La Monte Young's piano composition The Well Tuned Piano is based on a 12- tone, justly tuned scale that features the “flat” 7th partial in its design. A complete description of the scale can be found in La Monte Young's The Well- Tuned Piano (Gann) by Kyle Gann. Young's scale (Table 15-4) is a seven limit tuning, which means that ratios defining the scale degrees contain no prime numbers greater than seven.
Table 15-4. Step ratios in Young's scale.
Eb E F F# G# G A
1/1 567/512 9/8 147/128 1323/1024 21/16 189/128
Bb B C# C D Eb
3/2 49/32 441/256 7/4 63/32 2
This tuning has a number of interesting features. First, it contains a wide variety of interval sizes, ranging from the smallest, a 27 cents half-step between E and F, up to 204 cents, the just major second between E-flat and F. The scale's steps are also grouped into a pentatonic clustering around E-flat, F, G, B-flat and C in the standard chromatic scale. Another interesting feature of the scale is its use of the “flat” seventh harmonic to establish its consonant thirds relations. Young's tuning uses a 9/7 major third between E and G-sharp, G and B, A and C-sharp, D and F-sharp, and a 7/6 minor third between F-sharp and A, B and D, and C-sharp and E. Another unusual feature is that its G-sharp is lower than its G, and C-sharp is is lower than C! These pairs of tones were reordered to create 3/2 perfect fifths between the F-sharp, C-sharp and G- sharp spellings.
Example 15-11. Young's scale.
(new tuning :name 'young :ratios '((1 ef) (567/512 e) (9/8 f) (147/128 fs) (1323/1024 gs) (21/16 g) (189/128 a) (3/2 bf) (49/32 b) (441/256 cs) (7/4 c) (63/32 d) 2) :octaves '(-1 10)
Young's scale is based on an E-flat that is actually closer to D in the standard chromatic scale. But the sixth scale degree, A, is tuned to exactly 440.0 Hz, which means that the scale's lowest E-flat can be calculated by multiplying the lowest A in the standard chromatic scale by 128/189, the reciprocal of the sixth degree's ratio. The scale degree ratios in Example 15-11 are listed in monotonically increasing order, which means that the symbols that define the G-sharp and C-sharp note names appear before their natural versions. The last ratio in the list is 2, which establishes the octave ratio without defining a redundant note name.
Interaction 15-12. Values in Young's scale.
cm> (note 0 :in #&young) ef-1
cm> (hertz 'a4 :in #&young) 440.0
cm> (keynum 'c4 :in #&young) 70
cm> (hertz 'c4 :in #&young) 521.4814814814814
cm>
The following musical examples, implemented by Michael Klingbeil, demonstrate Young's tuning.
Example 15-12. A process to preview Young's scale.
(define (arp-chord notes rate dur amp) ;; arpeggiate a list of notes.
(process with transp = 0 for n in notes
for k = (keynum n :from #&young) output (new midi :time (now) :keynum (+ transp k) :duration dur
:amplitude amp) wait rate))
The following two examples demonstrate the basic scaler and intervalic relations in Young's scale.
Example 15-13. Scale and septimal interval relations.
(defineyoungs-scale
'(ef4 e4 f4 fs4 gs4 g4 a4 bf4 b4 cs5 c5 d5 ef5)) (definesept-7th ; 7/4 seventh
'(ef4 c5))
(definesept-min-3rds ; 7/6 thirds
'(e4 gs4 f4 g4 a4 cs5))
(definesept-maj-3rds ; 9/7 thirds
Interaction 15-13. Listening to the scale and intervals.
cm> (events (arp-chord youngs-scale 0.4 2 .6) "young.mid" :channel-tuning true) "young-1.mid"
cm> (events (play-tuning #&young 48 96 .15 .5) "young.mid")
"young-2.mid"
cm> (events (arp-chord sept-7th 0 3 .6) "young.mid")
"young-3.mid"
cm> (events (arp-chord sept-min-3rds 0 2 .6) "young.mid")
"young-4.mid"
cm> (events (arp-chord sept-maj-3rds 0 2 .6) "young.mid") "young-5.mid" cm> → young-1.mid → young-2.mid → young-3.mid → young-4.mid → young-5.mid
Example 15-14 defines some representative sonorities that Young uses in The Well Tuned Piano.
Example 15-14. Chords from The Well Tuned Piano.
(defineopening-chord '(ef3 bf3 c4 ef4 f4 bf4)) (definemagic-chord '(e3 fs3 a3 b3 d4 e4 g4 a4)) (definegamelan-chord '(fs3 a3 c4 e4)) (definetamiar-dream-chord '(b2 d3 g3 a3 b3 d4 g4 a4)) (definelost-ancestral-lake-region-chord '(g2 b2 d3 fs3 g3 a3)) (definebrook-chord '(bf3 c4 ef4 f4 g4 bf4 c5 ef5 f5 bf5)) (definepool-chord '(ef3 f3 fs3 bf3 c4 ef4 f4 bf4))
Interaction 15-14 demonstrates the Opening Chord, Magic Chord, Gamelan Chord, Tamiar Dream Chord, 89/89 Lost Ancestral Lake Region, The Brook, and The Pool chords:
Interaction 15-14. Sonorities from The Well Tempered Piano.
[Class]
cm> (events (arp-chord opening-chord 0.25 3 .6) "young.mid" :channel-tuning true) "young-6.mid"
cm> (events (arp-chord magic-chord 0.25 3 .6) "young.mid")
"young-7.mid"
cm> (events (arp-chord gamelan-chord 0.25 3 .6) "young.mid")
"young-8.mid"
cm> (events (arp-chord tamiar-dream-chord 0.25 3 .6) "young.mid")
"young-9.mid"
cm> (events (arp-chord lost-ancestral-lake-region-chord 0.25 3 .6)
"young.mid") "young-10.mid"
cm> (events (arp-chord brook-chord .15 5 .6) "young.mid")
"young-11.mid"
cm> (events (arp-chord pool-chord .15 5 .6) "young.mid") "young-12.mid" cm> → young-6.mid → young-7.mid → young-8.mid → young-9.mid → young-10.mid → young-11.mid → young-12.mid
Modes
A mode in Common Music is a scale that defines a transposable subset of a tuning. Adjacent degrees in the mode may be non-contiguous in the tuning.
mode
mode supports the following slot initializations: :name {string | symbol}
An optional name for the mode.
:tonic note
Sets the tonic note name of the mode. Defaults to the first note in the steps specification, or C if the steps are described as intervals.
:scale tuning
Sets the tuning system for the scale. Notes in the mode must also be notes in the tuning. Defaults to the standard chromatic scale.
:steps {notes | intervals}
One octave of the mode specified as a list of notes or list of intervals between notes. If steps is a list of notes the tonic (transposition offset) of
the steps are specified as a list of intervals the transposition is set to C. A mode can be transposed to a new tonic using the transpose function.
Functions like hertz, keynum, note and transpose accept modes as well as
tunings as scale inputs:
Interaction 15-15. Transposing modes.
cm> (new mode :name 'dorian :steps '(d e f g a b c d)) 1
cm> (keynum 'd4 :in #&dorian) 35
cm> (loop for i from 35 to 42
collect (note i :in #&dorian)) (d4 e4 f4 g4 a4 b4 c5 d5)
cm> (transpose #&dorian 'cs) 2
cm> (loop for i from 35 repeat 45
collect (note i :in #&dorian)) (cs4 d4 e4 es4 fss4 gs4 as4 b4 cs5) cm>
To demonstrate the use of modes we define a process that harmonizes random chords on Messiaen's Modes of Limited Transposition.
Example 15-15. Messiaen's Modes of Limited Transposition.
(new mode :name 'mode1 :steps '(c d e fs gs bf c)) (new mode :name 'mode2 :steps '(c df ef e fs g a bf c)) (new mode :name 'mode3 :steps '(c d ef e fs g af bf b c)) (new mode :name 'mode4 :steps '(c df d f fs g af b c)) (new mode :name 'mode5 :steps '(c df f fs g b c)) (new mode :name 'mode6 :steps '(c d ef fs gs as b c))
(new mode :name 'mode7 :steps '(c df d ef f fs g af a b c)) (definechords '((0 2 4 5) (0 1 3 5) (0 2 3 4)))
(define (messiaen mode start end rate) (let ((num (length chords)))
(process for m from start to end
for c = (list-ref chords (random num)) for l = (note (transpose c m) :in mode) each n in l
output (new midi :time (now) :keynum n
:duration (* rate 2)) wait (odds .15 (* rate .5) rate))))
The messiaen process selects chords at random from a list of possible chord
templates defined in the global variable chords. Each chord template is a list of intervals that will be transposed to specific key numbers in the mode specified to the process. The wait expression uses Common Music's odds function to
H. Taube 05 Oct 2003
randomly produce “added value” rhythms that are characteristic of much of Messiaen's music.
Interaction 15-16. Listening to Messiaen's Modes
cm> (events (messiaen #&mode2 35 50 .4) "messiaen.mid")
"messiaen-1.mid"
cm> (events (messiaen #&mode6 35 50 .4) "messiaen.mid")
"messiaen-2.mid" cm>
→ messiaen-1.mid
→ messiaen-2.mid
Chapter Source Code
The source code to all of the examples and interactions in this chapter can be found in the file scales.cm located in the same directory as the HTML file for this chapter. The source file can be edited in a text editor or evaluated inside the Common Music application.
References
Mathews, M. & Pierce, J. (1989). The Bohlen-Pierce Scale. In Mathews V., & Pierce, J. (Eds.), Current Direction in Computer Music Research (pp.165-174). Cambridge: The MIT Press.
NIL (1972). Harvard Dictionary of Music. In Apel, W. (Eds.), NIL (pp.NIL). Boston: Harvard University Press.
Gann, K. (1993). La Monte Young's The Well-Tuned Piano. Perspectives of New Music, 31(1), 134-162.
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16
Mapping
A mapping is between two things: it is the life of the argument, linking and identifying and relating. The life of music may have some relationship to this concept as well: how else would music be able to teach itself, unless there were a process at work that made connections, showed similarities, identified differences, and conveyed their significance?
— Edward Rothstein, Emblems Of Mind: The Inner Life of Music and Mathematics Proportion, juxtaposition and balance between musical ideas are fundamental relationships in music, and transformations that affect these properties permeate the craft of composition. Scaling and offsetting are two simple yet powerful ideas that composers use to transform musical ideas. In this chapter we examine some computer techniques related to scaling and offsetting and demonstrate how they can be used to control the procedural description of parameterized sound data.