Chromatic Scales with Linear Intervallic Expansion (LIE)
My initial focus is on abstract scales derived from Gaussian runsums (natural and triangular numbers) that produce recursive sequences (see Srinivasa Ramanujan – the man who heard infinity).
1/ Every scale that expands from a single PC will produce a descending retrograde inversion of the ascending scale.
“Morley’s categoricity theorem… a fundamental result in linear algebra is that any two vector spaces that have the same dimension and same field of scalars are isomorphic.” (Justin Tatch Moore PNAS )
2/ Every scale that expands from central adjacent PC’s (e.g., E & F) with +1 & -1 IC’s occurring simultaneously, will only utilise the PC’s of Messian’s Oct (0,1), [E, F, G, G#, Bb, B, C#, D,] as follows;
Central Point: (E+ F-) = +/- 1
Ascending: E +1 F +2 G +3 Bb +4 D +5 G +6 C# +7 G# +8 E +9 C# +10 B +11 Bb +12 Bb +13 B +14 C#…
Descending: F -1 E -2 D -3 B -4 G -5 D -6 G# -7 C# -8 F -9 G# -10 Bb -11 B -12 B -13 Bb -14 G# …
If E & F are regarded as the closest PC’s to a quartertone nil (zero) point of E+ or F-, then the scale would be written;
Ascending: F +1 F# +2 G# +3 B +4 Eb +5 G# +6 D +7 A +8 F+9 D +10 C +11 B +12 B +13 C +14 D…
Descending: E -1 Eb -2 C# -3 Bb -4 F# -5 C# -6 G -7 C -8 E -9 G -10 A -11 Bb -12 Bb -13 A -14 G …
Extended chromatic principles:
1/ All equal tempered/proportioned frequencies (chromatic pitches divided by semitonal/IC1 steps) constitute a potentially infinite chromatic world beyond Octave Equivalence (OE), Pitch Class (PC), & human hearing ranges.
- (a) all pitches within human hearing range are (a subset of this possible world) .
- (b) all pitches within any given instrument or ensemble range are a subset of a (acousmatic means may be employed to extend b to a)
2/ LIE scales are pitch collections derived from linear intervallic expansion (LIE), i.e. runsums (+1 +2 +3 +4…)
- (a) all scales have 3 possible starting points from which to expand – top, bottom, & middle of range (in both directions), analogous to modes, retrograde, & inversion (alternative potential weight/identities; +2 +3 +4…, +3 +6 +9…, 1/4 + 1/2 + 3/4…)
- (b) as with everything we perceive, all scales should (& potentially do) expand from a central point of reference, in that any outer edge may also be regarded as a central point, as anything out of range, beyond (above or below) the outer edges is potentially (numerically at least) viable. What David Lewen describes as “larger formal spaces that are theoretically conceivable while musically impractical.” (cited in Dmitri Tymoczko, 2010, p12)
- (c) scales may be derived from the range of the instrument(s) playing them
3/ All unused (non-collected) pitches within any given range are here defined as the “Outagraphic” (a term inspired by & derived from “outagraphy“, a surrealist photographic device that removes the principle object from the frame of reference, a concept attributed to the surrealist artist, musician & poet – Ted Joans); the Outagraphic might also be thought of in terms of “Counterpart Theory”, and the SORITES PARADOX, i.e., when is a heap not a heap?
If the main aesthetic is punctualist, then intervallic areas of the outagraphic might also act as a topos/region potentially populated by glissandi & microtonal elements, a get out of jail free card or frottage?
Lie Scales & The Outagraphic are thus a macquette, compositional code base, a chroma palette for temporal/rhythmic experimentation – structured improvisation! Scoring ostensibly tempered (but abstracted & extended) chromatic based music for orchestra’s and traditional ensembles is a means of rerouting the historical concepts of tonality and melody, timbre and temporality. In the performance of this music, the orchestra/ensemble becomes complicit (ironically, if you like) in the acknowledgement of it’s own demise, whilst potentially extending it’s contemporary repertoire (relevance), and simultaneously acting as a deceptive détournement – the medium is/was (and could be) the message.
Arguably, the same could be said for the deployment of any extended technique and alternative system, but by focusing on punctualism, any legato and glissando moments (accidental, aleatory and/or prescribed) may take on syntactical melodic significance within our encultured reification and deification of historically tonal, pantonal and chromatic equivalent systems. The concept of klangfarbonmelodie (coupled with the minimalists use of repetition) thus becomes a minor détournement in itself, while the listener searches for, creates, and/or imagines a syntax, unique internalised timbrel collections are able to flourish – providing a plethora of diverse signals for said listener. In our saturated world of the visual Hegemonic, tapping a violin with a shovel (someone elses fingernail would be spectacular) or bowing a urinal (I believe) carries less polemic significance than music about music, sound about sound (everything); although the image of Cardew feeding hay to a piano is a wonderful one, with hindsight, Nancarrow’s player piano pieces (for example) perhaps retain more polemic potential.
Bringing temporal elements into any system (Cage, Patch, Boulez et al) post “Total serialism”, Music of Changes, and John Coltrane, runs the risk of merely reinforcing the temporal status quo, the post industrial 9-5, therefore I have opted for the juxtaposition of all/any with none (justified with reference to Heisenberg’s “uncertainty principle”), which one might only get away with at fast tempi – as while we contemplate Feldman’s systems, the powers that be are stealing our lunch money. Change happens NOW! So how can one mention Coltrane and not abandon notation in favour of improvisation? Apparently, Coltrane carried two books around with him on tour, the Koran, and Nicola Slomynskies Thesaurus of scales…
SPATIO-TEMPORAL SOUND & MA
Sound itself, in the moment, is SPATIO-TEMPORAL (consisting of both frequential/harmonic/vertical & temporal/horizontal elements), a slice of presentism, auralising the space of temporality with the temporality of space. Whereas, for example, Toru Takemitsu’s – Far Away for Piano (1973) uses sustain to emphasize the “space between”, I’m experimenting with Punctualism & silence to reveal “MA” (Japanese expression), the space between spaces…
Wabi-Sabi – (a Japanese philosophy of aesthetics)
“Wabi is philosophy of spatiality, direction or path; Sabi is an aesthetic of objects and their possession of time. Both of them forms of space and time combination” (niniasverdrup.com).
OCTAVE EQUIVALENCE (OE) & Octave stretch:
OE tends to be generally universal, however, “Octave adjustment with tone pairs that are two or more octaves apart… appears to require some musical experience and talent… taking into account that the intervals between the pitches of a harmonic complex tone must be expected to be stretched by pitch shifts. Due to this stretch of the pitch pattern, the best match of the two pitch patterns is obtained for an oscillation-frequency ratio that, on the average, somewhat exceeds the value 2:1” (Ernst Terhardt, 2000).