================================================= ==================== Tonality ================= ================================================= A. What is tonality? 1. let's start with an old but yet a very good and relevant definition - "Tonic is the tone of ultimate rest, the other notes tend to move toward the tonic" (L. B. Meyer 1956). As we can see the definition emphasize that the hierarchy in TONALITY, refers to the STABLE and the DYNAMIC properties of tones. The tonic is the most stable tone in the SCALE. Usually it is the last note in a melody or music piece. The tonic or the notes of the tonic triad are the usually the beginning notes. 2. Lets listen to the first movement of Mozart's "Eine Kleine Nachtmusik" and try to perceive the feeling of tonality. (Mira the first movement is 5 minutes long - the same can be done with the Minuet - only 2 melodic lines about one minute without the trio) a. Does the end feels like the ultimate rest of the piece? b. Listen to the next examples suggesting some notes as possibilities for ending the Segment. Rate the ending notes from 1 (most unstable) to 5 (most stable)(do it fast and intuitive - between the examples you got only 2 seconds to rate. In the second time that you listen, rate it using 1 to 5 one time only. [midi-examples-1: the end of the piece with different notes as an ending] 2a. Now listen only to the first 4 bars of the first movement and try to answer the almost the same questions as above: a. Does it feels like we got to the ultimate rest? b. Listen to the next examples suggesting some possibilities for ending the Segment. Rate the ending notes from 1 (most unstable) to 5 (most stable)(do it fast - between the examples you got only 2 seconds to rate. the second time that you listen, rate it using 1 to 5 only one time only. [midi-examples-2: the begining of the piece with different notes as an ending] 3.More music theory The identity of a piece of Western tonal music is ordinarily assumed to derive from the way that pitch is organized within it... most music-theoretic or music-analytic writings of the last two hundred years have focused on the elucidation of music's harmonic and melodic dimensions (see Bent, 1980; Palisca, 1980), exploring these primarily in terms of configurations of pitches and their inter-relations as scales, keys and modes under the banner of TONALITY. (Pitch Schemata Ian Cross) Theory of tonal music deals with: i) what collections of notes belong together (in defining major and minor scales); (ii) which notes from these collections are more important than others and are thus likely to occur at the beginning or end of successions of notes, or melodies (in defining the tonal functions - tonic, dominant etc. - of the different scale notes); (iii) which notes from the collections may occur simultaneously with others (in defining triadic chord configurations); (iv) which simultaneities are functionally similar (in defining inversional and substitutive equivalence of chords); (v) which keys are more closely, and which are more remotely, related; (vi) the ways in which the tonal function of notes in a melody determines which notes will succeed one another (as in the rules of voice-leading); and (vii) the ways in which principles of combining notes in chords and melodies will interact. (Pitch Schemata Ian Cross) a.- The SCALE The western scale consists of 7 Pitch Classes that are chosen from the 12 Pitch classes that are the chosen 12 discrete tones parsing the octave. The hierarchy of scale degrees is in fact (obviously) hierarchy of Pitch-Classes. This hierarchy is represented in various ways: - The names of the scale degrees refer to the tonic or the dominant TONIC (I) - DOMINNAT(V) - SUBDOMINANT (IV) - SUPER TONIC (ii)- MEDIANT (iii) -SUBMDIANT (vi). In fact the names takes in account the intervals of the scale degree to the tonic and the dominant. - Leading Tones - dynamic move to stable ton mostly to tonic e.g. B->C - Consonance/dissonance: the intervals of the tonic triad and their inversions (complimentary intervals) are consonance the other intervals are dissonance. Consonance interval is stable; dissonance interval has dynamic quality and it should be resolved to a stable interval. We should emphasize that the Consonance/dissonance dichotomy, is part of tonal conventions and theory, But it arises a lot of problems from the human perception point of view. - Stable/dynamic scale degrees refers (as the Consonance/dissonance relations) To Harmony implications like: [Tonic-triad -------> dominant-triad] | | | | [Stable ------------> dynamic] b. The Triad The basic harmony is based on Triads - 3 pitch classes with the third interval between them (based on the scale). The triads built on the Tonic (I) Dominant (V) and subdominant (IV) degrees are major chords (- M3=major third, m3=minor third) I - C (+ M3) E (+ m3) G IV - F (+ M3) A (+ m3) C V - G (+ M3) B (+ m3) D The triads on the ii, iii, vi are minor chords: Ii - D (+ m3) F (+ M3) A iii - E (+ m3) G (+ M3) B vi - A (+ m3) C (+ M3) E c. A Little about Harmony Harmony in the common sense means the way tones sounds together like in an interval or in a chord. We use her the notion of harmony in a broader sense, dealing with the way that notes are interrelated, one after another or in the same time. The harmony is part of the succession of the melody, the accompaniment of it and the most influential factor on the structure of the melody. In the small range we are dealing with chords of the scale as properties of a given key, in a large range we are dealing with key relationships. In a given scale the chords has functions - that defined the progressions of chords. For example the cadence in a major scale is based on the progression of I -> IV -> V -> I, in the key of C major The chords are C -> F -> G -> C. Chords in the cadence can be substitute with chords of the same function like I -> ii -> V -> I here the ii=Dm in C major is in the same function as the IV=F, Both of them function as subdominants in the cadence. We can enlarge the 3 chords progression by adding chords like I -> vi -> IV -> ii -> V -> I. In C Major the chords are C -> Am -> F -> Dm -> G -> C. We can go on like this and develop the basic cadence to a very complex chord progression, we can add to the triads tensions as they put it in jazz, more notes (like 7, 9, 11, 13 and more) But at least for now we keep with the triads. There are very strict rules that articulates the "voice leading" while moving from chord to chord. But it is a subject for a special treatment. 4.Tonal Perception Usually the experiments in music perception meet music theory. In the experiments we find that the perception of key (scale) depends on the frequency of appearances and the duration of the pitch classes. One product of the experiences is a table of pitch class rating in a given scale The representation of this rating in numbers enable us to use then to create some rules of weight that would help us to generate melodies based on this rating. KRUMHANSL - in the book "COGNITIVE FOUNDATION OF MUSICAL PITCH" Basically, the theory of tonal music and the empiric phsycoacoustics experience e.g. Krumhansl experiences with tonality meets to confirm the concept of music tonality based on tonic as the most stable tone and the tonic triad tones as stable too. Here is a Graph that summarize the rating of pitch-classes in C major and C minor - scales referring to experiments-can be used as values to define WEITHED Rules to control the appearance of PC in a tonal melody in Major/Minor keys - while generating melody (or analyzing one). The "tonal hierarchies" evident in subjects' ratings of probe notes following major or minor contexts. The same treatment more or less, is given to the key relations between 2 scales: (Table 2.4 p.38.) The correlation's between 2 key profiles enhance the theoretical notion on interkey relations and the values can be used to define modulations possibilities. (for example the correlation between C major to A minor is .651, but between C major to D# minor is -.654) 5. Representation of Pitch Classes in Tonal environment using the (12 7)is a necessity. For example in the scale A major we have the pitch class C# (1, 0) while in F minor it is Db (1,1). MATHEMATICS a. AGMON in the article "A Mathematical Model of The Diatonic System" [Mira you got this Article): 1. mathematical properties of a set of integer-pair classes mode (12, 7)... a set of of scale steps, p3. From scale-steps to Diatonic Intervals 2. Using the P4 (5, 3) as a generator - "cyclic generator"- to generate the 12 pitch classes by multiply it in integers -6,........,6 -> p.6 3. A general formal definition of "diatonic system". p. 11 There were some researchers that gave mathematics or quasi mathematics description of the structure of the diatonic scale. Christopher Longuet-Higgins ) proposed a formal model of "tonal space" intended to elucidate the perception of pitch relations in music. Longuet-Higgins' model represents pitches as points on an infinite two-dimensional plane, single steps being perfect fifths within one dimension and major thirds in the other. Diatonic major scales thus constitute asymmetrical and uninvertible L-shaped blocks formed by groups of seven proximate pitches (see Figure 2). In this way movement between notes within a key and movement between keys can be represented as shifts within and between regions of the two-dimensional plane. Figure 2 Longuet-Higgins' model representing pitches as points on an infinite two-dimensional plane: single steps are perfect fifths in the y-dimension and major thirds in the x-dimension. The diatonic major scale on C is shown as an asymmetrical and uninvertible L-shaped region. . Blazano's group theoretic model of musical pitch relations (Balzano, 1982).Balzano treats the notes of the chromatic scale as analogous to the cyclic group of order 12 (i.e., the set of numbers 1 to 12, or 0 to 11 ). Within this approach, all octave-related notes are treated as functionally identical. This enables the notes of the chromatic scale - or chromatic set - to be depicted in two different circular, or cyclical, forms; these are directly analogous to the chroma [the 12 pitch classes y.] circle and to the circle of fifths (see Figure 8).This approach also relies on the idea that sets of notes can be regarded as functionally identical at the level of pitch class set if one set can be transformed into another by means of an operation such as transposition (and, in Forte's (1973) formulation, transposition together within inversion). Pitch class sets are unordered, being regarded as identical if they have the same membership irrespective of the order in which the members of the pitch class set are laid out. To give an example, the set [2,0,7] - or D, C, G - would be regarded as identical to the set [0,2,7], and to the set [2,4,9]. That is, within the group-theoretic representation of musical pitch, two sets of notes or pitch classes are identical when the same structural relations can be shown to hold for each set. Figure 8 Upper two circles show the chroma circle as note-names and as integers; lower two circles show the circle of fifths as note-names and as integers. Figure 9 Balzano's direct product group. Single steps are minor thirds in the y-dimension and major thirds in the x-dimension. A diatonic major set is shown based around C as an asymmetrical region. (Note that this two-dimensional representation constitutes an "unrolling" of a toroidal structure on the surface of which all points of the same integer are coincident.) IDEAS ***Weighted rules [Mira the articles are from the book Roger gave you - Machine Models of Music] F. P. Brooks, Jr., A. L. Hopkins Jr., P. G. Neumann, and W. V. Wright (1957). "An Experiment in Musical Compositions" (first publication in IRE Transactions on Electronic Computers, 1957) in Machine Models of Music, ed. S. M. Schwanauer & D. A. Lavitt. MIT press, Cambridge, Massachusetts. pp.23-42. An experiment that takes 37 hymes encoded them, and by using Markoff Analysis-Synthesis methods and a computer: The method in use is based on analyzing pitch patterns and generating a table of cumulative probabilities, which we prefer to consider as WEIGHTED RULES, how to continue those patterns. In the synthesis part there were some metric constrains G. M. Rader. (1974). "A method for Composing Simple Traditional Music by Computer", Association for Computing machinery. In Machine Models of Music, ed. S. M. Schwanauer & D. A. Lavitt. MIT press, Cambridge, Massachusetts. . pp.243-260. Generation process in 2 parts: 1) Generating Harmonic framework; 2) Generating melody within the Harmonic context. Using the harmonic degrees I to VI of a scale the steps are: Harmony PRODUCTION, APLICABILITY RULES, WEIGTH RULES Melody - " - " " - " " - " + specification + ***Key Algorithm [Mira I Xeroxed this article for you before I left] Temperly David.(1999). What's Key for Key? ..... .Music Perception, 17/1 pp. 65-100. a way to understand and compute scale hierarchy in tonal music. The trigger is Krumhansl and Schmukclers' algorithm ***Music Transformation [y.] - Melodic Descriptive view of the chords triadic-notes - auxiliary (aux) and passing tones [stable --------------> dynamic] thematic transformations in a Small scale (range): - stretching part of a motive (music-pitch-sequence) - keeping the rhythmic pattern of a motive changing the pitch up or down (transposition/moving in key) - no change in contour. - changing intervals, keeping the contour and the rhythm. - changing intervals and contour, keep the rhythm. - Adding passing notes between tow chords notes - doubling the durations or changing them in a way that keeps the structure of meter as it is. e.g. (the meter is tow four) [C1/4, E1/4] --> [C1/8, D1/8, E1/4] or [C3/16, D1/16, E1/4]. - Adding auxiliary notes to notes keeping the meter structure: e.g. [C1/4, E1/4] --> [C1/4, E1/16, D1/16, E1/8]. .......