Transcript
Page 1: Linear and Non-Linear Interval Space

I Introduction

A. Overview

This article examines two mathematical spaces which can be constructed using the twelve canon-ical interval classes. Each is derived from the minimum interval sets found in dyads and tri-chords. The first elaborates these germinal sets in a linear, isotropic format, while the second elab-orates them in an non-linear, anisotropic format.

Both spaces prescind from any hierarchical ranking of these minimum structures. They parsimo-niously inventory the intervals and trichords without indicating their relative prominence orvalue, whether acoustic, cognitive, or idiomatic.

The two representations diverge radically in the means by which they do so. The first replicatesthe inventory of minimum structures symmetrically around every pitch. The second dispersesthis 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 discretespace, in contrast to other musical parameters such as loudness which are mere continua. Ratiosrelate discontinuous frequencies, and thus striate the pitch continuum with discrete thresholds;while the difference between two frequencies can vary continuously, pitches also exhibit discretedifference insofar as they inhabit coordinates within this array of thresholds.

The interval spaces described in this article concern these discrete differences that derive fromrational 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 bearranged in order of increasing exponential difference by factors of three. The resulting relation-ships are captured in modular arithmetic. Whereas the traditional interval names (used occasion-ally in this paper as a more intuitive shorthand) measure continuous difference in discrete incre-ments (scalar steps), our arithmetic measures the authentically integral differences produced bythe most elementary ratios.

+1–1

±6

+4–4

+3

–2

–5 +5

0

+2

–3 (m3 down)(m3 up)

(M3 up)(M3 down)

(P4 down)(P4 up)

(Tritone)

(Octave)

(m2 down)(m2 up)

(M2 up)(M2 down)

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This arithmetic clearly does not rank the interval classes in terms of consonance, as it neglects fre-quency relationships involving higher prime numbers which effectively intersect those producedby compounding factors of three. However, it does illuminate certain group-theoretic relation-ships, which we now review briefly.

Octaves and unisons give us zero harmonic difference; they are the additive identity, 0, in ourarithmetic. If we assign ‘fifths’ the role of the multiplicative identity, +1, ‘fourths’ assume itsinverse, -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 doesnot divide into. A consequence of this incommensurability is that 5 effectively functions as a sec-ond unitary value: by taking every fifth value in our circle of interval classes, one derives an alter-nate ordering:

The values 1 and 5 are effectively interchangeable as multiplicative identities, that is, units: theyare both sufficient to generate and measure every other interval class. This dualism is evident inboth 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 representthe simplest harmonic relationship, then +5 and -5 are the ‘gradient’ units which represent thesmallest 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 half-steps, that is, five harmonic units at a time. If we assigned half-steps the value 1, then it wouldindicate the order of intervals as they increase by fourths or fifths. Each type of unit is equal tofive of the other unit: the smallest harmonic increment is equal to five gradient increments, andthe smallest gradient increment is equal to five harmonic increments.

+4–4 +1

+1

–1

–1

±6

±6

+4–4

+3 +3

–2-2 –5

–5

+5

+5

0

0

+2+2

–3–3

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C. A Preliminary Inventory of Proportional Classes

Three intervals coincide three pitches. While intervals compare pitches, trichords compare inter-vals. They present the array of proportions that comprise the genetic basis for any elaboration ofintervallic 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 classesof a trichord a “proportional class”. There are thirty-one of them. Because their inversion isambiguous in the absence of ordering, we will label them in terms of the absolute value of theirintervallic 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 proportionalclasses, which can be grouped in 7 inverse pairs:

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

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

A single interpolated pitch completes the trichord in each frame. The interpolated pitches them-selves 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 adja-cent frame, that is, to G and C, the result is a coderivative proportional class, (1,3,4). In general, ifthe interpolated pitches are shifted to occupy the adjacent frames, the result will be a co-deriva-tive proportional category that shares the framing interval. Below we show the interpolated pitchshifting across four adjacent frames to give us the proportional classes containing the framinginterval 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 pro-portional categories that contain 1: |1,2,3| - |1,3,4| - |1,5,4| - |1,5,6|. The lower row orders thecategories that contain 5: |5,2,3| - |5,3,4| - |1,5,4| - |1,5,6|. The square circuit on the left ordersthe categories that contain 3: |1,2,3| - |1,3,4| - |5,3,4| - |5,2,3|. The triangular circuit in the mid-dle orders the categories that contain 4: |1,3,4| - |1,5,4| - |5,3,4|. On the very left we have twocategories 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 result-ing series of neighborhoods. Finally, there are two non-redundant proportional classes which co-derive themselves: |1,5,6| and |2,4,6|. The latter is the only non-redundant proportional classthat contains no unitary interval classes; as such, it is co-derivationally inert, that is, disconnectedfrom every other proportional class.

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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 sin-gle interval class, or a constant degree of difference between consecutive interval classes. We willassemble 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 alwaysappears 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 represen-tation displays the minimum collection of pitches required to instantiate the category. Thenuclear representation displays a central pitch along with the neighborhood of adjacent pitchesprovided by the two inversions of the interval category. Lastly, the isotropic representation dis-plays 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 class-es 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 repre-sentation of an interval category.

To represent non-redundant proportional classes - those which include three unlike interval class-es - we will have to accommodate three independent linear dimensions. By arranging the threepitch classes of a trichord symmetrically, we arrive at a germinal representation that integrates thelinear dimensions on a plane:

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In the corresponding nuclear representation, these three linear dimensions intersect at a singlepitch class, and also form a circuit around six peripheral notes. The central pitch class is sur-rounded 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 thisexample, it is separated from two other pitches by the interval classes +1 and +3, -2 and -3, or -1and +2. Simultaneously, it completes three forms of the inverse proportional class; each upwardpointing triangle forms a different transposition of the +(1,2,3) proportional class, while eachdownward pointing triangle forms a different transposition of its inverse, the -(1,2,3) proportion-al class:

As with the interval classes, we will speak of “proportional categories” in order to conflate theproportional 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 - appearin the nuclear representation of a given proportional category as 120-degree angles:

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In the isotropic representation of a proportional category, each pitch is situated at the intersectionof 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 withthree inverses and a point with the others. A shared edge denotes two shared pitches; each edge-adjacent inverse replaces one pitch class such that the interval categories that separate it from eachside of its intervallic frame are swapped:

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

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C. Tetrachords and Intervallic Equilibrium

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

As with non-redundant trichords, we can obtain a germinal representation which assigns aunique directional orientation to each of the constituent interval and proportional categories byarranging four points equidistantly. Four intersecting triangles and six intersecting lines areimmediately 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:

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In a tetrachord each pitch complements one of four trichords; thus each tetrachord incorporatesfour proportional categories. Although the two inverse pairs of non-redundant tetrachords con-tain 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|

Proportional Format B

|5,3,4| |1,5,6| |1,2,3| |2,4,6|

Non-redundant tetrachords exclude co-derivative proportional categories: each of the constituenttrichords shares just one interval class with each of the others, just as in a tetrahedron, any two tri-angular faces share one edge. If any of the pitches in one of these constituent trichords arereplaced 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 dia-gram of co-derivative proportional categories twice, circling the four categories that compriseeach proportional format:

Proportional Format A Proportional Format B

The full inventory of proportional categories is only obtained in the corresponding nuclear repre-sentation, in which six intervallic paths and four proportional fields intersect symmetrically at asingle point:

1,2,3

|5,2,3|

|1,3,4|

5,3,4|1,5,4| 1,5,6 2,4,6

|1,2,3|

5,2,3

1,3,4

|5,3,4||1,5,4| 1,5,6 2,4,6

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Proportional Format A Proportional Format B

The peripheral pitch classes form a polyhedron known as a cuboctahedron. Buckminster Fullercoined the term ‘vector equilibrium’ to refer to this polyhedron when considered together withthe 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 rendersevery 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 asfour hexagons arrayed symmetrically around the nuclear pitch class. The remaining, co-deriva-tive categories appear as 120-degree angles, completing the full inventory of proportional cate-gories. Note that the category |1,5,4| is group-theoretically precluded from non-redundant tetra-chords. It only appears as a 120-degree angle in non-linear interval space because it never appearsin non-redundant tetrachords, only in transpositions of the latter which share a pitch. Any twotranspositions of a non-redundant tetrachord by the interval categories |1|, |5|, or |4| will con-tain the proportional category |1,5,4|.

The duality of proportional formats reflects the duality of harmonic and gradient units, that is, theability 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 andgradient units. To demonstrate, we can superimpose the gradient intervallic cycle upon the har-monic interval cycle, thereby assigning each interval class with a counterpart. Each interval classis equal to five of its counterparts (commutatively).

+4–4 +1

+1

–1

–1

±6

±6

+4–4

+3 +3

–2-2 –5

–5

+5

+5

0

0

+2+2

–3–3

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If each interval class in one of the interval equilibria is replaced by its counterpart, the other inter-val equilibrium results. Likewise, each proportional category in one interval equilibria has acounterpart 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 intervalequilibrium, and vice versa. If one of the two unitary interval categories appears in a given pro-portional domain, it has an unlike counter-domain; if both or neither of the two unitary intervalcategories appear, then it is its own counter-domain (to be precise, it inverts in the other intervalequilibrium: the triangulated plane is turned ‘upside-down’, switching the trichords with theirinverses).

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 whichseparates the two remaining pitches. Of the six interval categories in the tetrachord, there are thenthree pairs of disjunct categories. However, all six interval categories intersect in the nuclear pitchof an interval equilibrium. The latter combines each of these pairs in an additional rectilinearplane of symmetry:

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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 inter-section of the six linear paths and seven planar arrays found in the interval equilibrium. Thegeometry 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 ‘isotrop-ic interval matrix’. To construct it, we can start with one of the of triangular planar arrays. When

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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 newpitches form a planar array which is parallel to the first (hence of the same proportional catego-ry):

The pitches that complement the inverse trichords are located in the opposite direction - if thecomplementary pitches of one proportional class form a superjacent layer, then the complemen-tary pitches of the inverse proportional class will form a subjacent layer. Further planes can bestacked in both directions infinitely.

We can outline the atomic cells that constitute the isotropic vector matrix by connecting the pitch-es of the superjacent layer to the adjacent pitches in the first layer. Firstly, an array of tetrahedrapoints from the original triangular array towards the viewer. Secondly, an array of inverse tetra-hedra points from the superjacent triangular array away from the viewer. Finally, the pocketsremaining between the tetrahedra each comprise octahedra:

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Each trichord constitutes the face of a tetrahedron in one direction, and the face of an octahedronin 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 onone side of a given proportional class to enclose a tetrahedral space, and diverging on the otherside to make room for an octahedral space; the same proportional fields converge and diverge inopposite directions from the inverse proportional class.

The octahedra conjoin precisely the interval categories that are disjunct in the tetrachord; eachinterval category appears twice and is doubly linked to its estranged counterpart in the hexachordcorresponding the octahedron:

D. Summary

Linear interval space integrates four degrees of structural complexity, revealed in their germinalform in monads, dyads, trichords, and tetrachords. These four degrees of complexity are situat-ed respectively in zero, one, two, or three dimensions and are homologous to the elements ofEulerian topology, namely, vertexes, edges, and faces, and cells. Each germinal form is interval-lically non-redundant, but the corresponding dimensions are then saturated in redundancythrough the linear replication of this germinal form. Each new degree of non-redundancy isdeferred to a new topological element in a higher dimension.

One pitch gives us identity. A second unlike pitch gives us rational difference. A third unlikepitch spaced at unlike interval classes gives us proportionality. A fourth such pitch gives us acomplete inventory of the preceding elements. The fourth degree is the terminal, insofar as a pen-tachord necessarily involves structural redundancy, that is, replicated interval and proportionalcategories.

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

If interval classes and the proportional classes provide the genetic basis for pitch organization inthat they provide the possible differences in pitch and the possible proportions between these dif-ferences, the interval equilibrium conflates structural distinctions by encompassing every differ-ence and proportion at once. Thus, non-redundant tetrachords are useful simply as the minimumindex of the gamut of trichords.

The isotropic interval matrix presents no restrictions on the ordering of a pitch set. Rather, it rep-resents the interval classes as twelve cardinal directions surrounding every pitch. Thus, everyordering of every pitch set is a contiguous trajectory in the isotropic interval matrix. In this per-spective, the linear paths of the isotropic interval matrix are vectors representing operationswhich produce pitch sets or pitch sequences of arbitrary complexity.

These vectors can operate either as radial or compressive forces. Each interval category radiatesaway from a given point at a consistent rate of difference; further out from a pitch, these interval-lic vectors intersect to form concentric polygonal and polyhedral shells that constrain this linearmovement. 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 theisotropic interval matrix, namely, the interval equilibrium. To be more precise, since the intervalcategory |6| replicates the same pitch class on either side of a nuclear pitch class, each instanceof a tone row can be obtained in a nuclear pitch class along with eleven surrounding pitch class-es - a ‘dimpled’ interval equilibrium. Of course, taken in order, a given series doesn’t necessarilyfollow a contiguous trajectory around this form, but its pitch classes do form a relative localitywithin a composition which constitutes this total elaboration and compression of interval struc-ture. The monotonous recurrence of this single intervallic confers a harmonic uniformity on seri-al music; on the other hand, the sequential permutations of this omnipresent structure, by provid-ing fleeting glimpses of radial momentum, confer a pittance of variety at the most local scale ofmagnitude.

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III Non-Linear Interval Space

A. Non-Linear Pitch Sequences

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

Curvature is effected in this sequence through the covert presence of trichords: each successivestep between two adjacent pitches implies a third, mediating pitch, namely, what the first pitchwould arrive at if the preceding interval class in the sequence were repeated. The pitch it arrivesat instead is always separated by the same interval class, not from the initial pitch, but from themediating pitch.

Ordered trichords measure a change of interval. They can be thought of as minimum curves. Thethree interval classes assume distinct roles:

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

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Using the three interval positions within an ordered trichord as our terms, we now define a family of max-imally non-linear pitch sequences, that is, sequences that exclude any static degree of difference. Whereasthe linear elements in the isotropic interval matrix exclude curvature in favor of linearity, these continu-ally 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 dis-tance from the third pitch which is equivalent to the framing interval class between the first and thirdpitches, accelerating the curvature of the line by encompassing both of the preceding interval classes in asingle step. The second, third, and fourth pitches form a new trichord which can be extended analogous-ly, 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 inter-val classes which accelerates into itself. Two species of such non-linear sequences exist, each of which canbe inverted, for a total of four reversible sequences. Whereas the isotropic interval matrices inventory theproportional categories in an unordered format, these sequences inventory them in an ordered format;each of the six permutations of each trichord appears twice.

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The sum of two consecutive intervals classes is equal to the interval class that follows them; equiv-alently, the difference between two consecutive interval classes is equal to the value that precedesthem. 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 thesame, vel cetera. In contrast, non-linear sequences are derived through an operation on the vari-able quantities of the three positions within each ordered trichord, without any reference to a con-stant value, and so are not reducible to zero.

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

Among the sequences containing unitary intervals, henceforth ‘twelvefold sequences’, we canobserve that the A sequences monopolize the null interval category |0|, while the D sequencesmonopolize the ‘tritone’, |6|. When a unison is followed by any other interval class, the result-ant framing interval class is identical to the latter. As a result, the A sequences also have themonopoly 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 inter-val class is a harmonic or gradient unit; the other redundant proportional classes appear in thenon-linear sequences of reduced cardinality. On the other hand, the D sequences contain everyinstance in which both unitary interval categories appear together in a trichord, that is, everyappearance of the |1,5,4| and |1,5,6| proportional classes.

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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 pri-mary, they are each uniquely located, such that they can each be assigned an indexical address:

However, these positions are interwoven due to a striking property: values separated by five stepsare 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 isidentical to the original. As a result, every position has a twin somewhere in the same sequencethat 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 +A120 +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 +A240 +5 +5 –2 +3 +1 +4 +5 –3 +2 –1 +1

+A01 +A20 +A15 +A10 +A05 +A24 +A19 +A14 +A09 +A04 +A23 +A18

+A01 +A02 +A03 +A04 +A05 +A06 +A07 +A08 +A09 +A10 +A11 +A120 +1 +1 +2 +3 +5 –4 +1 –3 –2 –5 +5

+A13 +A14 +A15 +A16 +A17 +A18 +A19 +A20 +A21 +A22 +A23 +A240 +5 +5 –2 +3 +1 +4 +5 –3 +2 –1 +1

–A01 –A02 –A03 –A04 –A05 –A06 –A07 –A08 –A09 –A10 –A11 –A120 –1 –1 –2 –3 –5 +4 –1 +3 +2 +5 –5

–A13 –A14 –A15 –A16 –A17 –A18 –A19 –A20 –A21 –A22 –A23 –A240 –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

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We can represent this self-intersection of a twelvefold sequence by stacking it in an orthogonaldimension, offsetting it by five indexical positions on each layer, to obtain a two-dimensionalmatrix 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 canmodulate a sequence of pitches, that is, measure the distances to a second sequence of pitches. Inthe above matrix, each row is transposed by the intervals in the +A sequence to obtain the nextrow, as is each column; thus the matrix is vertically and horizontally saturated by the +Asequence.

The addition and subtraction of non-linear sequences can be generalized; the sum or differencebetween successive values in two non-linear sequences always constitutes a third sequence whichis also non-linear. That is, the non-linear sequences form an abelian group. For instance, the table

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

___

If two superimposed twelvefold sequences are offset either once or twice by ±1, ±5, ±7, or ±11indexical degrees - that is, once or twice by an amount which 24 does not divide into - their dif-ference will also be one of the four twelvefold sequences; otherwise, the difference reduces to anon-linear sequence of lower cardinality. Hence one can superimpose up to three twelvefoldpitch sequences, offset by two equal indexical shifts, whose vertical relationships instantiate threetwelvefold interval sequences; any fourth pitch sequence will be vertical relationships of reducedcardinality to at least one of the other three sequences.

As a result, a trichords are the largest structures which effectively exclude proportions of reducedcardinality. They circumscribe the domain of strictly twelvefold organization, that is, organiza-tion that incorporates harmonic and gradient units. Every three by three square on the self-inter-secting 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 ofthree values intersects itself in a three by three matrix which matches the perimeter of purelytwelvefold organization just described. When it is applied as an intervallic operator on a twelve-fold sequence it modulates the latter into one of the other three twelvefold sequences. That is, thesum of the binary sequence and a twelvefold sequence is always one of the other three twelvefoldsequences, according to the relative indexical position of the addenda:

+A-A

+D-D

+A1

±[6]01 +A03 –D08 ±[6]03 +[4]03 +A20 ±[6]10 +D06 –A17 ±[6]15 –D24 +D11

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

±[6]09 –A15 +A08 ±[6]14 –[4]03 –D20 ±[6]18 –D18 +D17 ±[6]08 +D12 –A11

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

1 2 3 4 5 6 7 8 9 10 11 12

+A-A

+D-D

+A1

±[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

13 14 15 16 17 18 19 20 21 22 23 24

Page 22: Linear and Non-Linear Interval Space

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

+ A =0 0 00 6 60 6 6

0 +1 +1 +2 +3 +5 -4 +1 -3 -2 -5 +5 0 +5 +5 -2 +3 +1 +4 +5 -3 +2 -1 +10 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±60 -5 -5 +2 -3 -1 -4 -5 +3 -2 +1 +1 0 -1 -1 -2 -3 -5 +4 -1 +3 +2 +5 -5

+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±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6 ±6 0 ±6±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+[2]3

-D1

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

+A1+[2]2

+D13

Page 23: Linear and Non-Linear Interval Space

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

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

+ A =0 6 60 6 60 6 6

Page 24: Linear and Non-Linear Interval Space

These double points can be filtered by three-by-three binary matrices so as to simultaneouslyselect from the two pitch classes at each point, and outline the limits of purely twelvefold organ-ization. These minimum matrices are filters which can rove through the twelvefold matrix to cre-ate sequences of twelvefold intersections of minimum curves.

Since each pitch class is accompanied by the pitch class one tritone away, we can arbitrarilyrestrict our selection of pitches to one of any two self-complementing hexachords simply by mod-ulating to the appropriate twelvefold sequence at a given point in time, or equivalently, by filter-ing it through the preferred binary matrix. In general, sequences of a given cardinality transposethe notes in a twelvefold sequence within a symmetrical subset of pitch classes, effectively mak-ing it possible to isolate any self-complementing subset of interval or pitch classes and articulatethem non-linearly.

Earlier we observed that each interval equilibrium can be derived from the other by substitutinggradient units for harmonic units, and vice versa. Modulation by a trinary sequence effects thissubstitution 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 indexicaldegrees, that is, halfway through the cycle. The twelvefold sequences can be visualized on thesurface of a Moebius strip, where each pair of unitary counterparts is allocated to the opposingfaces of a given position along the strip:

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

+5-5+2

–4

+1

+5 –5

+1

+4

–1+1

–2

+4

+5

-1 +1

+5

-4

+D 6

–3 –3+1+5

+4

–2

+5

+1 +5

–1

+2

+1+5

–4

+2

-5

+1 +5

+1

-2

+A 0

–3 +3

Page 25: Linear and Non-Linear Interval Space

Finally, the fourfold non-linear sequences govern transposition by minor thirds, allowing sort ofmajor/minor translation, and the sixfold non-linear sequences articulate the twelvefold sequencesas the alternation of whole-tone scales (which alternation is identical to the alternation of the bina-ry 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 thatthe one of its intervals frames the other two. We can elaborate this arborescent structure furtherby bifurcating the constituent intervals, the new constituents that result from this bifurcation, andso forth. Furthermore, just as one bifurcates an interval by applying an intervallic operation to thefirst pitch of an ordered dyad, one can bifurcate each interval in a pitch sequence using the sameoperations that produced trichords from dyads. Trichords are the seed for all further structuralelaboration in precisely this sense.

To construct a linear arborescence, we can take one of the linear dimensions from the isotropicinterval matrix, and interpolate a new pitch after each existing pitch at a consistent interval. Wearrive at a familiar pattern:

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

-4+4-4

–4

+4

-4 +4

+4

+4

-4+4

+4

+4

-4

-4 +4

-4

-4

[12]0

0 0

Page 26: Linear and Non-Linear Interval Space

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

In general, linear pitch arborescences elaborate a trajectory along one of the linear dimensions ofthe isotropic interval matrix by deviating periodically along a different linear dimension, deviat-ing periodically from the deviations, and so forth.

In contrast to these linear arborescences, we now derive a genus of arborescences from our non-linear sequences that maximize curvature on every order of difference. Thus our beloved arbitri-um will be flanked with exemplary references: pure intervallic inertia on the one side, and pureintervallic acceleration on the other.

As noted above, the operation that derives trichords from dyads is strictly equivalent to the oper-ation that derives arborescences from pitch sequences; thus, the task of seeking a non-lineararborescence is referred to the task of deriving a trichord from each interval of the non-linearsequences. We must be able to locate a third pitch which is implicit in each pair of consecutivepitches.

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

However, the duplication of this intermediate pitch introduces a redundancy, thereby deprivingus of our goal of total curvature. This redundancy becomes more pervasive with every bifurca-tion. After two iterations - that is, after we bifurcate each dyad of a non-linear pitch sequencebased on the preceding trichord, then bifurcate each of the new dyads similarly - the emergent lin-earity is already striking:

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

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

etc.

Page 27: Linear and Non-Linear Interval Space

And in fact, our non-linear sequences do associate non-adjacent positions. For as we noted earli-er, these sequences intersect themselves such that every interval class has a twin elsewhere in thesame sequence:

Thus, each interval class can be bifurcated into the two interval classes that precede its twin. Eachof these interval classes can be similarly bifurcated, ad infinitum:

Each indexical position thereby branches into infinitely detailed, non-linear curve which can bearticulated to an infinitely small degree of resolution. Each of these perfectly smooth and dynam-ic curves is associated with a specific minimum curve: the ordered trichord created by the firstpitch to bifurcate the indexical position in question. Conversely, each ordered trichord is elabo-rated recursively with the inventory of 24 curves that correspond to the indexical positions of theoriginal non-linear sequence. As with the original non-linear sequence, no one trichord is verti-cally originary or prior to the others; each position subdivides into all the others through a seriesof bifurcations.

+A01 +A02 +A03 +A04 +A05 +A06 +A07 +A08 +A09 +A10 +A11 +A120 +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 +A240 +5 +5 –2 +3 +1 +4 +5 –3 +2 –1 +1

+A01 +A20 +A15 +A10 +A05 +A24 +A19 +A14 +A09 +A04 +A23 +A18

Page 28: Linear and Non-Linear Interval Space

These curves form an abelian group which is homologous that of the foundational non-linearsequences. So, for instance, if two curves belonging to adjacent indexical positions were super-imposed, the resultant vertical intervals would form another twelve-fold curve. But if an A anda D curve belonging to the same indexical position were superimposed, the resultant verticalintervals 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 right-branching pedigrees which return to a given curve every 3, 6, or 12 generations. For instance, D5is 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 exactsame pedigrees in reversed positions, as each pitch in the original non-linear sequence is effective-ly attached to the pitches which precede it in the prograde forms. If we label the antecedent pedi-grees A through E and the consequent pedigrees Y and Z, they correspond to the indexical posi-tions of the prograde arborescences in the following pattern:

Page 29: Linear and Non-Linear Interval Space

The sequence of antecedent pedigrees repeats every eight positions, just as the trinary non-linearsequence repeats every eight values; both articulate the twelvefold non-linear sequences in a tri-nary translational symmetry. As we noted above, modulation by the trinary sequence can effectsubstitution 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 thateffects unitary substitution. Here are the antecedent pedigrees as viewed on the moebius strip ofunitary duals:

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

-4+4-4

–4

+4

-4 +4

+4

+4

-4+4

+4

+4

-4

-4 +4

-4

-4

[12]0

0 0D1

C3B3

C5

B1

C1 D3

B2

D5

C6D4

A2

D2

A3

D6 C4

A1

C2

E6 E2

E4

E1

E3 E5

18Z08

01Z01

02Z12

03Y01

04Y06

05Z03

06Z02

07Y03

08Y08

09Z05

10Z04

11Y05

12Y10

13Z07

14Z06

15Y07

16Y12

17Z09

19Y09

20Y02

21Z11

22Z10

23Y11

24Y04

18C4

01E1

02C6

03A1

04D2

05E2

06D1

07B3

08C3

09E3

10C2

11A3

12D4

13E4

14D3

15B1

16C5

17E5

19A2

20D6

21E6

22D5

23B2

24C1

Page 30: Linear and Non-Linear Interval Space

D. Summary

We can now take stock of the many different kinds of sequences found within these intervalspaces.

1. The most static possible sequences of intervals other than the ‘null sequence’ ofunisons are the 6 linear sequences found in the isotropic interval matrix.

2. Sequences with static curvature move through the isotropic interval matrix in ahomogenous twist.

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

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

5. Non-linear arborescences form curves which intervallically accelerate at everyscale of magnitude. While linear sequences are 'smooth', these non-lineararborescences are 'rough' at an arbitrarily small degree of resolution: there arealways further twists at greater magnifications. Although the sequence of inter-val classes at any given scale of magnitude is unpredictable, it is replicated at allother scales of magnitude. The minutest fluctuations seen at the greatest levels ofmagnification and relatively macroscopic scales share the same non-linear pat-tern.

6. Successive intervals from a non-linear sequence can be substituted on any scaleof magnitude within a linear arborescence, and single constant interval couldreplace any scale of magnitude within a non-linear arborescence. That is, onecould filter either one of the two genuses of arborescence through the other atgiven scale of magnitude.

7 The arborescence created by bifurcating the interval classes of a non-linearsequence identically to the directly preceding trichords generates local linearityby echoing each segment of the non-linear progression. It is globally non-linearand locally linear. Each bifurcation introduces greater redundancy. At highdegrees of magnification, it is effectively a linear arborescence which is isomor-phic to the non-linear arborescence, exhibiting five antecedent and two conse-quent pedigrees in the same relations.

8. Finally, one can use non-linear sequences and arborescences of lower cardinalityas intervallic operators to modulate between non-linear curves.

Page 31: Linear and Non-Linear Interval Space

Any two ordered trichords that appear within the same non-linear sequence come to be adjacentsomewhere in the arborescence constructed from it. They always appear with their directantecedent or consequent in the original non-linear sequence whenever the appropriate indexicalposition is bifurcated. Beyond that, one curve is adjacent to another in one of four ways: 1) it pre-cedes the curves found in the the left-pedigree of its consequent; 2) it follows the curves found inthe 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 posi-tion 1 precedes 2, 8, 10, 16, 18, and 24, follows 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, and 24, beginstogether 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, etcetera; as a result any sequence of pitches derived from the magnification of any portion of acurve can be parsed into a binary antecedent-consequent pairs at any scale of magnitude. Thisimplies that non-linear pitch organization can be molded into any permutation of arborescentstructure, by selectively magnifying portions of the curve and eliding dominance relations wher-ever necessary.

IV Conclusion

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

Their opposition is thoroughgoing enough that we could use choose any number of designationsother than “linear” and “non-linear” to refer to them. For instance, we could call linear intervalspace “cardinal proportional space” and non-linear interval space “ordinal proportional space”. The former accommodates proportionality by separating the twelve interval classes into twelvecardinal 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 order-ings. While it does not privilege any specific ordering of a pitch set, it does register each of themas a unique trajectory. In fact, it is an image of the entire universe of sequences. Meanwhile, ordi-nal proportional space positions each basic intervallic permutation uniquely by precluding linearordering.

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 posi-tion, while each pitch in anisotropic pitch space is an intervallically distinct position. Isotropicpitch space achieves consistency by allowing each structural element to saturate a given dimen-sion with redundancy, deferring each new degree of non-redundancy to a higher dimension. Onethe other hand, the foundational sequences of anisotropic pitch space register each structural ele-ment parsimoniously, while the related arborescent sequences disperse their recurrence to thegreatest possible degree.

Page 32: Linear and Non-Linear Interval Space

Antinomies notwithstanding, our two spaces are linked in their premises and implications. Theyare both the result of an elaboration of minimum structures. The dyads and trichords whichinstantiate intervallic and proportional categories provide genetic material which is methodicallyelaborated 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 inter-vals. Anisotropic pitch space arranges these pitch sets in sequences by adding consecutive inter-vals.

Both genuses represent the intervals and trichords parsimoniously: isotropic pitch space is non-hierarchical, while anisotropic pitch space is hierarchical but uniformly recursive. Thus they areneutral with respect to preferences for certain intervallic configurations that may exist in certainstyles or in the nature of acoustics or cognition. This is a radical omission, as the intervals associ-ated 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 wereabsolute and the acoustic reality of pitches was negligible. Of course, this is an entirely paradox-ical scenario, as pitch space is discrete in the first place only because we can perceive the simplefrequency ratios found in overtones.

So as a cognitive model these interval spaces present a radical distortion, but perhaps a valuableone. For if each interval is distinct to an acute listener, then these spaces facilitate the creation ofmusic which is liminal to cognition. They present condensed transpositional templates whicheffectively treat each pitch as the punctual representation of an entire tonic region. This is in manyways what free atonality asked for but serialism failed to offer. For serialism treated pitch class-es, not interval classes, parsimoniously. Insofar as people hear ‘relative pitch’ and are complete-ly indifferent to the absolute frequency of pitch classes, the effect was to statistically neutralize ourintuition of pitch structure rather than provoking it. On the contrary, serialism instead had thesalutary effect of making parameters other than pitch more salient by comparison. Non-linearinterval space, on the other hand, totalizes the hyper-dynamism glimpsed in the transition fromlate romanticism to free atonality.

So while these pitch spaces are worthless to the cognitive scientist, for the composer they havevalue as drafting tools - as a straight-edge and a french curve, so to speak. Together, they providea 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 navigationof cardinal proportional space to be easy and ordinal proportional space to be difficult. The for-mer provides a kind of spontaneous structural fluency, while the diverging tracks of ordinal pro-portional space generally necessitate some reference to tables. Aquinas’ notion that only god isconscious of all things at once while man must think in order is germane here, and not withoutirony.


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