomenology, and Modes of Perception" (1986).
While Lewin's rigorous formal theory may seem forbidding, his writing is marked by a sense of poetry and a critical awareness of disciplinary issues and cultural biases. He often makes clear which dense sections can be skipped by readers unfamiliar with mathematics, and connects his abstract theory to practical musical considerations, such as performance and music perception. For example, in Musical Form and Transformation: Four Analytic Essays, Lewin provides ear-training exercises to develop an ability to hear more difficult musical relationships. His work has influenced later theorists, such as Richard Cohn
Richard Cohn
Richard Cohn is a music theorist and Battell Professor of Music Theory at Yale. Early in his career, he specialized in the music of Béla Bartók, but more recently has written about Neo-Riemannian theory as well as metric dissonance.-External links:*...
, Robert Morris, Henry Klumpenhouwer
Henry Klumpenhouwer
Henry Klumpenhouwer is a musicologist and professor at the University of Alberta. A former PhD student of David Lewin and the inventor of Klumpenhouwer networks, which are named after him. He is the editor of Music Theory Spectrum.-Bibliography:...
, John Clough, Brian Hyer, and Norman Carey and David Clampitt. Posthumously, in 2003, a symposium on David Lewin's theories was conducted at the Mannes Institute for Advanced Studies in Music Theory.
Publication list
- "Re Intervallic Relations Between Two Collections of Notes." Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
3/2 (1959): 298–301 - "The Intervallic Content of a Collection of Notes, Intervallic Relations between a Collection of Notes and its Complement: an Application to Schoenberg’s Hexachordal Pieces." Journal of Music Theory 4/1 (1960): 98–101
- "A Metrical Problem in Webern's Op. 27." Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
6/1 (1962): 125-132 - "A Theory of Segmental Association in 12 tone music." Perspectives of New Music 1/1 (1962): 89-116
- "Berkeley. Arnold Elston Quartet. Seymour Shifrin Quartet No. 2." Review in Perspectives of New Music 2/2 (1964): 169-175
- "Communication on the Invertibility of the Hexachord." Perspectives of New Music 4/1 (1965): 182-186
- "Is it Music?" Proceedings, First Annual Conference of the American Society of University Composers (1966): 50-53, on computer music.
- "Congruence-Invarian Measures in Uniform Spaces." Transactions of the American Mathematical Society 124/3 (1966): 50-53
- "On Certain Techniques of Re-Ordering in Serial Music." Journal of Music Theory 10/2 (1966): 276-287
- "A Study of Hexachord Levels in Schoenberg's Violin Fantasy." Perspectives of New Music 6/1 (1967-8): 18-32
- "Moses und Aron: Some General Remarks, and Analytic Notes for Act I, Scene I." Perspectives of New Music 6/1 (1967–8): 18–32; repr. in The Garland Library of the History of Western Music, ed. E. Rosand, xii (New York, 1965): 327–43
- "Inversional Balance as an Organizing Force in Schoenberg’s Music and Thought." Perspectives of New Music: 6/2 (1967–8): 1–21
- "Some Applications of Communication Theory to the Study of Twelve-Tone Music." Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, 12 (1968): 50–84 - "Some musical jokes in Mozart's Le Nozze di Figaro." In Studies in Music History : Essays for Oliver Strunk, edited by Harold Powerspp. 443–47; reprinted in "Figaro’s Mistakes", Current Musicology, no.57 (1995): 45–60; reprinted in Studies in Music with Text, Oxford University Press, 2006.
- "Behind the Beyond … a Response to Edward T. Cone", Perspectives of New Music, vii (1968–9), 59–69
- "Toward the Analysis of a Schoenberg Song - Op.15 no.1", Perspectives of New Music, xii/1-2 (1973–74), 43–86
- "On Partial Ordering", Perspectives of New Music, xiv/2 (1976), 252-257
- "On the Interval Content of Invertible Hexachords", Journal of Music Theory, xx/2 (1976), 185-188
- "A Label-Free Development for 12-PC Systems", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, xxi/1 (1977), 29-48 - "Some Notes on Schoenberg's Op. 11", In Theory Only, iii/1 (1977), 3-7
- "Forte’s Interval Vector, my Interval Function, and Regener’s Common-Note Function", Journal of Music Theory, xxi (1977), 194–237
- "A Communication on Some Combinational Problems", Perspectives of New Music, xvi/2 (1978), 251-254
- "Two Interesting Passages in Rameau's Traité de l'harmonie", In Theory Only, iv/3 (1978), 3-11
- "A Response to a Response On PCSet Relatedness", Perspectives of New Music, xviii/1-2 (1979–80), 498-502
- "On Generalized Intervals and Transformations", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, xxiv/2 (1980), 243-251 - "Some New Constructs Involving Abstract PCSets, and Probabilistic Applications", Perspectives of New Music, xviii/1-2 (1979–80), 433-444
- "Some Investigations into Foreground Rhythmic and Metric Patterning", Music Theory: Special Topics, ed. R. Browne (New York, 1981), 101–37
- "On Harmony and Meter in Brahms's Op.76 No.8", 19-Century Music, iv/3 (1981), 261-265
- "A way into Schoenberg's opus 15, number 7", In Theory Only, vi/1 (1981) 3-24
- "Comment: "On Joel Lester, 'Simultaneity structures and harmonic functions in tonal music', In theory only 5/5: 3-28, and Marion Guck, 'Musical images as musical thoughts: the contribution of metaphor to analysis', In theory only 5/5: 29-42", In Theory Only v/8 (1981) 12-14
- "Vocal Meter in Schoenberg’s Atonal Music, with a Note on a Serial Hauptstimme", In Theory Only, vi/4 (1982), 12–36
- "A Formal Theory of Generalized Tonal Functions", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, xxvi (1982), 23–60 - "An example of serial technique in early Webern", Theory and Practice, vii/1 (1982) 40-43
- "On extended Z-triples", Theory and Practice, vii/1 (1982) 38-39
- "Auf dem Flusse: Image and Background in a Schubert Song", 19th-Century Music, vi (1982–3), 47–59; rev. as Auf dem Flusse … Schubert: Critical and Analytical Studies, ed. W. Frisch (Lincoln, NE, 1986), 126–52
- "Transformational Techniques in Atonal and Other Music Theories", Perspectives of New Music, xxi (1982–3), 312–71
- "Brahms, his Past, and Modes of Music Theory", Brahms Studies: Washington DC 1983, 13–27
- "An Interesting Global Rule for Species Counterpoint", In Theory Only, vi/8 (1983), 19–44
- "Amfortas’s Prayer to Titurel and the role of D in Parsifal: the Tonal Spaces of the Drama and the Enharmonic C/B", 19th-Century Music, vii (1983–4), 336–49
- "Studying with Roger", Perspectives of New Music, xxiii/2 (1982–3), 152-154
- "On Formal Intervals Between Time-Spans", Music Perception: An interdisciplinary journal, i/4 (1984) 414-23
- "On Ellwood Derr's 'Deeper Examination of Mozart's 1-2-4-3 Theme.'" In Theory Only viii/6 (1985) 3
- "Music Theory, Phenomenology, and Modes of Perception", Music Perception, iii (1986), 327–92
- Generalized Musical Intervals and Transformations (Yale University Press: New Haven, CT, 1987; reprint Oxford University Press, 2007)
- "On the 'ninth-chord in fourth inversion' from Verklärte Nacht", Journal of the Arnold Schoenberg Institute, X/1 (1987) 45-64
- "Concerning the inspired revelation of F. J. Fétis", Theoria, ii (1987) 1-12
- "Some Instances of Parallel Voice-Leading in Debussy", 19th-Century Music, xi (1987–8), 59–72
- "Klumpenhouwer NetworkKlumpenhouwer networkA "Klumpenhouwer Network", named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is, "any network that uses T and/or I operations [ transposition or inversion] to interpret interrelations among pcs" [ pitch class sets]...
s and Some Isographies that Involve Them", Music Theory SpectrumMusic Theory SpectrumMusic Theory Spectrum is a peer-reviewed, academic journal specializing in music theory and analysis. It is the official journal of the , and is published by University of California Press in Berkeley, California. The journal was first published in 1979 as the official organ of the SMT, which had...
, xii (1990), 83–120 - "Some Problems and Resources of Music Theory" Journal of Music Theory PedagogyJournal of Music Theory PedagogyJournal of Music Theory Pedagogy is a peer-reviewed, academic journal specializing in the teaching and pedagogy of music theory and analysis. It began publication in 1987, under the auspices of The Gail Boyd de Stwolinski Center for Music Theory Pedagogy, at the University of Oklahoma.The...
, v/2, (1991) 111-132 - "Musical Analysis as Stage Direction", Music and Text: Critical Inquiries, ed. S.P. Scher (Cambridge, 1992), 163–76
- "Women’s Voices and the Fundamental Bass", Journal of Musicology, x (1992), 464–82
- "Some Notes on Analyzing Wagner: The Ring and Parsifal", 19th-Century Music, xvi (1992–3), 49–58
- "A Metrical Problem in Webern’s Op.27", Music Analysis, xii (1993), 343–54
- Musical Form and Transformation: Four Analytic Essays (Yale University Press: New Haven, CT, 1993; reprint Oxford University Press, 2007)
- "A Tutorial on Klumpenhouwer Networks, using the Chorale in Schoenberg’s Opus 11 No.2", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, xxxviii (1994), 79–101 - "Comment on John Roeder's article", Music Theory Online, 0/6(1994)
- "Generalized Interval Systems for Babbitt’s Lists, and for Schoenberg’s String Trio", Music Theory SpectrumMusic Theory SpectrumMusic Theory Spectrum is a peer-reviewed, academic journal specializing in music theory and analysis. It is the official journal of the , and is published by University of California Press in Berkeley, California. The journal was first published in 1979 as the official organ of the SMT, which had...
, xvii (1995), 81–118 - "Cohn Functions", Journal of Music Theory, xl (1996), 181–216
- "Some Notes on Pierrot Lunaire", Music Theory in Concept and Practice, ed. J.M. Baker, D.W. Beach and J.W. Bernard (Rochester, NY, 1997), 433–57
- "Conditions Under Which, in a Commutative GIS, Two 3-Element Sets Can Span the Same Assortment of GIS-Intervals; Notes on the Non-Commutative GIS in This Connection", Integral 11 (1997) 37-66
- "The D major Fugue Subject from WTCII: Spatial Saturation?", Music Theory Online, iv/4 (1998)
- "Some Notes on the Opening of the F Fugue from WTCI", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, 42/2 (1998), 235-239 - "Some Ideas about Voice-Leading Between PCSETS", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, 42/1 (1998), 15-72 - "All Possible GZ-Related 4-Element Paris of Sets, in All Possible Commutative Groups, Found and Categorized", Integral 14-15 (2000–2001) 77-120
- "Special Cases of the Interval Function Between Pitch-Class Sets X and Y", Journal of Music TheoryJournal of Music TheoryThe Journal of Music Theory is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl in 1957....
, 42/2 (2001), 1-29 - "Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles", Music Theory SpectrumMusic Theory SpectrumMusic Theory Spectrum is a peer-reviewed, academic journal specializing in music theory and analysis. It is the official journal of the , and is published by University of California Press in Berkeley, California. The journal was first published in 1979 as the official organ of the SMT, which had...
, 45/1 (2002), 196-230 - "Some Compositional Uses of Projected Geometry", Perspectives of New Music, 42/2 (2004), 12-65
- "Studies in Music and Text" (New York: Oxford University Press, 2006)
External links
- Harvard Gazette: Obituary and Faculty Memorial.