# Linear and Non-Linear Interval Space

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Jackson Moore explores two models of intervallic harmony - harmony structured in terms of intervals instead of pitches - and isolates several mathematical objects of basic importance.TRANSCRIPT

<p>I IntroductionA. 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.</p>
<p>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</p>
<p>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.</p>
<p>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) 3 2 4 (Octave) 1 0 +1 (P4 down)</p>
<p>(M2 down) (m3 up)</p>
<p>+2 (M2 up) +4</p>
<p>(M3 down)</p>
<p>+3 (m3 down) (M3 up)</p>
<p>(m2 up)</p>
<p>5</p>
<p>(Tritone)</p>
<p>6 +5</p>
<p>(m2 down)</p>
<p>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.</p>
<p>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: 1 +2 5 0 0 +5 +1 2 4 +5</p>
<p>-2</p>
<p>+2</p>
<p>3 3 4</p>
<p>+4</p>
<p>+3 +3 +4</p>
<p>1 5</p>
<p>6 +1 6</p>
<p>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.</p>
<p>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.</p>
<p>C. A Preliminary Inventory of Proportional Classes</p>
<p>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.</p>
<p>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:</p>
<p>Five additional proportional classes duplicate an interval class:</p>
<p>Beyond these proportional classes, there remain 14 intervallically non-redundant proportional classes, which can be grouped in 7 inverse pairs:</p>
<p>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:</p>
<p>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:</p>
<p>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.</p>
<p>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:</p>
<p>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.</p>
<p>II Linear Interval SpaceA. 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:</p>
<p>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.</p>
<p>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:</p>
<p>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:</p>
<p>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:</p>
<p>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:</p>
<p>In the isotropic representation of a proportional category, each pitch is situated at the intersection of all three linear dimensions:</p>
<p>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:</p>
<p>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:</p>
<p>C. Tetrachords and Intervallic Equilibrium</p>
<p>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:</p>
<p>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:</p>
<p>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:</p>
<p>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</p>
<p>|5,2,3| Proportional Format B</p>
<p>|1,5,6|</p>
<p>|1,3,4|</p>
<p>|2,4,6|</p>
<p>|5,3,4|</p>
<p>|1,5,6|</p>
<p>|1,2,3|</p>
<p>|2,4,6|</p>
<p>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 Proportional Format B</p>
<p>|1,2,3| 5,2,3</p>
<p>1,3,4</p>
<p>|5,3,4|</p>
<p>|1,5,4|</p>
<p>1,5,6</p>
<p>2,4,6</p>
<p>1,2,3</p>
<p>|1,3,4| 5,3,4</p>
<p>|5,2,3|</p>
<p>|1,5,4|</p>
<p>1,5,6</p>
<p>2,4,6</p>
<p>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:</p>
<p>Proportional Format A</p>
<p>Proportional Format B</p>
<p>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 t...</p>