Linear And Non-linear Interval Space

I Introduction A. Overview

This article examines two mathematical spaces which can be constructed using the twelve canonical interval classes. Each is derived from the minimum interval sets found in dyads and trichords. The first elaborates these germinal sets in a linear, isotropic format, while the second elaborates them in an non-linear, anisotropic format. Both spaces prescind from any hierarchical ranking of these minimum structures. They parsimoniously inventory the intervals and trichords without indicating their relative prominence or value, whether acoustic, cognitive, or idiomatic.

The two representations diverge radically in the means by which they do so. The first replicates the inventory of minimum structures symmetrically around every pitch. The second disperses this inventory maximally, but recursively. As a result they have distinct topologies. B. A Preliminary Inventory of Interval Classes

Our ability to discern simple frequency ratios distinguishes pitch as an authentically discrete space, in contrast to other musical parameters such as loudness which are mere continua. Ratios relate discontinuous frequencies, and thus striate the pitch continuum with discrete thresholds; while the difference between two frequencies can vary continuously, pitches also exhibit discrete difference insofar as they inhabit coordinates within this array of thresholds.

The interval spaces described in this article concern these discrete differences that derive from rational or harmonic frequency relationships. They can be derived from the very simplest ratios, using factors of two and three alone. Frequencies differing by factors of two form pitch classes, while twelve degrees of rational or harmonic difference between these pitch classes can be arranged in order of increasing exponential difference by factors of three. The resulting relationships are captured in modular arithmetic. Whereas the traditional interval names (used occasionally in this paper as a more intuitive shorthand) measure continuous difference in discrete increments (scalar steps), our arithmetic measures the authentically integral differences produced by the most elementary ratios. (P4 up)

(M2 down) (m3 up)

(M3 down)

–2

(Octave) –1

0

+1

(m2 up)

+2 (M2 up)

+3 (m3 down)

–3

–4

(P4 down)

–5

±6 +5

(Tritone)

+4

(M3 up)

(m2 down)

This arithmetic clearly does not rank the interval classes in terms of consonance, as it neglects frequency relationships involving higher prime numbers which effectively intersect those produced by compounding factors of three. However, it does illuminate certain group-theoretic relationships, which we now review briefly. Octaves and unisons give us zero harmonic difference; they are the additive identity, 0, in our arithmetic. If we assign ‘fifths’ the role of the multiplicative identity, +1, ‘fourths’ assume its inverse, -1. We could swapping their respective roles without effecting our observations.

Of the values given in modulo twelve arithmetic, +5 and -5 are the only values that twelve does not divide into. A consequence of this incommensurability is that 5 effectively functions as a second unitary value: by taking every fifth value in our circle of interval classes, one derives an alternate ordering:

-2

–1 +2

–5

0 0

+5

–3 –3 –4

+4

–1 –5

+1 –2

+2

+3 +3

±6 +1 ±6

–4 +5

+4

The values 1 and 5 are effectively interchangeable as multiplicative identities, that is, units: they are both sufficient to generate and measure every other interval class. This dualism is evident in both of the interval spaces we will examine. Five is the arithmetic value of a half-step. Thus, if +1 and -1 are ‘harmonic’ units which represent the simplest harmonic relationship, then +5 and -5 are the ‘gradient’ units which represent the smallest continuous difference in frequency that attains an integral harmonic difference.

In the above diagram, the inner circle indicates the order of intervals as they increase by halfsteps, that is, five harmonic units at a time. If we assigned half-steps the value 1, then it would indicate the order of intervals as they increase by fourths or fifths. Each type of unit is equal to five of the other unit: the smallest harmonic increment is equal to five gradient increments, and the smallest gradient increment is equal to five harmonic increments.

C. A Preliminary Inventory of Proportional Classes

Three intervals coincide three pitches. While intervals compare pitches, trichords compare intervals. They present the array of proportions that comprise the genetic basis for any elaboration of intervallic structure, as will be made clear in the course of this article.

If the two pitch classes of a dyad constitute an interval class, we can call the three interval classes of a trichord a “proportional class”. There are thirty-one of them. Because their inversion is ambiguous in the absence of ordering, we will label them in terms of the absolute value of their intervallic constituents. Twelve proportional classes include the interval class 0, and so duplicate a pitch class:

Five additional proportional classes duplicate an interval class:

Beyond these proportional classes, there remain 14 intervallically non-redundant proportional classes, which can be grouped in 7 inverse pairs:

Any two non-redundant proportional classes share at least one interval class, and so can be derived from one another by the replacement of a single pitch class. In seven instances, two proportional classes share two interval classes, as represented in the following diagram:

We will refer to neighbors within this diagram as ‘co-derivative’ proportional classes. We can illustrate this relation by repeatedly transposing a trichord by one of its constituent intervals. The latter then establishes a series of intervallic frames:

A single interpolated pitch completes the trichord in each frame. The interpolated pitches themselves are separated by the same interval, and frame inverses of the given trichord.

If we take the E which is interpolated within the first frame above, and instead relate it to the adjacent frame, that is, to G and C, the result is a coderivative proportional class, (1,3,4). In general, if the interpolated pitches are shifted to occupy the adjacent frames, the result will be a co-derivative proportional category that shares the framing interval. Below we show the interpolated pitch shifting across four adjacent frames to give us the proportional classes containing the framing interval category, |1|, in the order they appear in the co-derivation diagram:

Each interval class follows a different distribution in the diagram. The upper row orders the proportional categories that contain 1: |1,2,3| - |1,3,4| - |1,5,4| - |1,5,6|. The lower row orders the categories that contain 5: |5,2,3| - |5,3,4| - |1,5,4| - |1,5,6|. The square circuit on the left orders the categories that contain 3: |1,2,3| - |1,3,4| - |5,3,4| - |5,2,3|. The triangular circuit in the middle orders the categories that contain 4: |1,3,4| - |1,5,4| - |5,3,4|. On the very left we have two categories that contain 2. When any of these intervals is used to frame a series of transpositions, the associated sequence of co-derivative categories containing that interval appears in the resulting series of neighborhoods. Finally, there are two non-redundant proportional classes which coderive themselves: |1,5,6| and |2,4,6|. The latter is the only non-redundant proportional class that contains no unitary interval classes; as such, it is co-derivationally inert, that is, disconnected from every other proportional class.

II Linear Interval Space A. Intervallic Paths and Proportional Fields

Linear pitch sequences are those generated from a constant, for instance, a single pitch class, a single interval class, or a constant degree of difference between consecutive interval classes. We will assemble our linear interval space out of pitch sequences generated with a single interval class. These sequences saturate a single linear dimension by reiterating a single interval:

Insofar as these lines can be traversed in either direction, the interval class in question always appears with its inverse. We will refer to this pair as an interval category.

The diagrams above represent the interval category |2| in three forms. The germinal representation displays the minimum collection of pitches required to instantiate the category. The nuclear representation displays a central pitch along with the neighborhood of adjacent pitches provided by the two inversions of the interval category. Lastly, the isotropic representation displays this neighborhood for each pitch class. Some proportional classes are already visible in a single linear dimension. If we assume that as a ‘null’ interval class, 0 has no linear distance, but simply relates a point to itself, proportional classes that include 0 can be captured in the germinal representation of an interval category. Proportional classes that include a redundant interval class can be captured in the nuclear representation of an interval category. To represent non-redundant proportional classes - those which include three unlike interval classes - we will have to accommodate three independent linear dimensions. By arranging the three pitch classes of a trichord symmetrically, we arrive at a germinal representation that integrates the linear dimensions on a plane:

In the corresponding nuclear representation, these three linear dimensions intersect at a single pitch class, and also form a circuit around six peripheral notes. The central pitch class is surrounded by six trichords of which it is a constituent:

This nuclear pitch takes on one of the three possible functions in the proportional class; in this example, it is separated from two other pitches by the interval classes +1 and +3, -2 and -3, or -1 and +2. Simultaneously, it completes three forms of the inverse proportional class; each upward pointing triangle forms a different transposition of the +(1,2,3) proportional class, while each downward pointing triangle forms a different transposition of its inverse, the -(1,2,3) proportional class:

As with the interval classes, we will speak of “proportional categories” in order to conflate the proportional classes and the inverses that co-occur with them in linear interval space. Co-derivative proportional categories - those sharing two out of three interval categories - appear in the nuclear representation of a given proportional category as 120-degree angles:

In the isotropic representation of a proportional category, each pitch is situated at the intersection of all three linear dimensions:

Just as each pitch is adjacent to six others, each trichord is adjacent to six transpositions of itself, and six transpositions of its inverse. Out of these twelve adjacent trichords, it shares an edge with three inverses and a point with the others. A shared edge denotes two shared pitches; each edgeadjacent inverse replaces one pitch class such that the interval categories that separate it from each side of its intervallic frame are swapped:

A shared point denotes one shared pitch class - each of these inverses simply inverts the interval classes that separate it from the other two pitch classes:

C. Tetrachords and Intervallic Equilibrium

Six intervals coincide four notes. Hence, an intervallically non-redundant tetrachord will contain the full inventory of interval categories. There are only four such tetrachords, which form two pairs of inverses:

As with non-redundant trichords, we can obtain a germinal representation which assigns a unique directional orientation to each of the constituent interval and proportional categories by arranging four points equidistantly. Four intersecting triangles and six intersecting lines are immediately visible in the resulting tetrahedron:

If we extend just three of the linear elements in the tetrahedron, we see that as with trichords, tetrachords are interwoven with their inverses in linear interval space:

In a tetrachord each pitch complements one of four trichords; thus each tetrachord incorporates four proportional categories. Although the two inverse pairs of non-redundant tetrachords contain the same gamut of interval categories, their proportional inventories differ: Proportional Format A

|5,2,3|

|1,5,6|

|1,3,4|

|2,4,6|

|1,2,3|

|2,4,6|

Proportional Format B

|5,3,4|

|1,5,6|

Non-redundant tetrachords exclude co-derivative proportional categories: each of the constituent trichords shares just one interval class with each of the others, just as in a tetrahedron, any two triangular faces share one edge. If any of the pitches in one of these constituent trichords are replaced by the complementary fourth pitch, two of the interval classes necessarily change as well, for each pair of pitches is separated by a unique interval category. Below we reproduce our diagram of co-derivative proportional categories twice, circling the four categories that comprise each proportional format: Proportional Format A

|1,2,3|

1,3,4

5,2,3

|5,3,4|

|1,5,4|

Proportional Format B

1,5,6

2,4,6

1,2,3

|1,3,4|

|5,2,3|

5,3,4

|1,5,4|

1,5,6

2,4,6

The full inventory of proportional categories is only obtained in the corresponding nuclear representation, in which six intervallic paths and four proportional fields intersect symmetrically at a single point:

Proportional Format A

Proportional Format B

The peripheral pitch classes form a polyhedron known as a cuboctahedron. Buckminster Fuller coined the term ‘vector equilibrium’ to refer to this polyhedron when considered together with the internal lines that connect its vertices to a central point. After Fuller, we will use the term ‘interval equilibrium’ to evoke the essential property of this figure for our purposes: it renders every non-zero interval class, hence every other pitch class, adjacent to the nuclear pitch class. The four proportional classes of each tetrachord appear in the associated interval equilibrium as four hexagons arrayed symmetrically around the nuclear pitch class. The remaining, co-derivative categories appear as 120-degree angles, completing the full inventory of proportional categories. Note that the category |1,5,4| is group-theoretically precluded from non-redundant tetrachords. It only appears as a 120-degree angle in non-linear interval space because it never appears in non-redundant tetrachords, only in transpositions of the latter which share a pitch. Any two transpositions of a non-redundant tetrachord by the interval categories |1|, |5|, or |4| will contain the proportional category |1,5,4|. The duality of proportional formats reflects the duality of harmonic and gradient units, that is, the ability of either |1| or |5| to function as a unit that can multiply into every other interval. In fact, the relationship of the two interval equilibria is equivalent to the relationship of harmonic and gradient units. To demonstrate, we can superimpose the gradient intervallic cycle upon the harmonic interval cycle, thereby assigning each interval class with a counterpart. Each interval class is equal to five of its counterparts (commutatively).

-2

–1 +2

–5

0 0

+5

–3 –3 –4

+4

–1 –5

+1 –2

+2

+3 +3

±6 +1 ±6

–4 +5

+4

If each interval class in one of the interval equilibria is replaced by its counterpart, the other interval equilibrium results. Likewise, each proportional category in one interval equilibria has a counterpart in the other interval equilibrium by virtue of this substitution:

If |5| appears in a given proportional field, |1| appears in its counterpart in the other interval equilibrium, and vice versa. If one of the two unitary interval categories appears in a given proportional domain, it has an unlike counter-domain; if both or neither of the two unitary interval categories appear, then it is its own counter-domain (to be precise, it inverts in the other interval equilibrium: the triangulated plane is turned ‘upside-down’, switching the trichords with their inverses).

Each edge in a tetrahedron is separated from an opposing perpendicular edge. Equivalently, within a tetrachord, any given interval class is disjunct from one other interval class: that which separates the two remaining pitches. Of the six interval categories in the tetrachord, there are then three pairs of disjunct categories. However, all six interval categories intersect in the nuclear pitch of an interval equilibrium. The latter combines each of these pairs in an additional rectilinear plane of symmetry:

In total, then, the interval equilibrium has seven planes of symmetry:

C. The Isotropic Interval Matrix

In the terminal elaboration of linear interval space, every pitch is identically situated at the intersection of the six linear paths and seven planar arrays found in the interval equilibrium. The geometry of this isotropic expanse is known in crystallography as the ‘face-centered cubic lattice’; again following Fuller’s more evocative term ‘isotropic vector matrix’, we will call it the ‘isotropic interval matrix’. To construct it, we can start with one of the of triangular planar arrays. When

we juxtapose each of the trichords of a given inversion with the complementary fourth pitch class, situating the new pitches equidistantly from each of the three pitches they complement, the new pitches form a planar array which is parallel to the first (hence of the same proportional category):

The pitches that complement the inverse trichords are located in the opposite direction - if the complementary pitches of one proportional class form a superjacent layer, then the complementary pitches of the inverse proportional class will form a subjacent layer. Further planes can be stacked in both directions infinitely. We can outline the atomic cells that constitute the isotropic vector matrix by connecting the pitches of the superjacent layer to the adjacent pitches in the first layer. Firstly, an array of tetrahedra points from the original triangular array towards the viewer. Secondly, an array of inverse tetrahedra points from the superjacent triangular array away from the viewer. Finally, the pockets remaining between the tetrahedra each comprise octahedra:

Each trichord constitutes the face of a tetrahedron in one direction, and the face of an octahedron in the other direction; octahedra are facially adjacent to tetrahedra in all directions, and vice versa. We can visualize the three planar orientations which are not parallel to the page converging on one side of a given proportional class to enclose a tetrahedral space, and diverging on the other side to make room for an octahedral space; the same proportional fields converge and diverge in opposite directions from the inverse proportional class. The octahedra conjoin precisely the interval categories that are disjunct in the tetrachord; each interval category appears twice and is doubly linked to its estranged counterpart in the hexachord corresponding the octahedron:

D. Summary

Linear interval space integrates four degrees of structural complexity, revealed in their germinal form in monads, dyads, trichords, and tetrachords. These four degrees of complexity are situated respectively in zero, one, two, or three dimensions and are homologous to the elements of Eulerian topology, namely, vertexes, edges, and faces, and cells. Each germinal form is intervallically non-redundant, but the corresponding dimensions are then saturated in redundancy through the linear replication of this germinal form. Each new degree of non-redundancy is deferred to a new topological element in a higher dimension. One pitch gives us identity. A second unlike pitch gives us rational difference. A third unlike pitch spaced at unlike interval classes gives us proportionality. A fourth such pitch gives us a complete inventory of the preceding elements. The fourth degree is the terminal, insofar as a pentachord necessarily involves structural redundancy, that is, replicated interval and proportional categories.

While pitches and intervals are ubiquitous in musical discourse, the crucial role accorded to trichords in this geometry bears explanation. The significance of trichords essentially rests on the significance of intervals, which is patent. The identity of a musical passage depends on its intervallic constitution, not on its pitch constitution; thus we hear the transpositions of a musical passage as being identical insofar as the intervallic relationships within it are the same. Two transpositions only sound different only insofar as they are juxtaposed so that we hear a further interval between them. Redundancy refers precisely to this juxtaposition. And if we try to parse a musical passage into the smallest intervallically identical forms recurring in various transpositions, we’re left with trichords; for as we have seen, trichords constitute the highest degree of structural complexity that displays a significant degree of variegation in the absence of intervallic redundancy. If a pitch class is like any other pitch class until it is located within an intervallic framework, the specific relationships that define this framework are ultimately articulated in trichords.

If interval classes and the proportional classes provide the genetic basis for pitch organization in that they provide the possible differences in pitch and the possible proportions between these differences, the interval equilibrium conflates structural distinctions by encompassing every difference and proportion at once. Thus, non-redundant tetrachords are useful simply as the minimum index of the gamut of trichords. The isotropic interval matrix presents no restrictions on the ordering of a pitch set. Rather, it represents the interval classes as twelve cardinal directions surrounding every pitch. Thus, every ordering of every pitch set is a contiguous trajectory in the isotropic interval matrix. In this perspective, the linear paths of the isotropic interval matrix are vectors representing operations which produce pitch sets or pitch sequences of arbitrary complexity. These vectors can operate either as radial or compressive forces. Each interval category radiates away from a given point at a consistent rate of difference; further out from a pitch, these intervallic vectors intersect to form concentric polygonal and polyhedral shells that constrain this linear movement. That is, the linear movement away from a pitch can be shunted in a new direction, exchanging radial energy for an elaboration of structure.

Orthodox serialism, for instance, unvaryingly utilizes only the most compressive form within the isotropic interval matrix, namely, the interval equilibrium. To be more precise, since the interval category |6| replicates the same pitch class on either side of a nuclear pitch class, each instance of a tone row can be obtained in a nuclear pitch class along with eleven surrounding pitch classes - a ‘dimpled’ interval equilibrium. Of course, taken in order, a given series doesn’t necessarily follow a contiguous trajectory around this form, but its pitch classes do form a relative locality within a composition which constitutes this total elaboration and compression of interval structure. The monotonous recurrence of this single intervallic confers a harmonic uniformity on serial music; on the other hand, the sequential permutations of this omnipresent structure, by providing fleeting glimpses of radial momentum, confer a pittance of variety at the most local scale of magnitude.

III Non-Linear Interval Space A. Non-Linear Pitch Sequences

The isotropic interval matrix weaves the twelve linear sequences constructed from an intervallic constant. These sequences are exemplary in that they prescind from proportionality: they are precisely the sequences that have no curvature, that is, no difference between successive interval classes. We could also construct linear sequences of constant curvature, in which successive interval classes change by a consistent value. Whereas the linear elements of the isotropic interval matrix exhibit constant first-order difference and zero second-order difference, these sequences exhibit constant second-order difference and zero third-order difference:

Curvature is effected in this sequence through the covert presence of trichords: each successive step between two adjacent pitches implies a third, mediating pitch, namely, what the first pitch would arrive at if the preceding interval class in the sequence were repeated. The pitch it arrives at instead is always separated by the same interval class, not from the initial pitch, but from the mediating pitch. Ordered trichords measure a change of interval. They can be thought of as minimum curves. The three interval classes assume distinct roles:

They form a minimum hierarchical constituency in which a framing interval is bifurcated into two constituent intervals. Any two of these intervals determines the third. Trichords are proportions in the sense that each ordered trichord corresponds to a unique approximation of the extreme and mean ratio: if x, y, and z are the three interval-classes in a trichord, then we have three reversible orderings corresponding to the proportions x:y::y:z, y:z::z:x, and z:x::x:y.

Using the three interval positions within an ordered trichord as our terms, we now define a family of maximally non-linear pitch sequences, that is, sequences that exclude any static degree of difference. Whereas the linear elements in the isotropic interval matrix exclude curvature in favor of linearity, these continually twisting sequences exclude linearity in favor of curvature in every order of difference. To construct such a sequence, we can start with any ordered trichord. A fourth pitch is added at a distance from the third pitch which is equivalent to the framing interval class between the first and third pitches, accelerating the curvature of the line by encompassing both of the preceding interval classes in a single step. The second, third, and fourth pitches form a new trichord which can be extended analogously, and so forth. Such a sequence displays a perpetually accelerating curvature:

After applying this operation 24 times, one arrives back at the original trichord, forming a cycle of interval classes which accelerates into itself. Two species of such non-linear sequences exist, each of which can be inverted, for a total of four reversible sequences. Whereas the isotropic interval matrices inventory the proportional categories in an unordered format, these sequences inventory them in an ordered format; each of the six permutations of each trichord appears twice.

The sum of two consecutive intervals classes is equal to the interval class that follows them; equivalently, the difference between two consecutive interval classes is equal to the value that precedes them. Thus, the differences between successive values of the sequence is the self-same sequence: the sequence of intervals, curvatures, third-order differences, and so forth, are identical.

Linear sequences all reduce to a constant, that is, zero on some order of difference: a pitch repeats, always changes by the same interval class, the change in consecutive interval classes remains the same, vel cetera. In contrast, non-linear sequences are derived through an operation on the variable quantities of the three positions within each ordered trichord, without any reference to a constant value, and so are not reducible to zero.

While intervallic redundancy was the basic criterion of linear interval space, cardinality is the basic criterion of non-linear interval space. Non-linear pitch sequences include both redundant and non-redundant trichords; rather, what they exclude are trichords which appear in symmetrical sixfold pitch space, fourfold pitch space, trinary pitch space, binary pitch space, and unary pitch space (whole-tone scales, diminished chords, et cetera). If a trichord does not contain a unitary interval category, |1| or |5|, then it does not appear in the sequences above. Pitch manifolds of lower cardinality have their own non-linear sequences:

Among the sequences containing unitary intervals, henceforth ‘twelvefold sequences’, we can observe that the A sequences monopolize the null interval category |0|, while the D sequences monopolize the ‘tritone’, |6|. When a unison is followed by any other interval class, the resultant framing interval class is identical to the latter. As a result, the A sequences also have the monopoly on redundant proportional classes: |1,1,2| and |5,5,2| have a duplicate interval class, while |0,1,1| and |0,5,5| have a duplicate pitch class as well. In each case the redundant interval class is a harmonic or gradient unit; the other redundant proportional classes appear in the non-linear sequences of reduced cardinality. On the other hand, the D sequences contain every instance in which both unitary interval categories appear together in a trichord, that is, every appearance of the |1,5,4| and |1,5,6| proportional classes.

B. Non-Linear Interval Matrices

No specific trichord is necessary to germinate these non-linear sequences - any ordered trichord, operated upon appropriately, will derive one of them. While no single ordered trichord is primary, they are each uniquely located, such that they can each be assigned an indexical address: +A01 +A02 +A03 +A04 +A05 +A06 +A07 +A08 +A09 +A10 +A11 +A12 0 +1 +1 +2 +3 +5 –4 +1 –3 –2 –5 +5 +A13 +A14 +A15 +A16 +A17 +A18 +A19 +A20 +A21 +A22 +A23 +A24 0 +5 +5 –2 +3 +1 +4 +5 –3 +2 –1 +1 –A01 –A02 –A03 –A04 –A05 –A06 –A07 –A08 –A09 –A10 –A11 –A12 0 –1 –1 –2 –3 –5 +4 –1 +3 +2 +5 –5 –A13 –A14 –A15 –A16 –A17 –A18 –A19 –A20 –A21 –A22 –A23 –A24 0 –5 –5 +2 –3 –1 –4 –5 +3 –2 +1 –1 +D01 +D02 +D03 +D04 +D05 +D06 +D07 +D08 +D09 +D10 +D11 +D12 ±6 –1 +5 +4 –3 +1 –2 –1 –3 –4 +5 +1

+D13 +D14 +D15 +D16 +D17 +D18 +D19 +D20 +D21 +D22 +D23 +D24 ±6 –5 +1 –4 –3 +5 +2 –5 –3 +4 +1 +5 –D01 –D02 –D03 –D04 –D05 –D06 –D07 –D08 –D09 –D10 –D11 –D12 ±6 +1 –5 –4 +3 –1 +2 +1 +3 +4 –5 –1

–D13 –D14 –D15 –D16 –D17 –D18 –D19 –D20 –D21 –D22 –D23 –D24 ±6 +5 –1 +4 +3 –5 –2 +5 +3 –4 –1 –5 However, these positions are interwoven due to a striking property: values separated by five steps are identical to values separated by one step. If we take every seventeenth value of a sequence (that is, if we move backwards five steps at a time) the resultant twenty-four value sequence is identical to the original. As a result, every position has a twin somewhere in the same sequence that bears the same value. There are 9 pairs as well as 6 solitary positions that twin themselves. +A01 +A02 +A03 +A04 +A05 +A06 +A07 +A08 +A09 +A10 +A11 +A12 0 +1 +1 +2 +3 +5 –4 +1 –3 –2 –5 +5 +A13 +A08 +A03 +A22 +A17 +A12 +A07 +A02 +A21 +A16 +A11 +A06 +A13 +A14 +A15 +A16 +A17 +A18 +A19 +A20 +A21 +A22 +A23 +A24 0 +5 +5 –2 +3 +1 +4 +5 –3 +2 –1 +1 +A01 +A20 +A15 +A10 +A05 +A24 +A19 +A14 +A09 +A04 +A23 +A18

We can represent this self-intersection of a twelvefold sequence by stacking it in an orthogonal dimension, offsetting it by five indexical positions on each layer, to obtain a two-dimensional matrix which places each pitch class at both of its indexical positions simultaneously:

Non-linear sequences exist not only as pitches, but as sequences of intervallic operators which can modulate a sequence of pitches, that is, measure the distances to a second sequence of pitches. In the above matrix, each row is transposed by the intervals in the +A sequence to obtain the next row, as is each column; thus the matrix is vertically and horizontally saturated by the +A sequence. The addition and subtraction of non-linear sequences can be generalized; the sum or difference between successive values in two non-linear sequences always constitutes a third sequence which is also non-linear. That is, the non-linear sequences form an abelian group. For instance, the table

below shows the sum of each of the twelvefold sequences and the +A sequence, at each indexical degree:

+A1 +A -A +D -D

+A1

___

+A -A +D -D

1

2

3

4

5

6

7

8

9

10

11

12

±[6]01 +A03 –D08 ±[6]03 +[4]03 +A20 ±[6]10 +D06 –A17

±[6]15 –D24 +D11

±[6]09 –A15 +A08 ±[6]14 –[4]03 –D20 ±[6]18 –D18 +D17

±[6]08 +D12 –A11

[1]

+A24 +A02 ±[6]14 +D09 +A11 ±[3]08 –D15 +[4]02 ±[6]08 +A06 +D14

±[2]02 +A12 +D02 ±[3]02 –D21 –D11 ±[6]20 +D03 –[4]05 ±[3]04 –A18 –A14 13

14

15

16

17

18

19

20

21

22

23

24

±[2]01 +D24 –D14 ±[6]06 –A17 +D23 ±[6]04 +A15 +[4]05 ±[6]24 –D24 –D06 ±[3]05 –D03 +A20 ±[6]11 –[4]06 +D08 ±[3]06 –A06 –D05

±[6]23 +A24 +A23

±[2]03 –D12 –A14 ±[6]22 +A21 +A23 ±[6]12 –A03 –[4]02 ±[6]16 –D24 +D18 ±[6]17 +D15 +D20 ±[3]07 +[4]06 –A08 ±[6]02 +A18 +A05 ±[3]03 –A12 –D23

If two superimposed twelvefold sequences are offset either once or twice by ±1, ±5, ±7, or ±11 indexical degrees - that is, once or twice by an amount which 24 does not divide into - their difference will also be one of the four twelvefold sequences; otherwise, the difference reduces to a non-linear sequence of lower cardinality. Hence one can superimpose up to three twelvefold pitch sequences, offset by two equal indexical shifts, whose vertical relationships instantiate three twelvefold interval sequences; any fourth pitch sequence will be vertical relationships of reduced cardinality to at least one of the other three sequences. As a result, a trichords are the largest structures which effectively exclude proportions of reduced cardinality. They circumscribe the domain of strictly twelvefold organization, that is, organization that incorporates harmonic and gradient units. Every three by three square on the self-intersecting twelvefold matrix represents a basic twelvefold intersection of minimum curves. The binary non-linear sequence is the minimum non-trivial non-linear sequence; this sequence of three values intersects itself in a three by three matrix which matches the perimeter of purely twelvefold organization just described. When it is applied as an intervallic operator on a twelvefold sequence it modulates the latter into one of the other three twelvefold sequences. That is, the sum of the binary sequence and a twelvefold sequence is always one of the other three twelvefold sequences, according to the relative indexical position of the addenda:

+A1 +[2]1 -A13

0 +1 +1 +2 +3 +5 -4 +1 -3 -2 -5 +5 0 +5 +5 -2 +3 +1 +4 +5 -3 +2 -1 +1 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6

0 -5 -5 +2 -3 -1 -4 -5 +3 -2 +1 +1 0 -1 -1 -2 -3 -5 +4 -1 +3 +2 +5 -5

+A1 0 +1 +1 +2 +3 +5 -4 +1 -3 -2 -5 +5 0 +5 +5 -2 +3 +1 +4 +5 -3 +2 -1 +1 +[2]3 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 -D1

±6 +1 -5 -4 +3 -1 +2 +1 +3 +4 -5 -1 ±6 +5 -1 +4 +3 -5 -2 +5 +3 -4 -1 -5

+ A1 0 +1 +1 +2 +3 +5 -4 +1 -3 -2 -5 +5 0 +5 +5 -2 +3 +1 +4 +5 -3 +2 -1 +1 +[2]2 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 +D13 ±6 -5 +1 -4 -3 +5+2 -5 -3 +4 +1 +5 ±6 -1 +5 +4 +1 +1 -2 -1 -3 -4 +5 +1

Hence, a twelvefold matrix can be modulated with the binary sequence to integrate all four twelvefold sequences. For each unique self-intersection of the binary sequence within a three by three matrix there is a unique integration of the four twelvefold sequences. One such configuration modulates the +A matrix so that the +A sequence is interwoven with two other twelvefold sequences in equal proportions:

0 0 0 0 6 6 0 6 6

+A=

A different binary configuration modulates the +A matrix so that the +A sequence is intersected by the other three twelvefold sequences:

0 6 6 0 6 6 0 6 6

+A=

These configurations can be combined in three dimensions to achieve an integration of all four sequences in equal proportions, but which retains the asymmetry of the binary non-linear sequence [2]. However, it is more practical to work with a tritone substitution at every point as a shorthand for a full integration of the four twelvefold non-linear sequences:

These double points can be filtered by three-by-three binary matrices so as to simultaneously select from the two pitch classes at each point, and outline the limits of purely twelvefold organization. These minimum matrices are filters which can rove through the twelvefold matrix to create sequences of twelvefold intersections of minimum curves. Since each pitch class is accompanied by the pitch class one tritone away, we can arbitrarily restrict our selection of pitches to one of any two self-complementing hexachords simply by modulating to the appropriate twelvefold sequence at a given point in time, or equivalently, by filtering it through the preferred binary matrix. In general, sequences of a given cardinality transpose the notes in a twelvefold sequence within a symmetrical subset of pitch classes, effectively making it possible to isolate any self-complementing subset of interval or pitch classes and articulate them non-linearly. Earlier we observed that each interval equilibrium can be derived from the other by substituting gradient units for harmonic units, and vice versa. Modulation by a trinary sequence effects this substitution in non-linear interval space by selectively transposing notes by the interval category |4|. In this case unitary substitution derives the self-same sequence, shifted by twelve indexical degrees, that is, halfway through the cycle. The twelvefold sequences can be visualized on the surface of a Moebius strip, where each pair of unitary counterparts is allocated to the opposing faces of a given position along the strip:

+A

+5

-5 –1

+1

0

+5

+1

-2 +2 –3

+5

+1

–2 +2

+5 +1

+4 –4

+1 +5

+3

+D

+1

+5 +1

+5

6

–5

–1

-4 +4

+1

+5

–4 +4

–3

-1

-5

+2 –2

+5 +1

–3

Subtracting a sequence from itself at a distance of twelve indexical degrees, that is, subtracting unitary counterparts, we can see that their difference turns out to be modulated by the trinary sequence:

[12] -4

+4 +4

-4

0

+4

-4 +4 0

-4 +4

-4

–4 +4

+4 -4

-4 +4

-4 +4

0

Finally, the fourfold non-linear sequences govern transposition by minor thirds, allowing sort of major/minor translation, and the sixfold non-linear sequences articulate the twelvefold sequences as the alternation of whole-tone scales (which alternation is identical to the alternation of the binary non-linear sequence between |0| and |6|.) C. Non-Linear Arborescences

We have already noted that an ordered trichord is a minimum hierarchical constituency, in that the one of its intervals frames the other two. We can elaborate this arborescent structure further by bifurcating the constituent intervals, the new constituents that result from this bifurcation, and so forth. Furthermore, just as one bifurcates an interval by applying an intervallic operation to the first pitch of an ordered dyad, one can bifurcate each interval in a pitch sequence using the same operations that produced trichords from dyads. Trichords are the seed for all further structural elaboration in precisely this sense. To construct a linear arborescence, we can take one of the linear dimensions from the isotropic interval matrix, and interpolate a new pitch after each existing pitch at a consistent interval. We arrive at a familiar pattern:

This rudimentary arborescence is the linear replication of a given proportional class, which we used earlier to explicate co-derivative proportional categories. The proportion is transposed by

one of its three intervals, each occurrence of which frames the other two intervals. As we have noted, the intermediate pitches interpolated within each frame are themselves distributed by the same interval, and frame the inversions of the proportional class.

In general, linear pitch arborescences elaborate a trajectory along one of the linear dimensions of the isotropic interval matrix by deviating periodically along a different linear dimension, deviating periodically from the deviations, and so forth. In contrast to these linear arborescences, we now derive a genus of arborescences from our nonlinear sequences that maximize curvature on every order of difference. Thus our beloved arbitrium will be flanked with exemplary references: pure intervallic inertia on the one side, and pure intervallic acceleration on the other. As noted above, the operation that derives trichords from dyads is strictly equivalent to the operation that derives arborescences from pitch sequences; thus, the task of seeking a non-linear arborescence is referred to the task of deriving a trichord from each interval of the non-linear sequences. We must be able to locate a third pitch which is implicit in each pair of consecutive pitches.

Non-linear sequences are constructed by appending each trichord with its framing interval. Thus it is clear that an intermediate pitch for each interval in these sequences is covertly contained in the preceding trichord. Each interval class in the first place arose as a framing interval class, so it is merely a matter of reinserting the note that was elided when the framing interval class was duplicated as a suffix.

However, the duplication of this intermediate pitch introduces a redundancy, thereby depriving us of our goal of total curvature. This redundancy becomes more pervasive with every bifurcation. After two iterations - that is, after we bifurcate each dyad of a non-linear pitch sequence based on the preceding trichord, then bifurcate each of the new dyads similarly - the emergent linearity is already striking: etc.

With each bifurcation, this arborescence reintroduces a greater degree of linearity to the non-linear sequence it elaborates. It effects the converse of linear arborescences. Just as the latter introduce local curvature in a global linearity by interpellating one constant within another, the above bifurcations effectively introduce local linearity in a global curvature by echoing each interval class.

It would appear that in order to prevent linearity from seeping into these arborescences we will have to associate the intervals in the original pitch sequences with ordered trichords which are not adjacent to them; as long as the framing interval of an ordered trichord is associated with the interval that immediately follows it, we will not be able to reinsert the intermediate pitch into the latter without introducing redundancy.

And in fact, our non-linear sequences do associate non-adjacent positions. For as we noted earlier, these sequences intersect themselves such that every interval class has a twin elsewhere in the same sequence: +A01 +A02 +A03 +A04 +A05 +A06 +A07 +A08 +A09 +A10 +A11 +A12 0 +1 +1 +2 +3 +5 –4 +1 –3 –2 –5 +5 +A13 +A08 +A03 +A22 +A17 +A12 +A07 +A02 +A21 +A16 +A11 +A06 +A13 +A14 +A15 +A16 +A17 +A18 +A19 +A20 +A21 +A22 +A23 +A24 0 +5 +5 –2 +3 +1 +4 +5 –3 +2 –1 +1 +A01 +A20 +A15 +A10 +A05 +A24 +A19 +A14 +A09 +A04 +A23 +A18 Thus, each interval class can be bifurcated into the two interval classes that precede its twin. Each of these interval classes can be similarly bifurcated, ad infinitum:

Each indexical position thereby branches into infinitely detailed, non-linear curve which can be articulated to an infinitely small degree of resolution. Each of these perfectly smooth and dynamic curves is associated with a specific minimum curve: the ordered trichord created by the first pitch to bifurcate the indexical position in question. Conversely, each ordered trichord is elaborated recursively with the inventory of 24 curves that correspond to the indexical positions of the original non-linear sequence. As with the original non-linear sequence, no one trichord is vertically originary or prior to the others; each position subdivides into all the others through a series of bifurcations.

These curves form an abelian group which is homologous that of the foundational non-linear sequences. So, for instance, if two curves belonging to adjacent indexical positions were superimposed, the resultant vertical intervals would form another twelve-fold curve. But if an A and a D curve belonging to the same indexical position were superimposed, the resultant vertical intervals would be only unisons and tritones - a binary curve. Each bifurcation splits an interval class into an antecedent and consequent interval class. Repeated bifurcation of antecedent or consequent interval classes reveals left-branching and rightbranching pedigrees which return to a given curve every 3, 6, or 12 generations. For instance, D5 is a part of the following left-branching pedigree:

The four prograde arborescences each contain 5 antecedent pedigrees of 3 or 6 generations each, and 2 consequent pedigrees of 12 generations each. The retrograde arborescences have the exact same pedigrees in reversed positions, as each pitch in the original non-linear sequence is effectively attached to the pitches which precede it in the prograde forms. If we label the antecedent pedigrees A through E and the consequent pedigrees Y and Z, they correspond to the indexical positions of the prograde arborescences in the following pattern:

01 05 09 13 17 21

02

E1

06

E2

10

E3

14

E4

18

E5

22

E6

C6

D1 C2

D3 C4

D5

03 07 11 15 19 23

04

A1

08

B3

12

A3

16

B1

20

A2

24

B2

D2

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Z01

C3

Z03

Z12

D4

Z05

Z02

C5

Z07

Z04

D6

Z09

Z06

C1

Z11

Z08 Z10

Y01 Y03 Y05 Y07 Y09 Y11

Y06 Y08 Y10 Y12 Y02 Y04

The sequence of antecedent pedigrees repeats every eight positions, just as the trinary non-linear sequence repeats every eight values; both articulate the twelvefold non-linear sequences in a trinary translational symmetry. As we noted above, modulation by the trinary sequence can effect substitution of unitary counterparts, that, is mutual substitution of gradient and harmonic units. The antecedent pedigrees correspond to specific positions in this sequence of |4| modulation that effects unitary substitution. Here are the antecedent pedigrees as viewed on the moebius strip of unitary duals: D4

A3 B 2

C2 D5

E6

C1

E4

E1

E3

C3

D6

D3

E5

B3 A2

[12]

C6

D1

C4

B1

A1

C5 D2

E2

-4

+4 +4

-4

0

+4

-4 +4 0

-4 +4

-4

–4 +4

+4 -4

-4 +4

-4 +4

0

Just as the substitution of gradient units (half-steps) for harmonic units (fifths) in the non-linear sequences results in the very same sequences shifted by twelve indexical positions, unitary substitution within one of these non-linear curves produces the curve located twelve indexical positions away. Among the antecedent pedigrees, the A pedigrees and C pedigrees correspond to +4 transposition, the B pedigrees and the D pedigrees correspond to -4 transposition, and the E pedigrees correspond to null transposition. Unitary substitution corresponds to a modulation by a distinct trinary curve for each of these three positions.

D. Summary

We can now take stock of the many different kinds of sequences found within these interval spaces. 1. The most static possible sequences of intervals other than the ‘null sequence’ of unisons are the 6 linear sequences found in the isotropic interval matrix. 2. Sequences with static curvature move through the isotropic interval matrix in a homogenous twist.

3. Non-linear sequences continually twist in new directions through the isotropic interval matrix in a heterogenous trajectory. While the linear elements of the isotropic interval matrix give us total inertia, these sequences provide total acceleration. Despite their chaotic appearance on within linear pitch space, these nonlinear sequences form an abelian group. As a result, superimposed non-linear sequences always produce vertical relationships which are also non-linear.

4. Linear arborescences repeat interval classes periodically. They repeat trajectories which themselves contain repeated trajectories, and so forth. Each scale of magnitude is potentially distinguished by a distinct constant.

5. Non-linear arborescences form curves which intervallically accelerate at every scale of magnitude. While linear sequences are 'smooth', these non-linear arborescences are 'rough' at an arbitrarily small degree of resolution: there are always further twists at greater magnifications. Although the sequence of interval classes at any given scale of magnitude is unpredictable, it is replicated at all other scales of magnitude. The minutest fluctuations seen at the greatest levels of magnification and relatively macroscopic scales share the same non-linear pattern. 6. Successive intervals from a non-linear sequence can be substituted on any scale of magnitude within a linear arborescence, and single constant interval could replace any scale of magnitude within a non-linear arborescence. That is, one could filter either one of the two genuses of arborescence through the other at given scale of magnitude.

7 The arborescence created by bifurcating the interval classes of a non-linear sequence identically to the directly preceding trichords generates local linearity by echoing each segment of the non-linear progression. It is globally non-linear and locally linear. Each bifurcation introduces greater redundancy. At high degrees of magnification, it is effectively a linear arborescence which is isomorphic to the non-linear arborescence, exhibiting five antecedent and two consequent pedigrees in the same relations. 8. Finally, one can use non-linear sequences and arborescences of lower cardinality as intervallic operators to modulate between non-linear curves.

Any two ordered trichords that appear within the same non-linear sequence come to be adjacent somewhere in the arborescence constructed from it. They always appear with their direct antecedent or consequent in the original non-linear sequence whenever the appropriate indexical position is bifurcated. Beyond that, one curve is adjacent to another in one of four ways: 1) it precedes the curves found in the the left-pedigree of its consequent; 2) it follows the curves found in the right-pedigree of its antecedent; 3) it starts together with the curves found in its right pedigree; or 4) it ends together with the curves found in its left pedigree. So, for instance, the indexical position 1 precedes 2, 8, 10, 16, 18, and 24, follows 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, and 24, begins together with 5, 9, 13, 17, and ends together with 2, 5, 6, 9, 10, 13, 14, 17, 18, 21, and 22.

A curve is magnified through the bifurcation of its intervals, and never through trifurcation, et cetera; as a result any sequence of pitches derived from the magnification of any portion of a curve can be parsed into a binary antecedent-consequent pairs at any scale of magnitude. This implies that non-linear pitch organization can be molded into any permutation of arborescent structure, by selectively magnifying portions of the curve and eliding dominance relations wherever necessary.

IV Conclusion

We end with some thoughts on the relationship between these two interval spaces: the ways in which they differ diametrically as well as their collective significance.

Their opposition is thoroughgoing enough that we could use choose any number of designations other than “linear” and “non-linear” to refer to them. For instance, we could call linear interval space “cardinal proportional space” and non-linear interval space “ordinal proportional space”. The former accommodates proportionality by separating the twelve interval classes into twelve cardinal directions. The latter accommodates proportionality by ordering it in sequences.

Of course, cardinal proportional space is not unordered - rather, it is a synthesis of linear orderings. While it does not privilege any specific ordering of a pitch set, it does register each of them as a unique trajectory. In fact, it is an image of the entire universe of sequences. Meanwhile, ordinal proportional space positions each basic intervallic permutation uniquely by precluding linear ordering.

We could also call linear interval space “isotropic pitch space” and non-linear interval space “anisotropic pitch space”. Each pitch in isotropic pitch space is in an intervallically identical position, while each pitch in anisotropic pitch space is an intervallically distinct position. Isotropic pitch space achieves consistency by allowing each structural element to saturate a given dimension with redundancy, deferring each new degree of non-redundancy to a higher dimension. One the other hand, the foundational sequences of anisotropic pitch space register each structural element parsimoniously, while the related arborescent sequences disperse their recurrence to the greatest possible degree.

Antinomies notwithstanding, our two spaces are linked in their premises and implications. They are both the result of an elaboration of minimum structures. The dyads and trichords which instantiate intervallic and proportional categories provide genetic material which is methodically elaborated with a simple operation until a complete set of connections between them coalesces. Isotropic pitch space arranges these pitch class sets symmetrically and then replicates the intervals. Anisotropic pitch space arranges these pitch sets in sequences by adding consecutive intervals.

Both genuses represent the intervals and trichords parsimoniously: isotropic pitch space is nonhierarchical, while anisotropic pitch space is hierarchical but uniformly recursive. Thus they are neutral with respect to preferences for certain intervallic configurations that may exist in certain styles or in the nature of acoustics or cognition. This is a radical omission, as the intervals associated with a given pitch’s overtones and undertones do maintain a fundamental predominance. The implication is that these interval spaces model an intuition that would exist if intervals were absolute and the acoustic reality of pitches was negligible. Of course, this is an entirely paradoxical scenario, as pitch space is discrete in the first place only because we can perceive the simple frequency ratios found in overtones. So as a cognitive model these interval spaces present a radical distortion, but perhaps a valuable one. For if each interval is distinct to an acute listener, then these spaces facilitate the creation of music which is liminal to cognition. They present condensed transpositional templates which effectively treat each pitch as the punctual representation of an entire tonic region. This is in many ways what free atonality asked for but serialism failed to offer. For serialism treated pitch classes, not interval classes, parsimoniously. Insofar as people hear ‘relative pitch’ and are completely indifferent to the absolute frequency of pitch classes, the effect was to statistically neutralize our intuition of pitch structure rather than provoking it. On the contrary, serialism instead had the salutary effect of making parameters other than pitch more salient by comparison. Non-linear interval space, on the other hand, totalizes the hyper-dynamism glimpsed in the transition from late romanticism to free atonality.

So while these pitch spaces are worthless to the cognitive scientist, for the composer they have value as drafting tools - as a straight-edge and a french curve, so to speak. Together, they provide a composer with the gamut of basic structural relationships integrated on their own terms, abstracted from any ranking. Practically speaking, I have found the memorization and navigation of cardinal proportional space to be easy and ordinal proportional space to be difficult. The former provides a kind of spontaneous structural fluency, while the diverging tracks of ordinal proportional space generally necessitate some reference to tables. Aquinas’ notion that only god is conscious of all things at once while man must think in order is germane here, and not without irony.