Prolongation and the Diatonic Lattice:
An Introduction for Use with the PFP Finder Applet
Using the Mozart String Quartet, K. 465, as an Example

Robert T. Kelley


For background information on prolongational analysis, see my Introduction to Schenkerian Analysis and schenkerguide.com. The term "Pervasively Fluent Passage" (PFP) is defined here. If you need more information in order to understand pervasive fluency, prolongation, and the purpose of the PFP Finder, please feel free to .

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Mozart: String Quartet in C major ("Dissonant"), K. 465, Introduction

Score

MIDI File

Diatonic Lattice:

(T = 10, L = 11)

Chord No.
  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

  |  |  |--L--L--L--L--L--L--L--L  |--T  |--T--T--T--L  |  |  |--T  |--T  |  |  |--T  |  |  |  |  |  |--T--T--T--T--T  |  |  |  |  |  |  |--L  |  |  |  |  |--L  |  |--T--L  |  |--T  |--L--T  |  |  |  |  |--L  |  |  |--L
--8--9--9--9  |--9  |--9  |--9  |--9  |--9  |  |  |--9--9--9--9--9--9--9--9  |--8  |--8--8--8--8--8--9  |  |  |  |  |--8--8--8--8--8--8  |  |  |  |  |  |  |  |--8--9  |  |  |  |  |--9--8  |--9--9--8--8--8--8--8--8--8  |
  |--7  |--7--7--7  |  |--7--7--7  |--7  |--6--7--7--7  |--7  |--7  |--7  |--7  |--7  |  |  |--7  |  |--7--7--7--7--7  |  |  |  |  |  |--7--7--7--7--7--7--7--7  |--7--7--7--7--7--7  |  |--7--7--7--7--7--7  |  |  |  |--7
  |  |--6  |  |  |--6--6  |  |  |--6  |  |  |--5  |--5--5--5  |  |--5--5--5  |--5  |--5  |--5--5--5--5--5--5  |--5  |--5--5--5--5--5--6  |  |--5  |  |  |--5--5--5--5--5--5  |  |  |  |  |  |  |  |  |  |  |--5--5--6--6  |
--3--3  |  |  |  |  |  |  |  |  |  |  |  |  |  |--4  |  |  |--4--4  |  |  |--4  |  |  |--4  |  |  |  |  |  |--3--3--3--3--3  |--3  |  |--3--3--3--3  |--3  |  |  |  |  |  |--4--4--3--3  |  |  |  |--3--4--4  |  |  |  |  |
  |  |--2  |--2--2--2  |--2  |--2--2--2--2--1--1  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |--2  |  |  |  |  |--2--2--2--2--2--2  |  |--2  |--2--2--2  |--2--2--2  |  |  |--2--2  |--2  |--2  |  |  |--2  |--2
--0--0--0--1  |  |  |--0  |--0  |  |  |--0  |  |--0  |--0--0--0  |--0  |--0--0--0--0--0--0--0--0--0--0  |  |  |  |  |  |--0  |  |  |  |  |  |--0--0--0--0--0  |--0--1  |  |--0--0--0--0  |  |--0--0--0  |--0  |--0--0--0  |

  |_____|_____|_______________________|_____|_____|_____|_______________________|_____|_________________|_____|________|_____|___________|________|___________|___________|_____|_____|__|_____|_________________|________|

  1  3  5 13 15 17 19 27 35 37 40 42 46 49 53 59 61 62 64 70 73

  |  |--L--T--T--T  |  |--T--T  |  |  |  |--L  |  |--L  |  |--L
--8--9  |  |  |  |--9--8  |  |--8--8  |  |  |  |--9--8--9--8  |
  |  |--7--7--6--7  |  |--7--7  |  |--7--7--7--7  |  |--7  |--7
  |--6  |  |  |  |--5--5--5  |--5--5  |  |--5  |  |  |  |--5  |
--3  |  |  |  |--4  |  |  |--3--3  |--3--3  |--4--3  |  |  |  |
  |--2--2--2--1  |  |  |  |  |  |--2--2  |--2  |  |--2  |  |--2
--0--0  |  |  |--0--0--0  |  |  |  |  |--0  |--0--0  |--0--0  |

  |___________|________|____________________|__|______________|

  1 15 27 53 59 73

  |--T  |--L  |--L
--8  |--8  |  |  |
  |--6  |--7--7--7
  |  |--5--5  |  |
--3  |  |  |--4  |
  |--1  |--2  |--2
--0  |--0  |--0  |

  |________|__|__|

  1 53 59 73

  |--L  |--L
--8  |  |  |
  |--7--7--7
  |--5  |  |
--3  |--4  |
  |--2  |--2
--0  |--0  |

How to Construct a Diatonic Lattice

How to Analyze Music on the Diatonic Lattice

Study of the Mozart string quartet example shown above will offer insights into how the use of chord concatenations may reveal the fundamental harmonic structure underpinning the passage, and how the harmonic structure is contrapuntally elaborated at the music's surface. First, one must be able to read the diatonic lattice as a representation of the pitch-class content of the music. (A chart of the pitch class numbers for the twelve chromatic pitches can be found here.) The music's pitch classes are distributed among the seven diatonic scale steps that define any tonal key. Multiple listenings to the MIDI file may thus be necessary before one may be able to imagine the music while reading the lattice representation.

Next, copy and paste the surface-level lattice given above into the PFP Finder Applet to see the PFPs. (An analysis may also be accomplished using MODVLO7. Simply convert the MIDI file into a Sequence file using Scala, and load it into MODVLO7.) (Introduction to Schenkerian Analysis provides a definition of pervasive fluency.) Consider my reasons for selecting the PFPs that I have used to concatenate each level rather than any of the many other PFPs that the PFP Finder reveals. Reasons may include eliminating melodic ornamentation around one harmony, concatenating chords with the same harmonic function (or the same chord) that begin and end a progression, and choosing the PFPs that are defined by all of the most significant (outer voice) surface-level passing and neighboring voice motions.

Take note of the passing and neighboring functions that create the PFPs that have been selected on the graph above for concatenation. Listen for these voice-leading connections when playing the MIDI file. Repeat this process on each hierarchical level given above. Finally, try to perceive all of the passing and neighboring motions at all levels and how they establish harmonic transience and stability as you listen to the MIDI file one more time.

Homomorphism from 5-limit Just Intonation to the Diatonic Lattice

Given that a musical passage can be rendered on the diatonic lattice, one can generalize that diatonic interpretation into 5-limit just intonation using the formula given below. In the formula, q is the just-intonation frequency ratio between the pitches of an interval found on the diatonic lattice; h12 is the directed pitch class interval between the notes (((pc2 - pc1)+12)mod 12); h7 is the directed step class interval between the notes (((latticeposition2 - latticeposition1)+7)mod 7); and h3 is (3 * (-11 * h12 + 19 * h7) + 5 * (7 * h12 - 12 * h7)) - nint((7 * h12 - 12 * h7)/5), where nint() is the nearest integer function, or rounding off.

q = (25/24)^h12 * (128/125)^h7 * (81/80)^h3

Likewise, any 5-limit ratio q has a specific diatonic interpretation presentable on the diatonic lattice by the formulae given below. The interval must be rendered as powers of 2, 3, and 5. For example, for q = 6/5, q = 2^1 * 3^1 * 5^(-1). Below, v2, v3, and v5 represent the exponents of this expansion of q.

h12 = 12*v2 + 19*v3 + 28*v5
h7 = 7*v2 + 11*v3 + 16*v5

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2002 Robert Kelley