Lewin - Special Cases of Interval Function Between Pitch-Class Sets X and Y

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Special Cases of the Interval Function between Pitch-Class Sets X and Y Author(s): David Lewin Source: Journal of Music Theory, Vol. 45, No. 1 (Spring, 2001), pp. 1-29Published by: on behalf of the Duke University Press Yale University Department of MusicStable URL: http://www.jstor.org/stable/3090647Accessed: 21-10-2015 00:38 UTC

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SPECIAL CASES OF THE

INTERVAL FUNCTION BETWEEN

PITCH-CLASS SETS X AND Y

David Lewin

1. Pcset pairs <X,Y> for which IFUNC(X,Y) is constant

1.1.1 Example 1 shows aspects of an imaginary passage for strings (bass clef) and winds (treble clef). The strings (at a piano dynamic, making fre- quent bow changes) sustain an "Ode to Napoleon" hexachord, labeled "OTN" on the example. The winds (playing louder) make figuration based on forms of a hexachord H, bracketed on the example, namely H itself, an inverted form h, the complement H* of H, the Tl 1-transpose of h, and so forth. When we explore the pc-intervals spanned between notes of OTN and notes of H, an interesting feature is manifest:

There are three ways to span interval 0 between OTN and H, namely F-to-F, G#-to-Ab, and C#-to-Db. There are three ways to span interval 1 between OTN and H, namely C-to-Db, C#-to-D, and E-to-F. There are three ways to span interval 2 between OTN and H, namely C-to-D, F-to- G, and C#-to-E. There are three ways to span interval 3 between OTN and H, namely C-to-Eb, F-to-Ab, and E-to-G. There are three ways to span interval 4 between OTN and H, namely A-to-Db, C#-to-F, and E-to- Ab. There are three ways to span interval 5 between OTN and H, namely A-to-D, C-to-F, and G#-to-Db. There are three ways to span interval 6 between OTN and H, namely A-to-Eb, G#-to-D, and C#-to-G. There are

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h

OTN

Example 1

three ways to span interval 7 between OTN and H, namely C-to-G, G#- to-Eb, and C#-to-Ab. There are three ways to span interval 8 between OTN and H, namely A-to-F, F-to-Db, and C-to-Ak There are three ways to span interval 9 between OTN and H, namely F-to-D, Gk-to-F, and E- to-Db. There are three ways to span interval 10 between OTN and H, namely A-to-G, F-to-Eb, and E-to-D. There are three ways to span interval 11 between OTN and H, namely A-to-Ab, G#-to-G, and E-to-E.

My "interval function" provides a useful format in which to sum- marize the tabulation concisely.1 Given abstract pcsets X and Y, IFUNC(X,Y) is the function which assigns, to each pc interval i, the number of ways in which i can be spanned between some note of X and some note of Y. Thus considering the various intervals between notes of OTN and notes of H, as tabulated in the paragraph above, we can write IFUNC(OTN,H)(0) = 3, IFUNC(OTN,H)(1) = 3, IFUNC(OTN,H)(2) = 3, ... and so forth through IFUNC(OTN,H)(11) = 3. Even more com- pactly, we can write

IFUNC(OTN,H) = (333333333333). Mathematically, it can be deduced that IFUNC(OTN,HFORM) must

also be (333333333333) for any set HFORM that is a transposed or inverted form of H. Mathematically, it can also be deduced that IFUNC(OTN,H*FORM) must also be (333333333333) for any set H*FORM that is a transposed or inverted form of H*. The curious reader can verify, by inspecting pertinent intervals between strings and winds on Example 1, that this tally does indeed obtain for IFUNC(OTN,h), IFUNC(OTN,H*), and IFUNC(OTN,hl 1).

Let us now inspect the various pcintervals spanned in Example 1 between notes of OTN and notes of the dyad {Eb,F}, the first two notes in the winds. We can hear that IFUNC(OTN,{Eb,F}) = (111111111111): interval 0 is spanned once, from F to F; interval 1 is spanned once, from E to F;... ; interval 10 is spanned once, from F to Eb; and interval 11 is spanned once, from E to Eb. Mathematically, it can be deduced that

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IFUNC(OTN, 2DYAD) must also = (111111111111) for any set 2DYAD that is of interval-class 2. The curious reader can verify this in the case of the 2DYAD {E,F }, that appears on Example 1 in the winds at the beginning of m. 2.

Let us consider now the first eight notes of the winds on Example 1. Since IFUNC(OTN,H) = IFUNC(OTN,h) = (333333333333), and IFUNC(OTN,{Eb,F}) = IFUNC(OTN,{E,F#}) = (111111111111), one would expect that IFUNC(OTN, {D,Ab,Db,G}) should = (222222222222). And that is indeed the case. In similar fashion, considering the first twelve notes in the winds, we infer (correctly) that IFUNC(OTN,{A,Bb,B,C}) = (222222222222).

1.1.2 We have now explored a number of pcset pairs <X,Y> whose IFUNC-values remain constant, over the twelve pcintervals. That is to say, pairs <X,Y> for which IFUNC(X,Y) has the form (nnnnnnnnnnnn), n being some integer. We shall call any such pair <X,Y> a constant- IFUNC pair. Thus <OTN,HFORM> is a constant-IFUNC pair; so is <OTN, H*FORM>; so is <OTN,2DYAD>; so is <OTN,{D,Ab,Db,G}>; so is <OTN,{A,Bb,B,C >.

Example 2 illustrates some other constant-IFUNC pairs. The example shows aspects of an imaginary passage for winds (treble clef) and harp (bass clef). The winds sustain the trichord labeled WINDS, and then its 3-transpose, the trichord WINDS3. The harp makes figuration using the tetrachord labeled HARP. <HARP,WINDS> is a constant-IFUNC pair; so is <HARP,WINDS3>; in each case the pertinent IFUNC is (111111111111).

1.1.3 From the constant-IFUNC pairs already observed we may infer yet other such pairs. For one thing, <Y,X> is a constant-IFUNC pair when-

WINDS WINDS3

( I I I . HARP HARP

Example 2

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ever <X,Y> is.2 For another, <X,Yk> is a constant-IFUNC pair whenever <X,Y> is, Yk being the k-transpose ofY.3 Then again, <X,Y*> is a constant-IFUNC pair whenever <X,Y> is, Y* being the complement ofY.4 It is also true that <X,J(Y)> is a constant-IFUNC pair whenever <X,Y> is, J being any pc-inversion operation.5

In the context of our study so far a theoretical question arises natu- rally.

1.2 QUESTION 1: What are necessary and sufficient conditions on a pair <X,Y> of pcsets, that they should be a constant-IFUNC pair?

Because <X,Y*>, <X*,Y>, and <X*,Y*> are such pairs if and only if <X,Y> is, we may restrict our attention to X andY of cardinality less than or equal to 6.

We can also restrict our attention to non-empty sets X and Y: <NULLSET,Y>, for any set Y, is trivially a constant-IFUNC pair: its IFUNC is (000000000000).

1.3 ANSWER 1A: The following-(a) through (g)-is an exhaustive list of constant-IFUNC pairs <X,Y> where the cardinality of X is between 1 and 6 inclusive, and the cardinality of Y is also between 1 and 6 inclusive. (Some pairs appear in more than one way on the list.)

(a) X (resp.Y) a whole-tone set; Y (resp.X) a T-or-I form of (01), (03), (05), (0123), (0134), (0125), (0127), (0145), (0156), (0167), (0235), (0136), (0237), (0347), (0147), (0158), (0257), (0358), (0369), (012345), (012356), (012367), (012567), (023457), (012457), (013467), (013458), (012578), (013478), (014589), (013568), (013469), (013689), or (024579). The essential feature of the sets on the long list is that each has the same number of common tones with one whole-tone set, as it has with the other.

(b) X (resp.Y) a form of (014589); Y (resp.X) a T-or-I form of (02), (06), (0123), (0167), (0235), (0136), (0246), (0257), (0268), (0369), (012346), (012678), (012357), (012467), (023568), (013569), (013679), (023579), or (02468t). The essential feature of the sets Y on the long list here is that given any augmented-triad set A, Y has the same number of common tones with T6(A), as it has with A.

(c) X (resp.Y) a form of (048); Y (resp.X) a T-or-I form of (0123), (0167), (0235), (0136), (0257), or (0369). TheY-sets on list(c) are exactly those that are on both list(a) and list(b). NoY-set on list(c) is a hexachord. The number of intervals spanned between (048) and a hexachord is 18; the number of intervals spanned between any constant-IFUNC pair X and Y must be a mulitple of 12.

(d) X (resp.Y) a form of (0167); Y (resp.X) a T-or-I form of (024),

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(048), (012345), (013458), (014589), (024579), or (02468t). The "essential feature(s)" of the sets on the Y-list here will be made clear later.

(e) X (resp.Y) a form of (0235) or (0369); Y (resp.X) a T-or-I form of (048), (012678), (014589), or (02468t). The "essential feature(s)" of these sets will be made clear later.

(f) X (resp.Y) a form of (012345), (013458), or (024579); Y (resp.X) a T-or-I form of (06), (0167), (0268), (012678), (013679), or (02468t). The "essential feature(s)" of these sets will be made clear later.

(g) X (resp.Y) a form of (012678); Y (resp.X) a T-or-I form of (03), (0134), (0235), (0347), (0358), (0369), (012345), (013458), (014589), (013469), or (024579). The "essential feature(s)" of these sets will be made clear later.

1.4 Answer 1A, as expressed by the above catalogue, is complete. But it is to some degree conceptually dissatisfying, because it seems so ad hoc. To the extent one looks on the matter from a Kantian (rather than Lin- naean) perspective, one asks if there is not some general principle that governs which specific pairs <X,Y> appear on the lists of 1.3(a)-(g), and which pairs do not.

There is indeed such a principle, and we shall formulate it presently, as "Answer 1B" in Part 2 of this paper. To formulate Answer 1B, we shall first need to define five "Fourier Properties" that a pcset X may or may not possess.

2. Five "Fourier Properties" which a pcset may or may not possess; the relevance of this to Answer 1A above (1.3)

2.1 I defined these five properties in 1959.6 Here we shall define the properties in ways that are to some degree different, though logically equivalent. We call them "Fourier Properties" because they involve mathematical considerations analogous to formal features of the way in which the mathematician Jean-Baptiste-Joseph Fourier analyzed periodic wave forms into weighted combinations of cosine and sine functions.7

We call the five properties FOURPROP(6), FOURPROP(4), FOUR- PROP(3), FOURPROP(2), and FOURPROP(1). We proceed now to define them.

2.2.1 Definition: Pcset X has FOURPROP(6) if X has the same number of notes in one whole-tone set, as it has in the other.

2.2.2 Definition: Pcset X has FOURPROP(4) if X has the same number of notes in common with each of the three diminished-seventh-chord sets.

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2.2.3 Definition: Pcset X has FOURPROP(3) if, for any augmented-triad set A, X has the same number of notes in common with T6(A), as it has in common with A.

2.2.4 Definition: Pcset X has FOURPROP(2) if, for any (0167)-set K, X has the same number of notes in common with T3(K), as it has in common with K.

2.2.5 Definition: Pcset X has FOURPROP(1) if X can be expressed as a disjoint union of tritone sets and/or augmented-triad sets. For example, X = {1,2,6,7,t} has FOURPROP(1): it can be expressed as the disjoint union of { 1,7} and {2,6,t}. X = {3,4,9,t} also has FOURPROP(1): it can be expressed as the disjoint union of {3,9} and {4,t}.

From the definitions above, one easily infers the following:

2.3.1 Any T-or-I form of set X enjoys exactly the same FOURPROPS, as does X.

2.3.2 X*, the complement of X, enjoys exactly the same FOURPROPS, as does X.

Hence, in tabulating which sets X have which of the FOURPROPS, it suffices to tabulate set classes /X/, and to restrict our attention to set classes for sets X of cardinality 6 or less.

The empty set X = NULL enjoys all five Properties, since NULL always has the same number of common tones-viz. 0-with any other set (diminished-seventh set, augmented-triad set, etc.), and since NULL can be expressed as a null-union of tritone-sets and augmented-triad-sets. We shall not concern ourself further with the empty set here.

Any singleton set (say X = {0}) enjoys none of the five Properties. That is easily verified from Definition 2.2.1 through 2.2.5.

Restricting attention, then, to sets of cardinality 2 through 6 inclusive, Table 1 shows which set classes of such sets enjoy which of the five Fourier Properties.

We are now ready to give our "Kantian" Answer 1B for our earlier Question 1. We repeat the question below, and then give Answer lB.

2.4 QUESTION 1: What are necessary and sufficient conditions on a pair of (non-empty) sets <X,Y>, that they should be a constant-IFUNC pair?

2.5 ANSWER IB: It is necessary and sufficient that for each number P = 6,4,3,2, or 1, either X or Y (or both) enjoy FOURPROP(P).

2.6(a) On Table 1 let us inspect the whole-tone set class, (02468t). We see that it has Fourier Properties 4, 3, 2, and 1. According to Answer 1B

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TABLE 1: SET CLASSES OF CARDINALITY 2 THROUGH 6

SET YES NO SET YES NO CLASS FPROPS FPROPS CLASS FPROPS FPROPS

(01) 6 4321 (02346) 2 6431 (02) 3 6421 (02458) 2 6431 (03) 62 431 (02469) 2 6431 (04) 64321 (01478) 1 6432 (05) 6 4321 all other pentachords 64321 (06) 31 642 (012345) 642 31 (012) 4 6321 (012346) 3 6421 (013) 64321 (012356) 6 4321 (014) 64321 (012456) 4 6321 (015) 4 6321 (012367) 6 4321 (016) 64321 (012567) 64 321 (024) 42 631 (012678) 431 62 (025) 64321 (023457) 64 321 (026) 64321 (012357) 43 621 (027) 4 6321 (013457) 64321 (036) 64321 (012457) 6 4321 (037) 64321 (012467) 3 6421 (048) 421 63 (013467) 6 4321 (0123) 63 421 (013458) 642 31 (0124) 64321 (012458) 64321 (0134) 62 431 (014568) 4 6321 (0125) 6 4321 (012478) 64321 (0126) 64321 (012578) 6 4321 (0127) 6 4321 (013478) 6 4321 (0145) 6 4321 (014589) 6421 3 (0156) 6 4321 (023468) 64321 (0167) 631 42 (012468) 4 6321 (0235) 632 41 (023568) 3 6421 (0135) 64321 (013468) 64321 (0236) 64321 (013568) 6 4321 (0136) 63 421 (013578) 4 6321 (0237) 6 4321 (013469) 62 431 (0146) 64321 (013569) 3 6421 (0157) 64321 (013689) 6 4321 (0347) 62 431 (013679) 31 642 (0147) 6 4321 (013589) 64321 (0148) 64321 (024579) 642 31 (0158) 6 4321 (023579) 3 6421 (0246) 3 6421 (013579) 64321 (0247) 64321 (02468t) 4321 6 (0257) 63 421 (0248) 64321 (0268) 31 642 (0358) 62 431 (0258) 64321 (0369) 632 41 (0137) 64321

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above (2.5), then, a whole-tone set will form constant-IFUNC pairs with just those sets Y that have FOURPROP(6). We can read such Y-set- classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(a), one of the categories for Answer 1A.

2.6(b) On Table 1 let us inspect the "Ode to Napoleon" set class (014589). We see that it has Fourier Properties 6, 4, 2, and 1. According to Answer 1B above (2.5), then, an Ode-to-Napoleon set will form constant-IFUNC pairs with just those sets Y that have FOURPROP(3). We can read such Y-set-classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(b), another of the categories for Answer 1A.

2.6(c) On Table 1 let us inspect the augmented-triad set class (048). We see that it has Fourier Properties 4, 2, and 1. According to Answer 1B above (2.5), then, an augmented-triad set will form constant-IFUNC pairs with just those sets Y that have both FOURPROP(6) and FOUR- PROP(3). We can read such Y-set-classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(c), another of the categories for Answer 1A.

2.6(d) On Table 1 let us inspect the (0167) set class. We see that it has Fourier Properties 6,3, and 1. According to Answer lB above (2.5), then, a set in the (0167) set class will form constant-IFUNC pairs with just those sets Y that have both FOURPROP(4) and FOURPROP(2). We can read such Y-set-classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(d), another of the categories for Answer 1A.

2.6(e) On Table 1 let us inspect the set classes for (0235) and (0369). We see that they have Fourier Properties 6,3, and 2. According to Answer 1B above (2.5), then, sets in either of these set classes will form constant- IFUNC pairs with just those sets Y that have both FOURPROP(4) and FOURPROP(2). We can read such Y-set-classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(e), another of the categories for Answer 1A.

2.6(f) On Table 1 let us inspect the set classes for (012345), (013458), and (024579). We see that they have Fourier Properties 6, 4, and 2. According to Answer 1B above (2.5), then, sets in either of these set classes will form constant-IFUNC pairs with just those sets Y that have both FOURPROP(3) and FOURPROP(1). We can read such Y-set- classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(f), another of the categories for Answer 1A.

2.6(g) On Table 1 let us inspect the set class for (012678). We see that it has Fourier Properties 4, 3, and 1. According to Answer 1B above (2.5), then, sets in this set class will form constant-IFUNC pairs with just those

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sets Y that have both FOURPROP(6) and FOURPROP(2). We can read suchY-set-classes off of Table 1. Doing so, we find that they are just those classes listed in 1.3(g), another of the categories for Answer 1A.

2.7 2.6(a) through (g) exhaust the "Linnaean" lists of 1.3(a) through (g). There are no more constant-IFUNC pairs to be listed (of cardinalities between 2 and 6 inclusive). We can see why this is so from a "Kantian" perusal of Answer 1B (2.5). That Answer tells us that, for <X,Y> to form a constant-IFUNC pair, it is necessary that for each number P = 6,4,3,2, or 1, either X or Y (or both) must enjoy FOURPROP(P). Consequently, either X or Y must enjoy at least three of the five Fourier Properties. Looking over Table 1, we can see that cases 2.6(a) through (g) above have indeed taken into account every set class on that Table whose sets possess three or more of the five properties. So there are no more constant- IFUNC pairs (of cardinalities between 2 and 6 inclusive).

2.8 Earlier on (inl.1.3), we noted that <X,J(Y)> is a constant-IFUNC pair whenever <X,Y> is, J being any pc-inversion operation. Note 5 promised that we would see later why the proposition was true. We are now in a position to demonstrate its truth. J(Y) will have exactly those Fourier Properties, that Y has (2.3.1). So if <X,Y> is a constant-IFUNC pair, X and Y together have Properties 6, 4, 3, 2, and 1 (via Answer 1B); then X and J(Y) together have Properties 6, 4, 3, 2, and 1; and so <X,J(Y)> is a constant-IFUNC pair (via Answer 1B).

3. Pcset pairs <X,Y> for which IFUNC(X,Y) has two alternating values

3.1 Let us turn our attention again to example 2. In that imaginary passage, the winds sustained first the trichord WINDS = { G,A,B }, and then the trichord WINDS3 = { Bb,C,D }, while the harp made figurations on the tetrachord HARP = {Dt,A,E,Bb}. The tetrachord can abstractly be parsed into two tritones, {D#,A} and {E,Bb }; in the music of example 2 the parsing is relevant because, in each tetrachord-presentation of the example, the tritone {D$,A} consistently lies in a higher register, and the tritone {E,Bb } consistently lies in a lower register.

We compute IFUNC({D#,A}, {G,A,B}) = (101010101010); IFUNC({D#,A}, {C,D,E}) = (010101010101); IFUNC({E,Bb}, {G,A,B}) = (010101010101); IFUNC({E,Bb}, {C,D,E}) = (101010101010).

These interval functions have a special property. Though their values are not constant, they do alternate, between only two alternate values. That is, they are all of form (mnmnmnmnmnmn), m and n being suitable integers.

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3.2 Definition: Given pcsets X and Y, we shall call the pair <X,Y> an alternating-IFUNCpair when IFUNC(X,Y) has the form (mnmnmnmnmnmn) for some integers m and n.

According to our definition, a constant-IFUNC pair is a fortiori an alternating-IFUNC pair-this being a particular case where m = n.

3.3 Further examples: (a) If W is a whole-tone set, then <W,Y> is an alternating-IFUNC

pair, for any pcset Y. Specifically, IFUNC(W,Y) = (mnmnmnmnmnmn), where m is the number of Y-notes that lie in W, and n is the number of Y-notes that lie in W*. The curious reader to whom this is not clear may verify it as an exercise.8

(b) If X is any hexachord in set class (012678), and if Y is any pentachord in set class (02346), or (02458), or (02469), then <X,Y> is an alternating-IFUNC pair. The reader can verify that IFUNC({C,C#,D,Ft, G,Ab}, {C,D,D#,E,F }) = (323232323232).

3.4 QUESTION 2: What are necessary and sufficient conditions on a pair <X,Y> of pcsets, that they should be an alternating-IFUNC pair?

3.5 ANSWER 2: It is necessary and sufficient that for each number P = 4,3,2, or 1, either X or Y (or both) enjoy FOURPROP(P).

As Table 1 tells us, any whole-tone set W already enjoys all four of Properties 4, 3, 2, and 1. That is why <W,Y> will always be an alterating-IFUNC pair, whatever the set Y (3.3(a)): it is not necessary for Y to contribute any of the four Properties at issue to the tally for Answer 2.

As Table 1 tells us, any hexachord X of set class (012678) enjoys Properties 4,3, and 1. And-as the Table also tells us-any pentachord Y of set class (02346), or (02458), or (02469) will enjoy Property 2. Thus, according to Answer 2, <X,Y> will be an alterating-IFUNC pair, as asserted in 3.3(b).

As Table 1 tells us, any tritone-dyad X enjoys Properties 3 and 1, while any (024)-trichord Y enjoys Properties 4 and 2. Thus, according to Answer 2, <X,Y> will be an alternating-IFUNC pair, as observed in the particular cases of (3.1) above.

4. IFUNCs of form (kmnkmnkmnkmn), etc.

4.1 We might ask further Questions about IFUNCS, by analogy with Question 1 and Question 2. For instance, we might ask, as a Question, what are necessary and sufficient conditions on a pair <X,Y> of pcsets, that they should have an IFUNC of form (kmnkmnkmnkmn), k, m, and n being integers. The reader will probably already suspect the Answer: It is necessary and sufficient that for each number P = 6,3,2, or 1, either X or Y (or both) enjoy FOURPROP(P).

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So for example,Table 1 tells that any tetrachord X in set class (0235) enjoys Properties 6, 3, and 2, while any pentachordY in set class (01478) enjoys Property 1. According to our Answer just above, <X,Y> will have an IFUNC of form (kmnkmnkmnkmn). The reader can indeed verify that IFUNC({C,D,EK,F}, {C,C#,E,G,G#}) = (122122122122). Another example: Table 1 tells that any tetrachord X in set class (0134) enjoys Prop- erties 6 and 2, while any tetrachord Y in set class (0268) enjoys Proper- ties 3 and 1. According to our Answer just above, <X,Y> will have an IFUNC of form (kmnkmnkmnkmn). The reader can indeed verify that IFUNC({C,Db,Eb,E}, {C,D,F#,G }) = (112112112112).

But I am not disposed to lavish much musical attention upon this mathematical phenomenon. When we listen to an alternating IFUNC, of form (mnmnmnmnmnmn), we do not find it difficult to distinguish between our hearing of even intervals (m of each), and our hearing of odd intervals (n of each). In contrast, listening to an IFUNC of form (kmnkmnkmnkmn), we may perhaps notice that we hear k of each of the intervals 0, 3, 6, and 9, but I do not think we will easily notice that we hear m of each of the intervals 1, 4, 7, and t, or n of each of the intervals 2, 5, 8, and e. I do not think we are predisposed to group intervals 1, 4, 7, and t in our ears, the way we are predisposed to group together the odd intervals in toto, or the even intervals in toto.

4.2 In similar fashion, we can explore mathematically pairs <X,Y> where either X or Y satisfy all Properties with the exception of Property 3, or of Property 2, or of Property 1. Similarly, I do not apprehend any reasonably immediate musical intuitions, that correspond to such cases.

For example, {C,C#,D } and C,D,E6,F}, together, satisfy all Proper- ties except for Property 1. Their IFUNC is (222211000011). The form of the IFUNC is perhaps mathematically suggestive, but it is hard to catch a pertinent musical intuition that corresponds to the suggestion. Like- wise, {C,D,E} and {C,C#,E,F#}, together, also satisfy all Properties except for Property 1. Their IFUNC is (121110101112). The form of the IFUNC is again perhaps mathematically suggestive, but it is again hard to catch a pertinent musical intuition that corresponds to the suggestion. It is also hard to see offhand what the two preceding IFUNCs, (222211000011) and (121110101112), have to do with each other, espe- cially in connection with some pertinent musical intuition.

5. Some observations on ways that IFUNCs can be projected musically

5.1 In Example 1 we examined interval functions involving abstract pcsets OTN, H, {Eb,F}, and others. As we studied the various IFUNCs, each abstract pcset was represented in the music of the example by a cer-

11



tain collection of pitches. In each case, each pitch class of the abstract pcset was represented by only one pitch in the pertinent music. Further- more, in each case the pitches representing abstract pcset "X" were articulated from those representing abstract pcset "Y" in a number of ways. I cite three here: the pitches representing X were lower than those representing Y; the pitches representing X were played by a different instru- ment(al group) than those representing Y; the pitches representing one pcset were sustained longer, while those representing the other pcset were detached and of shorter duration. The same observations pertain to Example 2.

In those respects, the pitches of Examples 1 and 2 project their pcsets in an overly cautious manner. Any one of the three cited "ways" could have sufficed. Then, too, there are plenty of other "ways" in which abstract pcsets X and Y could have their interval function projected musically. Example 3 illustrates the point.

5.2 The example presents aspects of an imaginary passage for solo piano. The example projects abstract pcsets {E,Bh} and {F,A,C$}, together with their interval function, in ways we shall explore presently.

First, we observe that the two pitch classes of {E,Bb} are projected by nine different pitched events. Given the texture of the passage, this does not pose any difficulty for our aural/mental construction of a pertinent 2-element pcset. But it does raise interesting issues pertaining to our perception of IFUNC({E,B 1}, {F,A,C }) in the passage. We shall defer considering that matter for the moment, and move on to observe other aspects of Example 3 that contrast with our observations about examples 1 and 2.

Specifically, Example 3 does not articulate {E,Bb} from {F,A,C#} by instrumentation, nor by consistently placing the notes of one set in a lower register than the notes of the other, nor by sustaining some pitches in marked contrast to others. Rather, here the one set is articulated from

P-- f P

P A ZE 7ampl 3

Example 3

12



the other by before/after relations in time, by rhythmic texturing, and by dynamic profiling-"the beginning of {E,B? }" has a piano dynamic, and after the crescendo and rest, "the beginning of {F,A,C#}" returns to the piano dynamic.

The interested reader will soon recall or discover other ways in which two abstract pcsets X and Y can effectively be projected by musical textures, so as to render a listener sensitive to the projection of IFUNC(X,Y).

5.3.1 We return now to the issue whose consideration we deferred a short time ago. How can we construct, in ear and mind, a perception of an IFUNC between the two-element abstract set {E,Bb} and the three- element abstract set {F,A,C }, when the music of Example 3 represents {E,Bb} by nine-not two-distinct pitch-events?

We can elaborate the issue as follows. Abstractly, IFUNC({E,Bb}, {F,A,C#}) = (010101010101). That signifies: there are no ways to span any even pc-interval between {E,Bb} and {F,A,C# }; there is one way to span any odd pc-interval between the pcsets. But, more concretely, there are five E-events in the first half of Example 3, along with the later F- event-should not that count as five occurrences of directed pc-interval 1? And there are four BL-events in the first half of Example 3, along with the later Ct-event-should not that count asfour occurrences of directed pc-interval 3? In similar fashion, should we not count five (perceptual) occurrences of interval 5 (from the various Es to A), four occurrences of interval 7 (from the various Bbs to F), five occurrences of interval 9 (from the various Es to Ct), and four occurrences of interval 11 (from the various Bbs to A)? Should we not represent our perceptual intuitions, in this context, by the "concrete" function (050405040504), rather than the "abstract" function (010101010101)?

5.3.2 To be sure, there is no need to devalue either of these intuitions- "intensions" is a better word-at the expense of the other. A better for- mulation of the issues at hand asks what the two functions might have to do with each other mathematically, that is pertinent to the situation, and then asks what such a mathematical relation in fact models, by way of relating the two intensions-the "abstract" one modeled by (010101010101) and the "concrete" one modeled by (050405040504).

5.4.1 A pertinent mathematical relation between the two functions is manifest when we normalize each function, multiplying all function-values by a constant factor that makes the normalized function-values sum to 1. The "abstract" interval function (010101010101) thereby normalizes to (O 0 1/6 0 1/6 0 1/6 0 1 1/6 1/6 0 1/6), while the "concrete," event- counting function (050405040504) normalizes to (O 5/27 0 4/27 0 5/27 0 4/27 0 5/27 0 4/27). 5/27 is greater than 1/6 by 1/54; 4/27 is less than 1/6

13



by 1/54; accordingly, for any interval-argument i, the difference between one normalized function-value and the other, at argument i, is consistently less than or equal to 1/54 in absolute value, and the two normalized functions are quite "close" to each other.9

5.4.2 What, now, does the closeness of the two normalized functions model, in the way of aural/psychological intensions?

The normalized IFUNC, (0 1/6 0 1/6 0 0 1/6 0 1/ 1/61 0 1/6), models an intuition of probability. If we select a pitch class x at random from {E,Bb} and a pitch class y at random from {F,A,C#}, then when we examine the pc interval i from x to y, the probability is 0 that we will find i to be an even interval, while the probability is equal (hence 1/6) that we will find i to be any of the six odd intervals.

The "concrete" normalized function (0 5/27 0 4/27 0 5/27 0 4/27 0 5/27 0 4/27), in contrast, models an intension of statistical observation or experience. Having listened to example 3, we have heard 27 pitch- intervals between the first and second halves of the example. Of those pitch-interval events experienced, none were in any even pc-interval class, 5/27 of them were in pc-interval class 1, 4/27 of them were in pc- interval class 3, and so forth.

To be sure, a listener would not consciously tally up individual pitch- interval sensations here so exactly. A psychologically more appropriate description would put it that none of the experienced interval-events was in any even pc-interval class, some were in each of the odd pc-interval classes, and there seemed to be a slight preponderance of events in interval classes 1, 5, and 9, over events in interval classes 3, 7, and 11.10

5.4.3 The mathematical closeness of the two normalized functions here thus models a goodness-of-fit, between an intuition of probabilistic expectation and an intension of certain experiences organized statisti- cally. Were the fit not so good, we would likely be much less willing to analyze the first half of m. 1, in Example 3, as projecting the abstract pcset {E,Bb}.

For instance, keeping the rhythm and dynamics of the example, suppose we heard eight Es over the first quarter-beat duration, then a loud high Bb, and then the event of the third beat. We would be much more likely to segment the pc analysis as proceeding "from {E}, to {Bb}, to {F,A,C# }," rather than "from {E,Bb} to {F,A,C# }." That intension would be modeled, in part, by the non-closeness of the normalized probability function (0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6) and the normalized statistical-experience function (0 8/27 0 1/27 0 8/27 0 1/27 0 8/27 0 1/27).

"In part." To clarify that qualification, the reader can ponder yet another variation on Example 3. Suppose we heard five Es and then four Bbs. Our normalized statistical-experience function would then again be

14



(0 5/27 0 4/27 0 5/27 0 4/27 0 5/27 0 4/27), but we would be somewhat more hesitant than in the case of Example 3, to accept the segmentation "{E,Bb}-then-{F,A,Ct }" rather than "{E}-then-{(B}-then-{F,A,C }." Somewhat more. These matters involve ways in which the five E-events and the four Bb-events are distributed, over time (and over registral space, and metrically, and dynamically, et al.) Relevant statistical mathematics exist to model such aspects of the situation. The matter is well worth pursuing, but a fuller pursuit would distract us unduly from the main thread of matters at hand in the present article. At present, the main thread is our observation, that IFUNCs can reflect statistical listening involving many pitch-events-per-pitch-class and many pitch-interval- events-per-pc-interval-class. Example 4, representing mm. 117-119 from Schoenberg's String Trio, illustrates the pertinence of the observation to analyzing actual music.

5.5 The lower instruments, together with the violin's {B,D}, project a form of the piece's basic hexachord. The lower instruments, together with the violin's {G,E}, project an inverted form of the piece's basic hexachord. Analytically, then, we "know" that a pertinent transforma- tional relation between {B,D} and {G,E} in this passage is by inversion. But in what way do we (can we) perceive that inversional relation? Are we not likely to hear with overwhelming force the transpositional relation T5 between {B,D} and {G,E}, instead?

Without contesting our perceptive awareness of T5 relations (and then T7 relations) within the violin part of Example 4, we can help ourselves become perceptively aware of an inversional relation as well, using IFUNC-listening. To do so, we listen first to the interval function between {C,Db,F,Gb} and {B,D}, then to the interval function between {C,Db,F,Gb} and {G,E}. The music gives us plenty of occasion to listen alternately to the two interval functions, comparing our statistical impressions of the two.

IFUNC({C,Db,F,Gb}, {B,D}) = (011110110011) IFUNC({C,Db,F,Gb}, {G,E}) = (011001101111)

As the violin's dyads alternate over the passage, we can become sta- tistically sensitive to the normalized IFUNCs at hand. Specifically, we can focus our attention on not experiencing pc intervals 3, 4, or 7 between {C,Db,F,G } and {G,E}, whereas we do experience them, in equal measure with other intervals, between {C,Db,F,Gb} and {B,D }. Likewise, we do not experience pc intervals 5, 8, or 9 between {C,Db,F,Gb and { B,D }, whereas we do experience them, in equal measure with other intervals, between {C,Db,F,Gb} and {G,E}. "Not experiencing intervals 5, 8, or 9" is an inversional experience to "not experiencing intervals 3, 4, or 7," when we also experience no interval-0 sensations in either contrapuntal

15



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field, and we also experience equal amounts of interval-i sensations for pc intervals i = 1, 2, 6, 10, and 11, in both contrapuntal fields. In short, our statistical sensations fit well with a recognition of the inversional relation between the two IFUNCs above-the one is the other read back- wards, starting at interval 0.

To be sure, there is also a T5/T7 relation between the IFUNCs. We can see that relation if we start at the second zero of IFUNC({C,Dl,F,G)}, {B,D})-that is, at the value IFUNC({C,Db,F,Gb}, {B,D})(5) = 0-and read cyclically to the right-we obtain IFUNC({C,Db,F,Gb}, {G,E}). The two IFUNCs are transpositions (rotations), the one of the other, as well as inversions. The point is, it is easier-for me, at any rate-to hear an inversional relation between IFUNCs here, than to hear a direct inversional relation between the violin dyads {B,D} and {G,E}.

That impression is helped by my hearing a "middleground" interval function as well in the passage, between {C,Db,F,Gb} and {B,D,G,E}- or rather a statistical ensemble of intervallic impressions between the lower strings and the violin part as a whole, a statistical ensemble whose normalized function is a good fit to the abstract interval function.11 IFUNC({C,Db,F,Gk}, {B,D,G,E}) = (022111211122); the interval function is "interval-complement symmetrical," in the sense that the value of the function is the same for interval -i as it is for interval i. The property comports well with inversional relationships among the pcsets involved, while T5/T7 relations are not manifest within the middleground IFUNC.12

Having explored some relations between IFUNCs and listener phe- nomenology, we return now to more abstract investigations regarding certain kinds of IFUNCs.

6. When is IFUNC(X,Y) the same function as IFUNC(X,Z)?

6.1 Example 5 recapitulates and continues the passage of Example 3, for solo piano. Many textural features of the music make us hear m. 2 of the example as a variation on m. 1. Pertinent interval functions support that hearing:

IFUNC({E,Bb}, {F,A,C}) = (010101010101); IFUNC({E,Bb}, {G,A,B}) = (010101010101).

That is, an interval function pertinent to hearing the second measure of the example is exactly the same as the analogous interval function for the first measure of the example. Example 6 helps us focus our ears on the phenomenon, suggesting interesting aspects of Es and Bbs that appear in different registers, as regards various pitch-intervals they contribute to our sensations of "variation" between m. 1 and m. 2.

17



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6.2 Since IFUNC({E,Bb}, {F,A,C }) = (010101010101), we can see that {E,Bb} and AUGTRIAD will have exactly the same IFUNC, when AUGTRIAD is any even transposition of {F,A,C#}. Likewise, since IFUNC({E,Bb}, {G,A,B}) = (010101010101), we can see that {E,Bb} and 2WHOLESTEPS will have exactly the same IFUNC, when 2WHOLESTEPS is any even transposition of { G,A,B }. But why should {G,A,B} make exactly the same IFUNC with {E,B,}, that {F,A,C#} does? The question suggests a broader general question that we can ask about IFUNCs.

6.3 QUESTION 3: Under what circumstances will pcsets X, Y, and Z be such that IFUNC(X,Y) = IFUNC(X,Z)?

6.4 ANSWER 3: We disallow X =NULLSET.13 With that qualification, it is necessary and sufficient that

(a) Y and Z should have the same cardinality and that (b) for any P = 6, 4, 3, 2, or 1, either X possesses Fourier Property P,

or else Y and Z are in Fourier Relation P (or both). To be sure, we have not yet defined what it means for sets Y and Z to

be "in Fourier Relation P." We shall proceed at once to give the pertinent definitions.

19

1 11



6.5.1 Definition: Pcsets Y and Z are in FOURREL(6) when the number of notes that Y has in common with one whole-tone-set W, minus the number of notes that Y has in common with the other whole-tone-set W', is the same as the number of notes that Z has in common with W, minus the number of notes that Z has in common with W'. Writing "YAW" for "the set of notes in common between Y and W," writing "card" for "the cardinality of," we can express the condition for Y and Z to be in FOUR- REL(6) more concisely: (card(YAW) - card(YAW')) = (card(ZAW) -

card(ZAW')).

6.5.2 Definition: Pcsets Y and Z are in FOURREL(4) when, for any diminished-seventh-chord set D, (card(YAD) - card(YAT4(D)) = (card(ZAD) - card(ZAT4(D)).

6.5.3 Definition: Pcsets Y and Z are in FOURREL(3) when, for any augmented-triad set A, (card(YAA) - card(YAT6(A)) = (card(ZAA) - card (ZAT6(A)).

6.5.4 Definition: PcsetsY and Z are in FOURREL(2) when, for any (0167)- set K, (card(YAK) - card(YAT3(K)) = (card(ZAK) - card(ZAT3(K)).

6.5.5 Definition: Pcsets Y and Z are in FOURREL(1) when, for any ic2- dyad J, (card(YAJ) - card(YAT6(J)) = (card(ZAJ) - card(ZAT6(J)).

6.6 We check out the criteria of Answer 3 (6.4), for the case where X = {E,Bb}, Y = {F,A,C#}, and Z = (G,A,B}. X is not null; Y and Z have the same cardinality. We have, then, to check out that criterion (b) of the Answer is satisfied: for any P = 6, 4, 3, 2, or 1, either X possesses Fourier Property P, or else Y and Z are in FOURREL(P) (or both).

Inspecting Table 1, we see that X = {E,BL} has FOURPROPs 3 and 1. For IFUNC(X,Y) = IFUNC(X,Z), then, the Answer tells us it will suffice forY = {F,A,C#} and Z = {G,A,B} to be in FOURRELs 6, 4, and 2.

Y = {F,A,C#} and Z = {G,A,B} are indeed in FOURREL(6), according to definition (6.5.1): setting W = {A,B,C#,E6,F,G}, then (card(YAW) - card(YAW')) = 3 - 0 = (card(ZAW) - card(ZAW')).

Y = {F,A,C#} and Z = { G,A,B} are indeed in FOURREL(4), according to definition (6.5.2): for any diminished-seventh-chord set D, Y has exactly 1 note in common with D, and so does Z. Hence (card(YAD) - card(YAT4(D)) = 1 - 1 = (card(ZAD) - card(ZAT4(D)).

Y = {F,A,C#} and Z = {G,A,B are indeed in FOURREL(2), according to definition (6.5.4): for any (0167)-set K, Y has exactly one note in common with K, and so does Z. Hence (card(YAK) - card(YAT3(K)) = 1 - 1 = (card(ZAK) - card(ZAT3(K)).

Y = {F,A,C# } and Z = { G,A,B } thus enjoy Fourier Relations 6, 4, and 2. Since X = {E,Bb} enjoys Fourier Properties 3 and 1, Answer 3 tells us

20



that IFUNC(X,Y) will be the same function as IFUNC(X,Z)-a fact we have already observed directly.

7. A simplifying observation

7.1 {F,A,C#} and {G,A,B} have a common tone, A. We can express {F,A,C# } as the disjoint union of {F,C#} and {A}. The various intervals that {E,Bb} makes with {F,A,C#} are then the intervals that {E,Bb} makes with {F,C#}, together with the intervals that {E,Bb} makes with {A}. In short,

(i) IFUNC({E,Bb},{F,A,C#}) = IFUNC({E,Bb},{F,C#}) + IFUNC({E,Bb},{A}).

In like manner,

(ii) IFUNC({E,B },{G,A,B}) = IFUNC({E,Bb},{G,B}) + IFUNC({E,Bb},{A}).

Comparing (i) and (ii), we see that IFUNC({E,Bb},{F,A,Ct}) will equal IFUNC({E,Bb},{G,A,B}) if, and only if, IFUNC({E,Bb),{F,C#}) = IFUNC({E,Bb},{G,B }).

For this case, one can see directly-without inspecting the Proper- ties-that IFUNC({E,Bb},{F,C#}) does indeed equal IFUNC({E,Bb}, {G,B }). That is because {E,Bb} is its own T6-transpose, while {G,B } is the T6-transpose of {F,C#}. Hence IFUNC({E,Bb}, {F,C#}) = IFUNC (T6{E,Bb }, T6{F,C#t}) (the interval from x-to-y being always the same as the interval from (x+6)-to-(y+6)), and that interval function equals IFUNC({E,Bb}, {G,B}).

7.2 In (7.1) above we reduced a certain case of IFUNC(X,Y) = IFUNC(X,Z), to a case of IFUNC(X,Y?) = IFUNC(X,Z?), where Y? = Y - (YAZ) and Z? = Z - (YAZ). In the reduced case, IFUNC(X,Y?) = IFUNC(X,Z?), the reduced sets Y? and Z? have no common tones. The work of the following section will show that such a reduction can always be carried out and so, in considering pairs <Y,Z> satisfying IFUNC(X,Y) = IFUNC(X,Z), it will suffice to consider "reduced" pairs <Yo, Z?> where Y? and Z? have no common tones.

7.3.1 The basic general observation, part 1: Let X, Y and Z be pcsets, let Y? be Y - (YAZ), let Z? be Z - (YAZ). Then IFUNC(X,Y) = IFUNC(X,Z) if, and only if, IFUNC(X,Y?) = IFUNC(X,Z?).

Demonstration:

(i) IFUNC(X,Y) = IFUNC(X,Y?) + IFUNC(X, (YAZ)). That is because Y? and (YAZ) are disjoint: the various dyads {x,y)

spanning interval i from X to Y are then the various i-dyads {x,y} from

21



X to Y?, together with the (different) various i-dyads {x,y} from X to (YAZ). Likewise,

(ii) IFUNC(X,Z) = IFUNC(X,Z?) + IFUNC(X, (YAZ)).

Comparing (i) and (ii) directly above, we see that the assertion of (7.3.1) follows logically.

7.3.2 The basic general observation, part 2: Let X be a pcset; let A and B be disjoint pcsets; let C be any pcset disjoint both from A and from B. Let Y = AuC; let Z = BuC. Then IFUNC(X,Y) = IFUNC(X,Z) if, and only if, IFUNC(X,A) = IFUNC(X,B).

Demonstration: C is the set of "common tones" between Y and Z just above-that is, C = YAZ. So A = Y - (YAZ) is the Yo of (7.3.1); likewise B = Z - (YAZ) is the Z? of (7.3.1). (7.3.2) then follows from (7.3.1).

8. A catalogue of reduced pairs <Y,Z> for which Y and Z are in the various Fourier Relations

8.1 By a "reduced pair" <Y,Z> I shall mean a disjoint pair of pcsets that have the same positive cardinality. The common cardinality of Y and Z will then be between 1 and 6 inclusive. (A pair of heptachords cannot be disjoint.) Here is the catalogue.

8.2.1 Disjoint singletons enjoy FOURREL(6) when they lie in the same whole-tone set. enjoy FOURREL(4) when they lie in the same dim7th set. enjoy FOURREL(3) when they lie in the same aug-triad set. enjoy FOURREL(2) when they lie a tritone apart. cannot enjoy FOURREL(1).

8.2.2 Disjoint dyads Y and Z

enjoy FOURREL(6) when they are both of odd interval-class, or when they are both of even ic and lie (disjointly) within the same whole-tone set.

enjoy FOURREL(4) when they both lie within the same dim7th set, or when there are different dim7th sets D and D' which each contain one note of Y and one note of Z.

enjoy FOURREL(3) when there are different augtriad sets A and A' which each contain one note of Y and one note of Z.

enjoy FOURREL(2) when Y is not of ic6 and Z = T6(Y), or when Y and Z are both of ic3.

enjoy FOURREL(1) when Y and Z are both tritones, and one is the T4-transpose of the other.

8.2.3 Disjoint trichords Y and Z

enjoy FOURREL(6) when Y and Z together make up a whole-tone

22



set, or when Y and Z each have two common tones with one whole-tone set and one with the other.

enjoy FOURREL(4) when there are different dim7th sets D and D' such that Y and Z each have two common tones with D and one with D'; or when the three notes of Y lie in the three different dim7th sets, the same being true of Z.

enjoy FOURREL(3) when there is an augtriad set A such that Y has two common tones with A and one with T6(A), while Z has one common tone with each of A, T3(A), and T9(A); or when the same is true with the roles of Y and Z reversed; or when there is an augtriad set A such that Y and Z each have one common tone with each of A, T3(A), and T9(A).

enjoy FOURREL(2) when one of (a) through (d) below is the case. (a) Y contains no tritone and Z = T6(Y). (b) For some T-or-I operation OP, Y = OP({0,1,4)) and Z =

OP({5,6,8}), or OP({6,8,e}), or OP({2,6,e}). (c) The same as (b) with the roles of Y and Z reversed. (d)For some pc numbers j and k where j-k is odd, Y =

{j,j+2,j+4} or {j,j+4,j+8}, while Z = {k,k+2,k+4} or {k,k+4,k+8 .

enjoy FOURREL(1) when either (a) or (b) below is the case. (a) Y and Z are different augtriad sets. (b) For some T-or-I operation OP, Y = OP({0,4,j}) and Z =

OP({2, j+t, j+2}), j being an odd number mod 12.

8.2.4 Disjoint tetrachords Y and Z enjoy FOURREL(6) when either (a) or (b) below is the case.

(a) One whole-tone set consists of three notes from Y and the other three from Z, while the other whole-tone set contains one note of Y and one (different) note of Z.

(b) Each whole-tone set contains two notes of Y and two (different) notes of Z.

enjoy FOURREL(4) when either (a) or (b) below is the case. (a) one of the three dim7th sets consists of two notes from Y

and two (different) notes from Z, while each of the other dim7th sets contains one note from Y and one (different) note from Z.

(b) two of the three dim7th sets each consist of two notes from Y and two (different) notes from Z.

enjoy FOURREL(3) when either (a) or (b) below is the case. (a) For some augmented-triad set A, Y has two notes in A

and two in T6(A), while Z has two notes in T3(A) and two in T9(A).

(b) For some augmented-triad set A, Y has two notes in A

23



and one in T6(A), while Z has 1 note in A and none in T6(A); if A' is the augmented-triad set containing the fourth note of Y, then Z has two notes in A' and one in T6(A').

enjoy FOURREL(2) when either (a) or (b) below is the case. (a) Y contains no tritone and Z = T6(Y). (b) Y is the disjoint union of two ic3-dyads, and so is Z.

enjoy FOURREL(1) when Y is the union of two disjoint ic-dyads, and so is Z.

8.2.5 Disjoint pentachords Y and Z enjoy FOURREL(6) when one whole-tone set consists of three

notes from Y and the other three from Z. (b) Each whole-tone set contains two notes of Y and two

(different) notes of Z. enjoy FOURREL(4) when, for some dim7th set D, D contains one

note of Y and one note of Z, while T4(D) and T8(D) each comprise two notes of Y and two notes of Z.

enjoy FOURREL(3) when either (a) or (b) below is the case. (a) For one whole-tone set W, YAW is an ic2-or-ic4 dyad,

while ZAW = W - (that dyad); for the other whole-tone set W', YAW' is a trichord in sc (024) or sc (026), while ZAW' is the note that forms an augmented triad with the ic4-dyad of YAW'.

(b) The same as (a), reversing the roles of Y and Z. enjoy FOURREL(2) when either (a) or (b) below is the case.

(a) For some dim7th set D, Y (resp.Z) = T4(D) plus one note of D, while Z (resp.Y) = T8(D) plus one note of D; the two cited notes of D lie a tritone apart.

(b) For some dim7th set D, Y (resp.Z) contains three notes of T4(D), while Z (resp.Y) contains three notes of T8(D); Y (resp.Z) contains the fourth note of T8(D), while Z (resp.Y) contains the fourth note of T4(D); Y and Z each contain one note of D, and the two cited notes of D lie a tritone apart.

enjoy FOURREL(1) when Y and Z are disjoint forms from set class (04817) (the unique pentachordal set class enjoying FOURPROP(1))-that is, when for one of the whole-tone sets W, Y comprises an augmented triad from W plus an ic6- dyad from W', while Z comprises the other augmented triad from W plus a different ic6-dyad from W'.

8.2.6 Disjoint hexachords Y and Z = Y* enjoy FOURREL(6) when Y has three notes in common with each

whole-tone set.

24



enjoy FOURREL(4) when Y has two notes in common with each dim7th set.

enjoy FOURREL(3) when for some augmented-triad set A, Y has two common tones with A, two with T6(A), one with T3(A), and one with T9(A).

enjoy FOURREL(2) when Y is a whole-tone set, or when Y contains no tritone (so that Z = Y* = T6(Y)).

enjoy FOURREL(1) when Y has FOURPROP(1))-that is, when Y is a disjoint union of three ic6-dyads, or of two augmented- triad sets.

8.3 The catalogue of (8.2.1) through (8.2.6) above can be verified by determined and systematic inspection, using definitions (6.5.1) through (6.5.5), together with the assumptions that Y and Z are disjoint, and that Y and Z have the same cardinality. To verify the various possibilities for FOURREL(2), in the catalogue of (8.2.1) through (8.2.6), it is helpful to lay out the various (0167)-sets in the form of Table 2. To verify the various possibilities for FOURREL(1), it is helpful to lay out the various ic2- dyads in the form of Table 3.

TABLE 2 0167 45te 8923 349t 7812 e056

TABLE 3

02 46 8t 35 79 el 68 tO 24 9e 13 57

8.4 Some examples will show how the the catalogue of (8.2) can be used.

8.4.1 Task: to find all non-null pcsets X satisfying IFUNC(X, {CDEFF#tGB }) = IFUNC(X, { C#tDEFGAB}).

Step 1: By removing common tones, we reduce the task: to find all non-null pcsets X satisfying IFUNC(X, {CDF#G#}) = IFUNC(X, {C#D#GA}).

Step 2: Inspecting the criteria for disjoint tetrachords on the catalogue (8.2.4), we see that {CDF#G# } and { C#D#GA} are in Fourier Relations 3 and 1, but not in Relations 6, 4, or 2.

Step 3: By Answer 3 (6.4), we conclude that the X we want are exactly those Xs that enjoy Fourier Properties 6, 4, and 2. We can read these Xs off Table 1; they are those Xs in set classes (012345), (013458), (014589), or (024579). ("Or their complements," but here each listed sc is self-complementary.)

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We have completed the task. For instance, we know that

IFUNC({CDEFGA}, {CDEFFNtG#tB) = IFUNC({CDEFGA}, {C#D#EFGAB ),

and that

IFUNC({F#GABbBD}, {CDEFF#GGB }) = IFUNC({F#GABbBD}, {C#D#EFGAB ),

and so forth.

8.4.2 Task: Let X be the octachord {C#EF#GG#AB B B. We want to find all pentachords Y and Z satisfying IFUNC(X,Y) = IFUNC(X,Z).

Step 1: Y and Z will be such if and only if IFUNC(X*,Y) = IFUNC(X*,Z), X* being the complement of X, namely {CDE,F} (via 2.3.2).

Step 2: Looking up the set class of X* on Table 1, we see that X* has Fourier Properties 6, 3, and 2. Accordingly, by Answer 3 (6.4), the pairs <Y,Z> we seek are just those pairs for which Y and Z are in Fourier Rela- tions 4 and 1.

Step 3: Let <Y?,Z?> be the reduced pair for pentachords Y and Z: Y? = Y - (YAZ), Z? = Z - (YAZ). Y? and Z? have the same cardinality, which is 5 or less. IfY? and Z? are null, then Y = Z and <Y,Z> = <Y,Y> is trivially a suitable pentachord-pair. We ignore that case and search for pairs Y and Z that are not the same set. So we suppose that Yo and Z? are disjoint singletons, dyads, trichords, tetrachords, or pentachords.

Step 4: Inspecting the criteria for disjoint sets of cardinality 1 through 5, on the catalogue of (8.2.1) through (8.2.5), we note the following:

Disjoint singletons Y? and Z? cannot enjoy Fourier Relation 1, let alone Relations 4 and 1. Disjoint dyads Y? and Z? cannot enjoy both Fourier Relation 4 and Fourier Relation 1. Disjoint trichords Y? and Z? will enjoy both FOURRELS 4 and 1 if

(a) Yo and Z? are distinct augmented triads, or if (b) For some T-or-I operation OP, Y? (resp.Zo) and Z? (resp.Y?)

are OP({045}) and OP({237 ). Disjoint tetrachords Yo and Z? will enjoy both FOURRELS 4 and 1 ifY? is a form of (0167) or (0268), and Z? is T3(Y?) or T9(Y?). Disjoint pentachords Y? and Z? cannot be in both FOURRELS 4 and 1.

Step 5: Accordingly, our task yields possibilities (a) through (c) below, and just those possibilities:

(a) Yo and Z? are distinct augmented triads; Y = Y?uC and Z = Z?uC, where C is some (any) dyad disjoint from both Y? and ZO.

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(b) For some T-or-I operation OP, Y? and Z? (resp. Z? and Y?) are OP(045) and OP(237); Y = Y?uC and Z = Z?uC, where C is some (any) dyad disjoint from both Y? and Z?.

(c) Yo is a form of (0167) or (0268); Z? = T3(Y?) or T9(Y?); Y = Y?uC and Z = Z?uC, where C is some (any) singleton set disjoint from both Y? and Z?.

We have completed the task. We know, for instance, that

IFUNC({ C#EF#GG#ABbB }, { CC#tFGG }) = IFUNC({C#EF#GG#ABbB }, {D#EG#ABb}),

via case (c) of Step 5 above.

9. A mathematical afterword

9.1 It would be impossible here to give a crash course on the relevant mathematics for readers knowing little of the pertinent mathematical background. That background includes the behavior of complex numbers, and the theory of Fourier transformations for certain complex- valued functions defined on certain mathematical groups. For readers already at least somewhat familiar with such matters, I will indicate how I have used the mathematics to obtain my music-theoretic results.

9.2 If f is a complex-valued function on the numbers mod 12, the "adjoint function" will be that function f* defined so that f*(n) is the complex conjugate of f(-n).

We identify the twelve pcs with the numbers 0 through 11 mod 12 by the usual fixed-DO convention: C is labeled by 0, C# by 1,.... B by 11.

Pcsets X, Y, and Z then correspond to their characteristic functions f, g, and h: f(n) = 1 if the pc n is in set X; f(n) = 0 if the pc n is not in set X.

IFUNC(X,Y), under these identifications, is the convolution function f** g:

(f** g)(n) = SUMm (f*(m) g(n-m)) = SUMm (f(-m) g(n-m)) = SUMp (f(p) g(n+p) ); the term f(p) g(n+p) vanishes unless p is in X and n+p is in Y, in which case the term contributes the number 1 to the sum. So the p-SUM counts how many times some note p of X has its n-transpose in Y. That number is precisely IFUNC(X,Y)(n).

Let F be the Fourier transform of f: F(n) = (1/12) SUMm (f(m) EXP(-2 n i mn/12)). (Here the symbol i refers to the complex number, not to a pc interval.) Let G and H be the Fourier transforms of g and h. Let F' (n) be the complex conjugate of F(n). Then F' times G (pointwise mul- tiplication) is the Fourier transform of f** g, which is IFUNC(X,Y).

Because f is real-valued, F = F*; accordingly F(-n) will be zero if and only if F(n) is zero. Since f is integer-valued, F(5) will be zero if and only

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if F(1) is zero. (The one number involves changing the sign for the square-root of three in the other number.) So F(1), F(5), F(7), and F(11) are all zero if any of them is zero. Similarly for G, and for G - H.

By computing, we verify that F(6) = 0 if and only if X has Fourier Property 6, that F(4) = F(8) = 0 if and only if X has Fourier Property 4, that F(3) = F(9) = 0 if and only if X has Fourier Property 3, that F(2) = F(10) = 0 if and only if X has Fourier Property 2, and that F(1) = F(5) = F(7) = F(11)= 0 if and only if X has Fourier Property 1. Similarly for G.

IFUNC(X,Y) = (f** g) will be a constant function if and only if its Fourier transform vanishes everywhere except at n = 0. So <X,Y> will be a constant-IFUNC pair if and only if F' (n) times G(n) = 0 for all non-zero n; that will be so if and only if for all non-zero n, either F(n) = 0 or G(n) = 0; and that will be so if and only if for all P = 6, 4, 3, 2, and 1, either X has FOURPROP(P) or Y has FOURPROP(P).

IFUNC(X,Y) = (f** g) will be a function of form (pqpqpqpqpqpq) if and only if its Fourier transform vanishes everywhere except at n = 0 and n=6. So <X,Y> will be an alternating-IFUNC pair if and only if F'(n) times G(n) = 0 for all n other than 0 and 6; that will be so if and only if for all n other than 0 or 6, either F(n) = 0 or G(n) = 0; and that will be so if and only if for all P = 4, 3, 2, and 1, either X has FOURPROP(P) or Y has FOURPROP(P).

By computing, we verify that (G-H)(6) = 0 if and only if Y and Z are in Fourier Relation 6, that (G-H)(4) = (G-H)(8) = 0 if and only ifY and Z are in Fourier Relation 4, that (G-H)(3) = (G-H)(9) = 0 if and only ifY and Z are in Fourier Relation 3, that (G-H)(2) = (G-H)(10) = 0 if and only if Y and Z are in Fourier Relation 2, and that (G-H)(1) = (G-H)(5) = (G-H)(7) = (G-H)(1 1)= 0 if and only ifY and Z are in Fourier Relation 1.

Thus (IFUNC(X,Y) = IFUNC(X,Z)), meaning (f** g) = (f** h), will be the case if and only if (f** (g-h)) = 0, and that will be the case if and only if F' times (G-H) (pointwise) = 0. For non-null X (non-zero F), that will be the case if and only if, for all n (including n=0), either F(n) = 0 or (G-H)(n) = 0, and that will be the case if and only if (cardY = cardZ (G(0) = H(0))) and (for P = 6, 4, 3, 2, or 1, either (X has FOURPROP(P)) or (Y and Z are in FOURREL(P)).

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NOTES 1. I introduced the term "interval function" in my article, "Intervallic Relations Be-

tween Two Collections of Notes," Journal of Music Theory 3.2 (November 1959), 298-301.

2. Say IFUNC(X,Y) = (nnnnnnnnnnnn). Then, given any interval i, there will be n ways in which the interval -i can be spanned between X and Y, from some x to some y. These will correspond to n ways in which the interval i can be spanned from Y to X, from some y to some x. So IFUNC(Y,X)(i) will = n. That being the case for any sample interval i, we conclude that IFUNC(Y,X) = (nnnnnnnnnnnn).

3. If there are n ways to span the interval i-k between X and Y, there will be n ways to span the interval i between X and Yk.

4. Let c be the cardinality of set X. If there are n ways to span the interval i between X and Y, there will be c-n ways to span the interval i between X and Y*. So IFUNC(X,Y*) will be (c-n,c-n,c-n,c-n,c-n,c-n,c-n,c-n,c-n,c-n,c-n,c-n).

5. This cannot be demonstrated in so straightforward a way as the other observations above, in case X does not invert into itself. Later on, we shall see that the proposition is in fact true.

6. "Intervallic Relations," 299-300. 7. Unfortunately, the mathematical considerations will essentially be gibberish to

readers without considerable mathematical background. For such readers, Part 9 of this article will provide the pertinent mathematics in highly telegraphic form.

8. Hint: IFUNC(W,Y) can be expressed as the sum of the various functions IFUNC(W, {y}), as the pitch class y runs through the various members of Y.

9. The mathematical criterion for the "closeness" of the functions is the appropriate criterion here, in the context that will soon be developed. (In general, there are many possible mathematical criteria for measuring the "closeness" of functions.)

10. The exact measurement of such "slightness," here 1/54, is nevertheless potentially relevant for modeling listener psychology. In other situations, a listener might experience other deviations as "more slight" or "less slight" than this one, or than one another. Comparing numerical measurements of "slightness" is a useful mathematical resource for modeling ranked intuitions of "slightness," as greater or lesser in one situation than in another.

11. One could also assert such statistical impressions-of intervals between the lower strings and the violin part-over m. 117 as a whole. Those (normalized) statistical impressions would then be reinforced and confirmed, in rhythmic diminution, over the first half of m. 118, and then further reinforced and confirmed, in yet further rhythmic diminution, over each succeeding quarter-note beat of the passage.

12. I say that the interval-complement symmetry of the IFUNC here "comports well with inversional relationships among the pcsets involved." Such inversional relations, that is, will in general produce an interval-complement symmetrical IFUNC. The converse, however, is not a general rule: it is not true that any interval-complement symmetry in an IFUNC is necessarily caused by inversional relations among the pcsets involved. Even when a pcset X does not invert into itself, for instance, IFUNC(X,X) will always be interval-complement symmetrical: if there are n ways to span the interval i from X to X, there will necessarily be n ways to span the interval -i from X to X.

13. We disallow X = NULLSET because IFUNC(NULLSET, Y) = IFUNC (NULLSET, Z) = (000000000000) trivially, for any sets Y and Z.

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Lewin - Special Cases of Interval Function Between Pitch-Class Sets X and Y

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