scholarly journals Modes, the Height-Width Duality, and Handschin’s Tone Character

2011 ◽  
Vol 17 (1) ◽  
Author(s):  
David Clampitt ◽  
Thomas Noll
Keyword(s):  
Music Theory ◽  
Algebraic Combinatorics ◽  
Combinatorics On Words ◽  
Mathematical Approach ◽  
Scale Step ◽  
Theoretical Results

The theory of well-formed modes is a modal refinement of the theory of well-formed scales. The mathematical approach is based on various results from the subdiscipline of algebraic combinatorics on words. Section 1 provides anchors and motivations for this investigation both in music theory and in mathematics and traces some earlier cross-connections. An overview of the theory is presented in terms of a dichotomy betweengenericandspecificlevels of description. Section 2 presents a first group of basic theoretical results. Height-width duality mediates between scale step patterns and fifth-fourth folding patterns. Both are encoded through divided words, on two-letter alphabets, such asaaba|aabandyx|yxyxy. The lettersaandbdenote ascending whole and half steps, and the letters x and y denote ascending perfect fifths and descending perfect fourths, respectively. These words are well-formed words; i.e., in the language of word theory, they areconjugatetoChristoffelwords and inherit a duality that is akin toChristoffel duality. Qualitative differences between the modal varieties of the same underlying scale can be detected and formalized through word-theoretical arguments. For example, a property we refer to asdivider incidencecharacterizes modes corresponding tostandardwords.Positivestandard words generalize the ascending authentic Ionian mode.Sturmian morphismsprovide a transformational meta-language for the study of well-formed modes. Section 3 revisits Jacques Handschin’s concept of tone character and defends it on the basis of the mathematical results against two criticisms that had been raised by Carl Dahlhaus. Section 4 explores distinctions among the modes based upon considerations of word theory, especially divider incidence and concomitant properties that support major-minor tonality. A concluding section connects these arguments with other lines of investigation.

scholarly journals Application of Entropy Compression in Pattern Avoidance

10.37236/3038 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Positive Answer ◽  
Pattern Avoidance ◽  
Algebraic Combinatorics ◽  
Combinatorics On Words ◽  
Letter Alphabet ◽  
Infinite Word ◽  
Entropy Compression

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f= h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We give a positive answer to Problem 3.3.2 in Lothaire's book "Algebraic combinatorics on words'", that is, every pattern with $k$ variables of length at least $2^k$ (resp. $3\times2^{k-1}$) is 3-avoidable (resp. 2-avoidable). This conjecture was first stated by Cassaigne in his thesis in 1994. This improves previous bounds due to Bell and Goh, and Rampersad.

scholarly journals Harmonic Closure: Music Theory and Perception

1992 ◽  
Vol 9 (4) ◽  
pp. 383-411 ◽  
Author(s):  
Burton S. Rosner ◽  
Eugene Narmour
Keyword(s):  
Music Theory ◽  
Explanatory Power ◽  
Correlational Analysis ◽  
Root Position ◽  
Controversial Topic ◽  
Scale Step

Music theorists have often disagreed about the material variables that determine the perception of harmonic closure. To investigate this controversial topic, we presented subjects with pairs of selected two-chord progressions. The subjects judged which member of each pair seemed more closed. Preferences varied across pairs of cadences and generally obeyed transitivity. Quantitative reformulation of theoretical harmonic variables permitted correlational analysis of the results. Three or four variables, including one or two that reflect learned stylistic structures, best explained our findings. Conventional harmonic factors of scale step, soprano position, and root position demonstrated surprisingly little explanatory power.

Analyzing Form and Measuring Perceptual Content in Mozart's Sonata K. 282: A New Theory of Parametric Analogues

1996 ◽  
Vol 13 (3) ◽  
pp. 265-318 ◽  
Author(s):  
Eugene Narmour
Keyword(s):  
Music Theory ◽  
Perceptual Content ◽  
Active Listening ◽  
Top Down ◽  
Traditional Methods ◽  
Bottom Up ◽  
On Line ◽  
Level I ◽  
Scale Step ◽  
Perceptual Sense

Hierarchic analysis in music necessarily separates form from content. However, in active listening, the two are indivisible. To illustrate this, I first analyze in Part 1 the opening movement in Mozart's Sonata K. 282 from the top down, using traditional methods in music theory. Arriving at the manifest level, I then dissect the music from the bottom up, relying on the implication-realization model (Narmour, 1977,1989,1990,1991a, 1992). The contrasting perspectives reveal in great detail some of the movement's richly complex structuring. More generally, they confirm the inextricable feedback between parametric content and the meaning of form, specifically with respect to the contrary functions of closure and nonclosure. Following these analyses, Part 2 forges a synthesis by developing an implicative theory of analogical structures for melody, harmony, duration, and meter. Because, in terms of bottom-up processing, the analytical symbology for tracking structures is commensurable, we can, in all four primary parameters, weight similarity (aa), difference (ab), closure (stability), and nonclosure (implication) with comparable numbers. Further, by adding in some essential stylistic properties from the top down (scale step, diatonic pitch set, tonal cadential closure), we are able to represent the overall rhythmic shape of the first phrase in a single twodimensional graph. Thereby, we recapture from hierarchic analysis the perceptual sense that, in on-line listening, form and content are synthetically one.

Algebraic Combinatorics on Words

2004 ◽  
Vol 70 (1) ◽  
pp. 154-155
Author(s):  
J.-P. Allouche
Keyword(s):  
Algebraic Combinatorics ◽  
Combinatorics On Words
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