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

Quick Navigation of this Document:

• Mozart String Quartet Example
  • Score
  • MIDI File
  • Diatonic Lattice
• How to Construct a Diatonic Lattice
• How to Analyze the Lattice
• Just Intonation and the Diatonic Lattice

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

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

• Place the pitch classes of each chord in the lattice so that they fill appropriate positions in a circular array of seven diatonic degrees in the vertical dimension.
• Any vertical rotation of the lattice is equivalent, but for ease of notation, I always place the tonic pitch on the lowest diatonic step.
• The melodic strands that arise among chord members as they move among these diatonic degrees will become visible along the horizontal dimension of the array.
• Certain strictures pertain to the distribution of pitch classes among the diatonic degrees, such that the ambiguous diatonic situations arising in chromatic harmony can be resolved in a way that makes sense:
  1. The scale degrees are to be spelled only as follows: ^1, ^b2, ^2, ^b3, ^3, ^4, ^#4, ^5, ^b6, ^6, ^b7, ^7. (These are the spellings of the pitch classes that lie the closest to ^1 and ^5 on the infinite line of fifths.)
  2. If this scale-degree spelling results in a non-tertian spelling of a diatonic chord achieved by modal mixture (e.g. III# as ^3, ^b6, ^7), respell the chord so that it is tertian, maintains all common tones, and also moves by diatonic-interval root progression, if possible.
  3. All dominant-function chords, including secondary dominants, diminished seventh chords, and altered dominants, take the scale of the tonicized key as reference.
  4. The only possible enharmonic respellings of common tones are the result of a functional reinterpretation of a held interval between two chords, e.g. m3 becomes A2, m7 becomes A6.
• According to Jones (2002), when a chord progression can be heard as a harmonic prolongation, diatonic passing and neighboring (restricted to only one neighboring scale degree away) patterns can be found extending from all members of both the initiating and terminanting chords of the prolongation. In other words, if you hear a progression as a composed-out harmonic progression used by the composer to expand a single chord serving a larger-scale harmonic function, then passing and neighboring motions will extend from every member of the initiating chord to some member of the terminating chord, and vice versa.
• Each progression that can be shown to have such passing and neighboring motions is called a pervasively fluent passage (or PFP). (Introduction to Schenkerian Analysis provides a more rigorous definition of pervasive fluency.)
• It is important to note that, while foreground prolongations should always be PFPs, not all PFPs will necessarily be prolongations that are worth exploring. The finding of prolongations should be left to the musical judgment of the analyst (not the algorithmic processes of a computer).
• The PFP Finder Applet is thus not intended so much to facilitate the finding of PFPs as it is intended to allow the user to experiment with the prolongations that he or she hears in the music. To that end, the MODVLO7 program is designed so that the passing and neighboring lines between any selected "concatenation" of two chords are retrievable, and a hierarchic interpretation according to the selected chord concatenations is representable.

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