Note on terminology: I will be using a modified version of Howard Hanson's notation for interval-classes (is "old-school"), as it implies functional definitions for each ic. I will distinguish this by calling them icF (functional interval-classes), but will rarely preface a icF with its symbol. ie: ic(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) = icF(U, +d, +s, -n, -m, +p, t, -p, +m, +n, -s, -d). Also, I alwaysreplace the base-10 numbers '10' and '11' with 'α' and 'β' respectively in base-12, instead of the more common practice of using 't' and 'e'.
Several years ago I began attempting to integrate (or reconcile) several aspects of the theories put forth by Hanson, Babbit, Perle, Hindemith, Schenker, Rochberg, Reimann, Schoenberg, Messiaen and Lewin (et al.), as well as parallelistic and tonal structures, and the minimalist additive forms. They are all extremely well known theorists and techniques in their own rights, and the multiplicity of techniques presented all contained some degree of musical 'truth' if you will (they all functioned), but were incompatible with each other (except Lewin). I was tired of sounding like the patchwork of "postmodernism". I needed something more integrated.
Lewin became the 'glue' for this synthesis, which eventually required a slight redefinition of basic concepts that I had taken for granted. In effect, I felt as if I needed to go back to Reimann and his contemporaries and start afresh. Lewin's theory works well because it is a very elegant language: it describes what is there, and makes no judgement calls. about function without a theorist to interpret the information provided to xem. I quickly began experimenting with Hindemith, composing small phrases which were then inflicted on my friends. I quickly began to realize that certain intervalic motions created tension or blockage, whereas their inverses caused resolution or forward motion. The results of this were that intervals that existed in simple ratios had a stronger tendency towards a motion characteristic, and can be judged as either generating (overtones of a prior tonic, AKA "negative" motion) or generated (resolving to a tonic, AKA "positive" motion), except for the tritone. Very generally they can be summed up thus:
Strongly Positive
^+p = ic(5)
| +m = ic(8)
| +n = ic(9)
| +s = ic(2)
| +d = ic(1)
| t = ic(6) <-------Neutral
| -d = ic(β)
| -s = ic(α)
| -n = ic(3)
| -m = ic(4)
v -p = ic(7)
Strongly Negative
This is observable by looking at the strong functions versus weak functions in traditional theory using full major (ie: pmn(+)) triads rather than more rigorous IFUNC vectors.
Strongly
+p = V -> I, the dominant is generated by the tonic, and is "returning" to it. V is the 3rd overtone of I.
-p = I -> V = T7(IV -> I), the tonic generates the dominant as overtones. IV is the not even an audible overtone of I.
AND
+p = I -> IV = T5(V -> I), therefore +p is functional, but not determined by on tonal area (rather it is a local and global deciding factor).
Weakly
+s = IV -> V = T5(bVII -> I), little resolution is found. bVII is the 7th overtone of I.
-s = II -> I, again little tension or release is found. II is the 9th overtone of I.
The difference between overtones 7 and 9 is negligible, especially considering that overtone 8 is just another octave repeat.
Also, thirds/sixths and seconds/sevenths can each be compressed into generalized classes of thirds and seconds as they are probably the result of mistuning, and many studis have shown that they are easily confused.
I'll be moving pretty fast from here on in...
Now, if we do take the IFUNC values of a strongly positive progression, we clearly see why it is strongly positive. A traditional perfect authentic cadence is described thus: IFUNC({2, 7, B}, {0, 4, 7}) = -s+s+p+p+n+(U = 0)+d+p+m = +p3mnsd-s = p3mns11d (= p3mnd, generally). NB: Superscripts are used to denote positive icFs, and subscripts are used to denote negative icFs. If no sub/superscript is present, one can assume that it is positive, and intervals that do not appear in the function are not written. Tritones do not get subscripts as they are neutral, as thus default to being written as "positive", though no such motion is implied. We see here that the perfect authentic cadence is strongly positive and driving in nature because of the multiple +p intervals, and only a single negative icF (which also happens to be quite weak). If we were to reverse the order of chords, we end up with a strongly negative progression.
The traditional function of the dominant is actually defined by the resolution of a tritone between two chords leading up to a tonic. Thus, I define the dominant function not as a chord with a root that lies -p from the tonic, but rather as a set of two chords that have an IFUNC(X, Y) containing at least one tritone (such as in the traditional (II or IV) -> (III or V or VII) via icF(t(5, B))). This is also similar to Reimann's idea of function. We can be numbed to the function by consistently presenting unresolving tritones, or by having a continual flow of then in a chain, but this simply places them within the context of a dominant tonal space where tension is regulated without dominant-function resolution.
Atonality and pantonality had been difficult to express in the same terms as tonal languages until Lewin. The most unfortunate thing about atonal/pantonal music is that it is only intelligible to those heavily educated in its 2D (and I would say, anti-vertical) structure. The average person hears their music in distinct horizontal and vertical planes, not tone-rows or mangled motifs. Expressing concepts that exist beyond tonal relations in a vertically-minded (harmonic) way requires some redefinition...
The problem inherent to both pantonality and atonality is that our ear tries to hear a tonic in everything. The only exception to the rule is broadband sounds, where there is such evenness and variety in pitch material, that no frequency can be said to ground the entire mass of sound, and yet there are hypothetically an infinite number o internal hierarchies.
I define atonality as as series of chords with as little motion (few moving notes), direction and/or tension as possible while avoiding a tonic. My technique is somewhat analogous to parsimony in NeoReimiann theory. IFUNC(X, Y) values are calculated only as IFUNC({x ∈ Y | x ∉ X}, X) (= IFUNC((Y \ X), X)). Each chord must have the same internal root held as a pedal tone (to further increase internal chordal stability, and help provide control tonal region), and same internal IFUNC(X) [ie: IFUNC(X) = IFUNC(Y) = IFUNC(Z)...], as well as equality of each IFUNC(X, Y) between chords (progressions have equal tensions with no (dominant-function-producing) tritones), and the internal IFUNC(X)s must have equal positive and negative intervals and no (dominant-function-producing) tritones. Thus the atonal IFUNC(X) = paambbnccseedqqt0(zero), and atonal IFUNC(X, Y) = paambbnccseedqqt0(zero)). Note: The sub/superscripts have the same letters in the IFUNC(X) and IFUNC (X, Y) versions above, but are not related... I was just lazy and cut-and-pasted. Hey, that rhymes!
I define pantonality as as series of chords with as much motion, direction, tension and plurality of function as possible to produce as many implied tonics as possible within each local and global unit. All tones in each verticality must move every time to remove any possibility for a common-tone tonic to emerge even temporarily (ie: IFUNC({x ∈ Y | x ∉ X}, X) = Ø (= IFUNC((Y \ X), Y) = Ø)). This makes 6 the maximum number of chord tones in 12-tone tuning (which would cause overlap with classical serialism as hexachords would always produce combinatorial aggregates with their chordal neighbors). There also must be at least one of each icF (+ and -) to allow for preceeding and proceeding chords to be heard as all functions simultaneously (especially as the balanced IFUNC(X, Y)s do not imply any particular direction, and no tonic allows for any icF(+/-) and tone to take precedence). My technique here is somewhat analogous to extravagance in NeoReimiann theory. Again, each chord must have the same internal IFUNC(X) [ie: IFUNC(X) = IFUNC(Y) = IFUNC(Z)...], as well as equality of each IFUNC(X, Y) between chords (progressions have equal tensions), and the internal IFUNC(X)s must have equal positive and negative intervals, and contain at least one (dominant-function-producing) tritone.Thus the pantonal IFUNC(X) = paambbnccseedqqtz, and pantonal IFUNC(X, Y) = paambbnccseedqqtz). Note again: The sub/superscripts still have the same letters in the IFUNC(X) and IFUNC (X, Y) versions above, but are not related... I was just lazy and cut-and-pasted. Hey, that still rhymes!
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