Interval Content vs. DFT

Abstract

Several ways to appreciate the diatonicity of a pc-set can be proposed: Anatol Vierù enumerates connected fifths (or semitones, as an indicator of chromaticity), Aline Honing similarly measures ‘interval categories’ against prototype pc-sets [8]; numerous generalizations of the diatonic scales have been advanced, for instance John Clough and Jack Douthett ‘hyperdiatonic’ [5] which supersedes Ethan Agmon’s model [1] and the tetrachordal structure of the usual diatonic, and many others. The present paper purports to show that magnitudes of Fourier coefficients, or ‘saliency’ as introduced by Ian Quinn in [9], provide better measurements of diatonicity, chromaticity, octatonicity...The latter case may help solve the controversies about the octatonic character of slavic music in the beginning of the XX$^{th}$ century, and generally disambiguate appreciation of hitherto mostly subjective musical characteristics.

1 Introduction

Tautologically, the most diatonic seven-note scale is the diatonic scale, i.e. any collection/pc-set translated from $\{0,2,4,5,7,9,11\}$ in $\mathbf Z_{12}$. Slightly less obviously, the most diatonic collection in five notes is certainly the pentatonic scale $\{0,2,4,7,9\}$. But how is one to compare, say, $\{0,2,3,5,7,8,11\}$, $\{0,2,4,5,7,9\}$ or $\{0,2,4,6,7,11\}$? The question asked here is “how can one measure (with some precise, computable definition) the diatonic character of a pc-set?” While we are at it, it costs nothing to ask this question while replacing ‘diatonic’ with ‘chromatic’ or ‘octatonic’ (other adjectives will appear subsequently). Indeed it is a vexed issue (see [11]) whether Stravinsky’s music is octatonic; alternatively, it would be nice to appreciate objectively the evolution of chromaticity throughout Wagner’s Tetralogy (with Tristan in between) and what remains of it in Parsifal – similar questions abound.

Of course several answers have been advanced. We will present some of them through a few examples, and move on to argue why the most recent one, Ian Quinn’s “saliency”, is the best so far.

Some knowledge of pitch classes and pitch-class sets theory is assumed, alongside with basic music theory – common scales and chords, alongside with familiarity with Western Music. More elaborate machinery will be developed in Sect. 1.2 and later.

1.1 Some Examples

Let us focus on four pc-sets occurring at the beginning of Stravinsky’s Rite of Spring. The first two descending motives articulate C B G E B A i.e. the pc-set $X=\{0,4,7,9,11\}$. Then D and C$\sharp$ are added, making up $Y=\{0, 1, 2, 4,7,9,11\}$; it turns into something messier with chromatic quarts in the bass, that cover the chromatic aggregate. I will complete the sample with the black-keyed motif in measures 9–12, playing C$\sharp$ F$\sharp$ D$\sharp$ with a G$\sharp$ thrown in at the end, i.e. $Z = \{1,3,6,8\}$, and the new descending motif in measures 15–17 playing $T=\{0,1,3,6,7,8,9\}$.

Undoubtedly X can be considered diatonic. After all, it is a subset of a major scale – better, two major scales. There is, or was, a large current in XX$^{th}$ century Music Theory that focuses on inclusion relationships – so-called set-complex theory in American Set Theory, but also the lesser known notion of ‘poor’ and ‘rich’ modes by Anatol Vierù [12]¹, an independent and fairly well contrived alternative to the previous theory. However, numerous ambiguities arise:

• 1. How much, exactly, is X diatonic? Can we grade it?

• 2. In particular, is it more or less diatonic than other 5-note pc-sets, like $\{0,2,4,7,9\}$ or $\{0,2,4,5,7\}$ which are also subsets of diatonic scales?

• 3. What of sets which are not exactly included in a diatonic mode (like Y, Z) but almost?

Possible answers, clinging to the set relationships of inclusion and intersection, take into account the (maximum) number of common notes between a pc-set and each and every diatonic collection; or the percentage of such common notes averaged over some common basis (the cardinality of the mode, or 7, for instance). In the chosen examples, Y shares six notes $\{0, 2, 4,7,9,11\}$ with C and G major, and six others $\{1, 2, 4,7,9,11\}$ with D major. On the other hand, Z is included in no less than four diatonic scales, (albeit far from the ones that ‘neighbored’ X or Y), so Z should be rated diatonic – but how much so, when we have so many diatonic contexts to choose from?² Meanwhile, T intersects three diatonic collections in five notes, five others in four notes and the remaining ones in no less than three notes. How diatonic is that? Is it actually more chromatic? Or octatonic?

I will not waste time advocating against  the set-theoretical approach, which fails because set-theory is too poor to take into account complex musical notions³, but rather let the more elaborate models speak for themselves.

The notion of interval vector (${{\mathrm{{\mathbf {iv}}}}}$) is more precise, and provides several illuminating informations on a pc-set.⁴ Simply put (following one of the latest of D. Lewin’s illuminating comments), it is the probability⁵ of hearing a given interval if two pcs are chosen at random in a given pc-set. Then

$$\begin{aligned} {{\mathrm{{\mathbf {iv}}}}}_X(k) = \# \{(a, b)\in X^2 \mid b-a = k\} =\# \bigl (X \cap (X+k)\bigr ) \end{aligned}$$

i.e. the number of occurrences of interval k between elements of X.⁶

Since a diatonic collection has maximal value for ${{\mathrm{{\mathbf {iv}}}}}(5) = {{\mathrm{{\mathbf {iv}}}}}(7) = 6$ (among 7-note scales), it is natural and (important in practice) fairly elementary⁷ to compute ${{\mathrm{{\mathbf {iv}}}}}_X(5)$ for any pc-set X and compare it against that value.

Fig. 1.

${{\mathrm{{\mathbf {iv}}}}}$ for the diatonic D, X, Y, Z and T

Already ${{\mathrm{{\mathbf {iv}}}}}$ provides some satisfying information (see Fig. 1):

• For X, ${{\mathrm{{\mathbf {iv}}}}}(5)=3$ is indeed the maximal coefficient; but it is far below the value for the diatonic scale, which might express the contextual ambiguity (too many different diatonic scales include X). On the other hand, ${{\mathrm{{\mathbf {iv}}}}}(1)=1$, the chromatic value, is quite small with only one semitone.

• For Y, ${{\mathrm{{\mathbf {iv}}}}}(5)=5$ is almost as large as in the case of a diatonic collection. Notice however that ${{\mathrm{{\mathbf {iv}}}}}(2)$ is just as large (many whole tones) and ${{\mathrm{{\mathbf {iv}}}}}(1)$ is greater than it would be for a diatonic collection.

• For Z, ${{\mathrm{{\mathbf {iv}}}}}(5)=3$ is the largest coefficient and also the maximal possible value for a 4-note scale, confirming the diatonic character despite the contextual indetermination of its many diatonic neighbors.

• Lastly, T is much more contrasted, with ${{\mathrm{{\mathbf {iv}}}}}(6)$ a clear maximum⁸ and other coefficients between 3 and 4.

This looks fairly close to musical perception, at least as far as diatonicity and chromaticity are concerned. However, let us take a closer look at two hexachords which share the same value for ${{\mathrm{{\mathbf {iv}}}}}(5)$ (see Fig. 2): $H=\{0, 2, 4, 5, 7, 11\}$ and $H' = \{0, 1, 5, 6, 7, 8\}$. The first one, H, is a subset of C major, the second $H'$ has only five pcs in common with C$\sharp$ and G$\sharp$ major and appears substantially more chromatic and less diatonic.⁹

Fig. 2.

${{\mathrm{{\mathbf {iv}}}}}$ for two hexachords

This provides evidence that, at least in some cases, the ${{\mathrm{{\mathbf {iv}}}}}$ is not good enough to discriminate between different degrees of diatonicity. This requires both elucidation and improvement.

Anatol Vierù went deeper still in his analysis of diatonicity (or chromaticity), and understood the importance of connectivity of fifths. In a diatonic (or pentatonic) collection, we face an uninterrupted sequence of fifths, e.g. F C G D A E B. In $H, H'$, there are two broken fifth sequences, respectively (5, 0, 7, 2), (4, 11) and (5, 0, 7), (6, 1, 8): the first collection H adheres more closely to the generating structure of the diatonic scale than $H'$. Hence Vierù’s definition of diatonicity and chromaticity:¹⁰

Definition 1

The diatonicity (resp. chromaticity) of a pc-set is the maximal number of consecutive fifths (resp. semitones) between elements of the pc-set.

In the above example, H gets 3 and $H'$ only 2, though the values of ${{\mathrm{{\mathbf {iv}}}}}(5)$ are the same (4). Will the reader agree that the first is roughly 50$\%$ more diatonic than the second? Notice that this value is less obvious to compute than the ${{\mathrm{{\mathbf {iv}}}}}$, unless one skillfully multiplies¹¹ the pc-set by 5 and reads the sorted result for chromaticity, which is a way of reading visually the value on the chain of fifths (cf. right half of Fig. 3): the first pc-set turns into $\{10,11,0,1,7,8\}$ and the second into $\{11,0,1,4,5,6\}$.

Fig. 3.

Vierù’s chromaticity is lesser in H than $H'$ (left) but diatonicity stronger for H, as read on 5H and $5H'$ (right)

Let us cut this even finer. We would like to express that $H=\{0, 2, 4, 5, 7, 11\}$ is more diatonic than $H''=\{0, 2, 4, 5, 7, 8\}$ (and $T = \{0, 1, 5, 6\}$ less than $T' =\{0, 3, 5, 8\}$) though the “Vierù indexes” are identical.

One possible, dual argument, would be that the covering chain of fifths is shorter in one case than the other: 5 0 7 2 (9) 4 11 vs 5 0 7 2 (9) 4 (11 6) 1 8 (Fig. 4). This compounds neatly the inclusion criterion, the first scale being a subset of a diatonic and not the second, but at the price of mixing two criterions and enhancing the computational complexity: should we then look up, first the lengths of connected by fifth-components, and then, in case of ex-aequo, the span of the including chain of fifths? This is getting excessively complicated.

Fig. 4.

Covering chain of fifths for $\{0, 2, 4, 5, 7, 11\}$, $\{0, 2, 4, 5, 7, 8\}$, $\{0, 1, 5, 6\}$ and $\{0, 3, 5, 8\}$

In [7, 8], Aline Honingh endeavors to compare any pc-set with the appropriate ‘prototype’: for instance a hexachord will be measured against the Guidonian hexachord, a pentachord against the pentatonic, etc. For neatness, the pc-sets are first reduced to so-called ‘basic-form’.¹² For instance, the two tetrachords in the last example would be compared with the prototype C D F G (numeric results depend on the choice of similarity measure), which may or may not favor 0 1 5 6 over 0 3 5 8. I will leave the reader to peruse further details in her papers, not because this measure lacks interest, but quite contrariwise (indeed it allows for instance to discriminate between Beethoven’s compositions early, middle, and late periods): it gets extremely close to the last, simplest, and overall best candidate.

I present here without any technicity the values of saliency as defined in [9] and used in numerous analyses henceforth. Saliency is defined as the magnitude of one easily computed complex number, here (in the case of diatonicity) the fifth Fourier coefficient of a pc-set (formulas, references and properties will follow in the next section). For now, let us appreciate the values of this evaluation of diatonicity for all the above examples and some more. On Fig. 5, we can picture the magnitudes of all Fourier coefficients of the aforementioned heptachords, with the diatonic scale first. We focus on the fifth magnitude (equal to the seventh), highlighted by a dotted horizontal line, and notice that the ranking is: diatonic, Z, Y, X and T with little difference between Y and X, and a larger discrepancy with T.

Fig. 5.

Saliency for the diatonic, Z, Y, X, and T

A similarly satisfying result also arises with the hexachords on Fig. 6, with an unambiguous ordering of diatonicities: $\{0, 2, 4, 5, 7, 11\}$ followed by $\{0, 1, 5, 6, 7, 10\}$, and last $\{0, 1, 5, 6, 7, 8\}$.

Fig. 6.

Saliency for the hexachords $H, H', H''$ (horizontal line)

Others examples support unequivocaly this experimental evidence: that the fifth saliency corresponds very closely with the intuitive perception of diatonicity. We must look into the mathematics to understand why this should be, and above all how this falls in with the competing measurements of diatonicity listed above.

1.2 Some Technical Definitions

I provide only a cursory outline; the reader of the present paper will only need to bear in mind that some easily computed¹³ quantities, called Fourier coefficients, feature interesting characterizations of those pc-sets which divide the octave as evenly as possible.¹⁴ For a very pedagogical introduction to Discrete Fourier Transform (DFT) of pc-sets, see [4]. For thorough discussion and details, see the recent reference [3] which purports to give the state of the art.

To each pc-set A considered as a subset of $\mathbf Z_{12}$, is associated firstly its characteristic function

${\mathbf 1}_A: x\mapsto {\left\{ \begin{array}{ll} 1 &{} \text { if }x\in A \\ 0 &{} \text { if }x\notin A \end{array}\right. }$ and second the Discrete Fourier Transform $\mathcal F_A= \widehat{{\mathbf 1}_A}$ of this function, the DFT of the set:

$$\begin{aligned} \mathcal F_A: t\mapsto \sum _{x \in A} e^{-2i \pi x t/ 12}. \end{aligned}$$

This function is a sum of complex numbers of the form $e^{i \theta }$ which can all be construed as vectors $(\cos \theta , \sin \theta )$ of length 1, whose direction is given by the phase $\theta$. The value $\mathcal F_A(k)$ is called the $k^{th}$ Fourier coefficient. We will mainly be concerned with its magnitude, i.e. the length of the sum of these vectors.¹⁵

Here is a list of elementary though useful results without proofs:

• The set A can be reconstructed from the knowledge of the Fourier coefficients $\mathcal F_A(k)$.

• $\mathcal F_A(12-k) = \overline{\mathcal F_A(k)}$ (conjugate complex number).

• $\mathcal F_A(t) = -\mathcal F_{\overline{A}}(t)$ for $t\ne 0$ ($\overline{A}$ is the complement of A).

• $\mathcal F_A(0) = \# A$.

• $\sum |\mathcal F_A(k)|^2 = 12\times \# A$.

• The Fourier transform of the (12-dimensional) interval vector ${{\mathrm{{\mathbf {iv}}}}}_A$ is the square of the magnitude of $\mathcal F_A$:

  $$\begin{aligned} \forall k\in \mathbf Z_{12}\quad \widehat{{{\mathrm{{\mathbf {iv}}}}}_A}(k) = |\mathcal F_A(k)|^2. \qquad (\sharp ) \end{aligned}$$

Slightly more technical is the Huddling Lemma in [2]: in laymen’s terms it states that, the closer the angles $\theta _k$, the larger the sum $\sum _k e^{i \theta _k}$ (the vectors pull roughly in the same direction, coordinating their efforts). We will only need a simple case:

Proposition 1

When the cardinality of A is fixed, $|\mathcal F_A(1)|$ reaches maximal value when the elements of A are consecutive [i.e. when A is a chromatic chunk].

For us the most important result is

Corollary 1

When the cardinality of A is fixed, $|\mathcal F_A(5)|$ reaches maximal value when the elements of A are consecutive in the chain of fifths.

Proof

This follows from the relation $\mathcal F_A(5) = \mathcal F_{5 A}(1)$, which results from $5\times 5 = 1\mod 12$: hence the elements of 5A must be consecutive, which is equivalent to the condition stated.

This is but a special case of Quinn’s result:

Among all pc-sets with same cardinality d , the maximum magnitude for $\mathcal F_A(d)$ is obtained when A is a Maximally Even Set (ME set).

ME sets admit many equivalent definitions [2, 5]. We will need only to remember the most important ME sets in $\mathbf Z_{12}$:

• 1. The octatonic scale for $d=8$.

• 2. The diatonic scale for $d=7$.

• 3. The whole-tone scale for $d=6$.

• 4. The pentatonic scale for $d=5$.

Quinn aimed at a landscape of chords (starting from experimental knowledge) and sketched first the highest peaks. From some kind of continuity principle, it was natural to infer that the height of a chord close to a summit would still be high. Hence the definition of saliency, as a quality of proximity to a ME-set (that Quinn called ‘prototype’):

Definition 2

The d-saliency of a chord A is $|\mathcal F_A(d)|$.

• 1. Among d-chords, saliency is maximal for d-ME sets.

• 2. Remember if convenient that $|\mathcal F_A(d)| = |\mathcal F_A(12 - d)| = |\mathcal F_{\overline{A}}(t)|$, hence both diatonic and (non hemitonic) pentatonic scales have maximum saliency for index 5 (namely $2+\sqrt{3} \approx 3.73$).

• 3. For any (reasonable) distance on the set of pc-sets, a pc-set close to a ME set has saliency close to maximal.

• 4. Any pc-set (with given cardinality) distributes its saliencies according to its geometry: the sum of the squares of all saliencies is a constant. This echoes the idea in [8] that the distribution of [IC] categories throughout a piece tells of its local character.

All this provides fairly good mathematical justification, corroborated by empirical knowledge, for defining

Definition 3

• The chromaticity of a pc-set A is $|\mathcal F_A(1)|$ (remembering Proposition 1).

• The diatonicity of a pc-set A is $|\mathcal F_A(5)|$.

• The octatonicity of a pc-set A is $|\mathcal F_A(4)|$.

Some other values have actually been used for musical analysis: J. Yust calls ‘quartal quality’¹⁶ the magnitude $|\mathcal F_A(2)|$ which is, for instance, maximal among octachords for Tristan’s motif pc-set $\{2, 3, 4, 5, 8, 9, 10, 11\}$; while the ‘major-thirdishness’ $|\mathcal F_A(3)|$, for want of a better term (‘augmentedness’?) is maximal for an augmented triad, or for Schönberg’s Napoleon hexachord $\{0, 1, 4, 5, 8, 9\}$.

Remembering the equation $\sum |\mathcal F_A(k)|^2 = 12 \# A$, it could be argued that the proper measure should be the squared magnitude – perhaps averaged by the cardinality – since the sum of all these values is a constant. Also, it is the squared value that appears in the DFT of the intervallic function. I will keep to the original definition for the present paper, but would not be surprised if the squared value were to supersede it in the future (following [17]).

2 DFT vs. ${{\mathrm{{\mathbf {iv}}}}}$

2.1 Theoretical Advantage

DFT is a change of (orthogonal) basis among many (polynomials, wavelets...). The major advantage¹⁷ of expressing a (musical: pc-set, rhythm...) phenomenon in a basis of exponential functions is in the following:

Proposition 2

The DFT exchanges convolution product $*$ and termwise product $\times$. Namely, if f, g are two maps from $\mathbf Z_{12}$ to $\mathbf C$ and $\widehat{f}, \widehat{g}$ their DFTs, then

$$\begin{aligned} \widehat{f*g} (k)= \widehat{f} (k) \times \widehat{g}(k). \end{aligned}$$

This is crucial because ${{\mathrm{{\mathbf {iv}}}}}$ is a convolution product:

$$\begin{aligned} {{\mathrm{{\mathbf {iv}}}}}_A(k) = \sum {\mathbf 1}_A(t) {\mathbf 1}_A(t-k) = \sum {\mathbf 1}_A(t) {\mathbf 1}_{-A}(k-t) = ({\mathbf 1}_A*{\mathbf 1}_{-A}) (k) \end{aligned}$$

and more generally, any coincidence measure or correlation (say, the number of elements of A that lie in any diatonic scale i.e. any transposition $D+k$ of $D = \{0,2,4,5,7,9,11\}$) can also be read on a convolution product:¹⁸

$$\begin{aligned} \sum {\mathbf 1}_A(t) {\mathbf 1}_{D+k} (t) = \sum {\mathbf 1}_A(t) {\mathbf 1}_{D} (t-k) = ({\mathbf 1}_A*{\mathbf 1}_{-D})(k). \end{aligned}$$

Now the convolution product is a...convoluted operation¹⁹ while termwise product is straightforward. Cognitively speaking, this means that complicated operations become obvious in Fourier space (i.e. computing on Fourier coefficients) and perhaps suggests that the human mind processes some equivalent of Fourier coefficients.

2.2 Multiplying Saliencies

For the sake of simplicity I present computations for diatonicity only²⁰, i.e. comparing a pc-set A with various transpositions of the Diatonic D and considering the fifth saliency. This is the core of the present article, making sense in a unified way of all previous diatonicity measures. We analyse first the link between coincidence and saliency. Coincidence with a prototype is a variant of Honingh’s measure: ${\mathbf 1}_A*{\mathbf 1}_B(k)$ is a high value when $A+k$ shares many common values with B. We are especially interested in the case when B is a diatonic scale, $B=D$ or $-D$ or $k-D$ etc.

Applying Proposition 2 yields immediately

$$\begin{aligned} \mathcal F_A(5) \times \mathcal F_{-D}(5) = \widehat{{\mathbf 1}_A*{\mathbf 1}_{-D}}(5): \qquad (\sharp ) \end{aligned}$$

the product of the (diatonic) saliencies of A and $-D$ is a Fourier coefficient of the coincidence function of A and the diatonic scale. Low values of the latter mean that bad correlation will limit the magnitude of $\mathcal F_A(5)$, i.e. the diatonicity of A. Conversely, when does this coincidence function ${\mathbf 1}_A*{\mathbf 1}_{-D}$ (replaced below by ${\mathbf 1}_A*{\mathbf 1}_{D}$ for simplicity’s sake) exhibit a high diatonicity? On the left-hand side of equation $(\sharp )$, it means simply that A is highly diatonic (large value of $|\mathcal F_A(5)|$). On the right-hand side, it means that the coincidence function ${\mathbf 1}_A*{\mathbf 1}_{D}$

• 1. has at least some large values

• 2. and is ‘diatonic’ (large fifth Fourier coefficient).

In order to understand how the simple computation of saliency supersedes all previous notions, let us analyse this last feature, which means (in the case of diatonicity) being strongly 5-periodic: the prototype, the diatonic scale D, is a chain of fifths, meaning that $D+5$ has $7-1=6$ common elements with D.²¹ From this follows an automatic quasi-periodicity of ${\mathbf 1}_A*{\mathbf 1}_D$ (see Fig. 7):

Fig. 7.

Coincidence between D and A or $A+5$ changes at most by 1

Proposition 3

$$\begin{aligned}&\text {The difference between the correlations}\quad |({\mathbf 1}_A*{\mathbf 1}_D)(k+5) - ({\mathbf 1}_A*{\mathbf 1}_D)(k) | \\&\text { is either} \text { 0 or 1}. \end{aligned}$$

Proof

These two convolution products expressed as sums share 6 common elements, plus another one than can be either 0 or 1. More precisely, setting $D = \{5m, m=0 \dots 6\}$ for simplicity, we get

$$\begin{aligned} ({\mathbf 1}_A*{\mathbf 1}_{D})(k)&= \sum _{m=0}^6 {\mathbf 1}_A(k-5m) = {\mathbf 1}_A(k-30) + \sum _{m=0}^5 {\mathbf 1}_A(k -5m) \\ ({\mathbf 1}_A*{\mathbf 1}_{D})(k+5)&= \sum _{m=0}^6 {\mathbf 1}_A(k + 5 -5m) = \sum _{m=0}^6 {\mathbf 1}_A(k -5(m-1)) \\&= {\mathbf 1}_A(k+5) + \sum _{m=0}^5 {\mathbf 1}_A(k -5m), \end{aligned}$$

hence the two values coincide when ${\mathbf 1}_A(k+5)={\mathbf 1}_A(k-30) (= {\mathbf 1}_A(k+6)$ modulo 12), and differ by one if not.

How then can $\widehat{{\mathbf 1}_A*{\mathbf 1}_D}(5)$ be as large as possible? On the one hand, the geometry of the diatonic itself partly ensures some periodicity of ${\mathbf 1}_A*{\mathbf 1}_D$ (Proposition 3), which boosts its diatonicity. How can we further increase this periodicity?

Let for example $k=0$ in the condition ${\mathbf 1}_A(k+5) = {\mathbf 1}_A(k+6)$ just derived: we will have ${\mathbf 1}_A(5) = {\mathbf 1}_A(6)$ when neither F nor F $\sharp$ are elements of A (or both), for instance when $A = \{0,2,4,7,9,11\}$ (appropriately chiming the first notes of ‘Do you know what if means’). But in order to enlarge the remaining sum $\sum _{m=0}^5 {\mathbf 1}_A(0 -5m)$, we will need as many elements of A as possible in the partial chain of fifths C D E G A B (each adds 1 to the value of the convolution product). This will certainly be satisfied when A features a long connected subsequence of the chain of fifths.²² We have just understood, not only how the saliency notion includes Vierù’s definition, but also why it is superior: Vierù’s measure is identical for H and $H''$ but in the latter case the elements of H are better huddled in the chain of fifths, providing a larger tally of large correlation values of the convolution product ${\mathbf 1}_{H}*{\mathbf 1}_D$ (coincidence of H with the prototypical diatonic scale). Let us check this by computing some numerical values. Listing the values of the convolution products from 0 to 11 yields

$$\begin{aligned} {\mathbf 1}_{H}*{\mathbf 1}_D = [{\mathbf 6, 2, 4, 3, 3, \mathbf 5, 2, \mathbf 5, 2, 4, 4, 2}] \text { and } {\mathbf 1}_{H''}*{\mathbf 1}_D = [{3, 3, 3, 3, \mathbf 5, 3, 4, 3, 3, 4, 3, \mathbf 5}]. \end{aligned}$$

For tetrachords $T = \{0, 1, 5, 6\}$ and $T'=\{0,3,5,8\}$, it is perhaps even clearer:

$$\begin{aligned} {\mathbf 1}_{T}*{\mathbf 1}_D = [{2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 4}] \text { and } {\mathbf 1}_{T'}*{\mathbf 1}_D = [2, 2, 3, 1, \mathbf 4, 1, 3, 2, 2, \mathbf 4, 0,\mathbf 4]. \end{aligned}$$

Notice in the latter case how the value 4 occurs thrice in a row (in fifth order: at positions 11, 4, 9), in agreement with the geometric constraint found above. Indeed the 5-saliency of $T'$ is greater than T’s. Similarly, H is more diatonic than $H''$ because of the sequence of high values (in fifth order) $\dots 4,5,6,5,4 \dots$

Of course, computing these correlation vectors with the diatonic would provide an effective and convincing measurement of diatonicity²³; but as we have demonstrated, the lone and straightforward value of saliency neatly subsumes the whole vector.

2.3 Inclusion and iv

It is redundant but perhaps useful to synthesize briefly the case of the crude inclusion as compared to saliency in the light of the above calculations. Inclusion of a pc-set inside (say) a diatonic scale is indeed a coincidence measure that can be pinpointed as one large coefficient in ${\mathbf 1}_A*{\mathbf 1}_{-D}$ (at least one value equal to the cardinality of A, some other large values according to Proposition 3). This is but a special case of the preceding discussion, wherein it was shown that significant diatonicity depends not only on the number of coincidences but also on their grouping, or ‘huddling’. The same goes for large values of ${{\mathrm{{\mathbf {iv}}}}}_A(5)$ (many fifths), which are only indicative of diatonicity when most of the fifths are neighbors in the chain.²⁴ The extremities of the smallest chain of fifths containing a given pc-set are of course directly related to the number of overlapping diatonic scales – i.e. tally of maximum values of the convolution product –, as foretold in Vierù’s notion of ‘rich modes’.

2.4 Musical Examples

To gain perspective, let us vie away from diatonicity. D. Tymoczko’s thoughtful analysis of Stravinsky in [11] draws interpretation of pc-sets towards specific classes of scales. To his credit, he acknowledges the numerous ambiguities, criticizes fuzziness in previous analyses and avoids dogmatic pronouncements. Still, dataless statistical sentences like ‘...[this] scale accounts for virtually all of the pitches present’ leave room for contestation (I highlighted the adjective). On the other hand, exact measurements of diatonicity as magnitude of $\mathcal F_A(5)$ – and all other saliencies – can be compared both within Stravinsky’s own music, as it varies within a single piece, and from one piece to another; furthermore, this objective indicator can be applied to other composers (notably Slavic) and provide objective comparisons of their relative degrees of diatonicity, chromaticity, or octatonicity.

The interest of such comparisons warrants general and systematic research that cannot be included in this short paper. Here is but a small sample.

(1) To assess the general appreciation allowed by measurement of saliencies, I have compared all six saliencies (from chromaticity to whole-toneness) on several pieces of The Rite of Spring and, as an external reference, the Dance of the Firebird. The pieces are imported as MIDI files and a time-window of fixed width moves over it for computation of the saliencies of its pc-sets. Figure 8 simply exhibits the mean values of these saliencies.²⁵

Fig. 8.

Mean values of saliencies on some Stravinsky pieces

The figures show ambiguity in many pieces, which satisfyingly reflects the diversity of experts’ interpretations! However, some clear-cut features do emerge:

• 1. Whole-tone character dominates The Dance of the Firebird.

• 2. The very first piece of The Rite of Spring is fairly diatonic.

• 3. The Dance of Spring is more clearly diatonic.

• 4. The Dance of Earth is mostly whole-tonish.

• 5. In other pieces, the balance (interplay?) between octatonic and diatonic is apparent – in line with Van der Toorn or Taruskin’s analyses (as quoted in [11]).

(2) To give a feeling of the variety of these characters in the flow of the pieces, I provide some excerpts of saliencies as functions of time. On Fig. 9, following the first minute or so of the first movement of The Rite of Spring, the saliencies are squared (so that their sum is a constant²⁶), and thus it is easily seen which character predominates in a given passage.

It best to look at Fig. 9 while listening to the The Rite’s beginning. One can practically see the indecisive first bars (motif X) flash a spurt of chromaticism (when the C$\sharp$ interferes ca. $6''$) before settling for diatonicism (when the D is added to make up $Y=\{0,1,2,4,7,9,11\}$). Then the chromatic fourths around $15''$ boost $a_1$; $Z = \{1,3,6,8\}$ occurs between $36''$ and $40''$, flirting with a pentatonic i.e. largely diatonic character; finally, the last ambivalent motif T is played after $1'$, a short surge of chromaticism in a ‘quartal’ episode (large $a_2$).

Fig. 9.

Variations of saliencies in first minute of The Rite of Spring

This last moment exemplifies that other segmentations could, and should, be applied to music as it is perceived (as opposed to the music read on the score), for here T is clearly perceptible against the bass, though the numerical computation mixed everything together. Indeed, analyzing separate instruments, or voices, or groups, if justified on perceptual grounds, can lead to finer analyses, see examples in [11, 15], and would undoubtedly constitute an easy improvement of saliency analysis.²⁷

2.5 Phase and Tonality

The (random) colors on these pictures could be adjusted to reflect the phase (direction of vectors) of the Fourier coefficient, which reflects a generalization of tonality (for $a_5$ it can be checked against the values for 12 major scales or triads, for $a_6$ it would be against the two whole-tone scales, etc...). Detection of the character of a passage (diatonic, octatonal etc.) can be compounded by pinpointing which (say) diatonic paradigm is involved, by computation of the phase. This is a simple way to detect tonality, and its generalizations (which whole-tone, or octatonic, scale is prevalent, etc.). More about this in [3], Chap. 6.

2.6 Possible Applications to Dodecaphonic Music

A hasty reasoning might conclude that the calculations above are meaningless in dodecaphonic music, since the Fourier coefficients of the chromatic aggregate are nil. It is not so. It is certainly true of Nicolai Obouhow’s “harmonie totale”²⁸, but usually false in classical serial music when an appropriate time-span is used for the window of analysis, because the tone-row is often stated horizontally, not vertically; furthermore, at least in the second Viennese school, composition using the two halves (tropes) of the row are frequent. Of course a trope can be any hexachord, with distinctive saliencies, however (essentially this is Babbitt’s theorem) the saliencies of both tropes of a row are identical. For instance, analyzing both tropes in Alban Berg’s Lyrische Suite op. 28 and Violin Concerto op. 34 shows very strong diatonic components, see [3], p. 122. I fancy that this is a general feature of Berg’s serial music (as opposed to Webern or Schönberg, say) but my ongoing computations have been impeded by the lack of available Midi files for XX$^{th}$ century music.

3 Conclusion

From the perspective developed here, one gets a feeling that many worthy researchers have groped for years more or less in the same direction, feeling for the right definition of diatonicity without knowing exactly where it lay. Then came Ian Quinn, and lo! the Holy Grail was there for everyone to grasp.

Not only does saliency pinpoint the character (or lack thereof) of a piece of music, the other component of the Fourier coefficients (the phase) also points its precise direction (the tonality, in the diatonic case).

Precise measurements can, at long last, supersede empirical (at best, with bevies of bored and fallible test subjects) or completely subjective (at worst, and all the more virulent for it) evaluations.

Moreover, this kind of analysis is valid for a huge repertoire, since all that was said here mostly for the diatonic character stands just as well for the 5 other characters. It is hoped that saliency diagrams, pictures and movies will be developed for many pieces of music in the very near future. Indeed, it is only a slight exaggeration to fancy deaf people enabled at last to appreciate music, simply by looking at ‘Fourier clocks’ ticking as the Fourier coefficients vary throughout a piece!²⁹ It is an urgent task to develop some appropriate software for this kind of streaming analysis, picturing the Fourier flow of music on the fly.