Position Finding with Tritones in Nondiatonic Music

Yuri Broze

Introduction

In 1981, Richmond Browne proposed that the unique tritone1of the diatonic set might be an important feature used by listeners to gain their tonal bearings (Browne, 1981). The rarity of this particular interval could make it a potentially useful feature of the musical surface to aid in what Browne termed ``position finding'' --identifying the relation of given pitches to a surrounding diatonic environment. For example, the pitch string ``ti re fa'' is sufficient to enable most experienced listeners to intuit the location of the tonic pitch, as well as the location of all other members of the diatonic set.

The complex phenomenon of tonal comprehension, it is argued, must rely at least in part on the successful recognition of a given note's position within its tonal environment. In diatonic music, a note's identity might be determined by its particular intervallic context. For example, in the white-note diatonic set, ``D'' is surrounded on either side by whole steps (CD and DE) and minor thirds (BD and DF), and no other diatonic pitch shares this particular pattern. By matching a pitch's intervallic context to prior knowledge of the diatonic system's structure, it ought to be possible to infer its tonal identity--in the case of ``D'', scale degree $ \hat{2}$ in a major key. Furthermore, for tonal music, a skilled listener would then have the needed information to reproduce the entire diatonic scale, including, for instance, the tonic tone ($ \hat{1}$).

Because each member of a diatonic set is uniquely determined by its intervallic context, employing intervallic pattern-matching might be a particularly lucrative strategy of position-finding. Given the remarkable complexity of diatonic music and the speed with which we can comprehend it, one suspects that our mental processing of music must rely on particularly effective methods to extract tonal information. It is reasonable to expect that the most informative features of music would be exploited to facilitate the process of position-finding. The present article will explore certain implications of such hypothetical techniques in the context of tonal musical works.

Tritone-based Position Finding

The diatonic set is an example of a deep scale--its interval vector <254361> contains unique multiplicities for each interval class (Gamer, 1967). The diatonic tritone is the interval with only a single instance in the set, and therefore only two pitch classes could possibly partake in the tritone relationship. For diatonic scales embedded in a 12-tone universe, only scale degrees $ \hat{4}$ and $ \hat{7}$ may participate in a tritone. By contrast, a perfect fifth occurs six times, making an isolated perfect fifth a particularly poor indicator of the tonal identity of its constituent notes.

There is a complication, however. Due to the inversional enharmonicity of the tritone, an isolated instance of [06] is in fact a tonally ambiguous superposition of two possible orientations. The 6-semitone span might represent either a diminished fifth or an augmented fourth; B and F might indicate C as a presumptive tonic, but the enharmonically-equivalent C$ \flat$ and F point toward G$ \flat$ as tonic. The situation is resolved by the addition of a single additional tone: the keys of C major and G$ \flat$ major share no other pitches in common.

Indeed, experimental work by Brown and Butler (1981) confirm that listeners are quite adept at determining tonic location from `minimal cue-cells' consisting of a tritone plus one additional pitch. They observed that a listener who had successfully identified the cue-cell's diatonic location would have all the necessary information to determine the identity of tonic. Indeed, successful position finding ought to allow several responses by a sufficiently skilled listener, including vocalizing the tonic pitch, assigning solfège syllables to cue-cell pitches, or extrapolating the position of the tonic in relation to one of the cue-cell pitches. All three possible responses were allowed.

This process might be better elucidated by considering the following diagram, which frames the dilemma in terms of tritone-division identification:

Figure 1: Diatonic Scale as $ (\alpha ,\delta )$
\begin{figure}\begin{center} \begin{tabular}{p{.7cm}\vert p{.7cm}\vert p{.7cm}\v... ...olumn{5}{c}{} & \multicolumn{2}{c}{F} \\ \end{tabular}\end{center} \end{figure}

Here, the diatonic set is described as the synthesis of two conjoint tritones, each with its own characteristic intervallic pattern of whole tones (WT) and half tones (H). The greek letter alpha ($ \alpha $) represents the tritone containing the augmented fourth, WT|WT|WT. Delta ($ \delta $) representes the tritone containing the diminished fifth, H|WT|WT|H. Thus, $ (\alpha ,\delta )$ could be called a tritone decomposition of the diatonic set. Because the diatonic scale has only a single instance of a tritone, its tritone decomposition is unique.

Returning to the task of position-finding, we can imagine that should a listener hear two pitches 6 semitones apart and assume they are embedded within a diatonic set, more information would still be required to distinguish the augmented fourth from the diminished fifth--that is, $ \alpha $ from $ \delta $. Comparing these two tritone divisions directly, it is easy to see how a single extra note is sufficient:

Figure 2: Comparison of $ \alpha $ and $ \delta $ Tritone Species
\begin{figure}\begin{center} \begin{tabular}{c} $\alpha$: \\ $\delta$: \end{t... ...lticolumn{1}{\vert c\vert}{H} \\ \hline \end{tabular}\end{center} \end{figure}

The $ \alpha $ and $ \delta $ tritones share only their boundary pitches in common--we might consider this to be a tritone subunit complementarity relationship. Therefore, one extra note completes the cue-cell, breaks the ambiguity, and the diatonic identity of the tritone becomes clear. It now becomes theoretically possible for a listener to name the tonic pitch, do.

While this might be an elegant approach to identifying tonal center, it is limited by relying on certain assumptions: First, that the diatonic scale is indeed the scale being used by the music. Second, that within the diatonic scale, the tonic is the major-mode (Ionian) tonic. Third, that the music is stable in its tonal center, and is not in the process of modulating to a different key.

Moreover, it is possible that minimal cue-cells are not particularly common in musical works, casting doubt on the strategy's relevance. This could be checked through empirical study. A preliminary exploration was conducted in which each verticality in the first movement of Beethoven's first string quartet was tallied and converted to pitch-class notation. Of the 2036 verticalities encountered, 184 include a harmonically sounding tritone (9%). If instrumental rests are treated as though the most recent sounding note is being sustained, however, 635 verticalities include a tritone (31%). These results are only preliminary, but they do suggest that the appearance of the tritone is reasonably common; we infer that the three-note cue-cells appear with some frequency in common practice period music.

We might also consider the reasonability of the major-mode assumption. According to Huron (2008), 73% of the themes in the Barlow & Morgenstern Dictionary of Music Themes are in the major mode. Similarly, 74% of 6,520 key designators in multi-part scores in the Humdrum database are in the major mode, and 88% of the Essen (Germanic) folksongs are in major mode. This suggests that the assumption of majorness is not necessarily misplaced.

Nonetheless, non-diatonic and modulating passages in real musical works seem to be normative. If this is so, pattern-matching and position-finding could be considerably more complicated than a straightforward cue-cell model. Not only are alternative scalar structures widespread (including the harmonic and melodic minors, for instance), real compositions are often characterized by both local tonicizations and broadscale modulations. Yet given the tritone's rarity and its important functional association with dominant harmonies, it remains possible that tritones could yet play an important role in the perception of tonality. Through a series of hypothetical examples, we will find that a tritone-classifying approach to position is surprisingly robust to wrong-scale pitfalls.

Position-Finding in Minor Scales

It is useful to explore some possible results of applying tritone-based position finding to non-diatonic scales. One would expect that the harmonic and melodic minor scales could present considerable challenges to position-finding since they violate the assumption of diatonicity. For clarity, this section will make use of C melodic minor and C harmonic minor, using scale degree numbers for position in major and minor scales. Each will be analyzed in terms of a tritone decomposition, using (whenever possible) the same $ \alpha $ and $ \delta $ units which correspond to the diatonic scale. Since this will not always be possible with non-diatonic scales, additional tritone species will be introduced as they are encountered, and summarized below.

Strict $ \alpha $/$ \delta $-matching strategy

Let us first imagine a hypothetical listener who uses a strict $ \alpha $/$ \delta $-identifying heuristic approach to position finding. Such a listener will only attempt to identify the tonic pitch if she encounters an explicitly-stated $ \alpha $ or $ \delta $ structure in its entirety, with no notes omitted. This could be considered to be a rather conservative approach, since it requires four or five distinct notes in scalar order before a conclusion will be reached. Now, consider the case of a piece exhibiting a common non-diatonic scale, the ascending melodic minor. A tritone-decomposition of the melodic minor requires the introduction of a new tritone-division, $ \beta_{2}$, consisting of the sequence WT|H|WT|H.

Figure 3: Melodic Minor as $ (\alpha ,\beta _{2})$
\begin{figure}\begin{center} \begin{tabular}{p{.7cm}\vert p{.7cm}\vert p{.7cm}\v... ...}{c}{} & \multicolumn{2}{c}{E$\flat$} \\ \end{tabular}\end{center} \end{figure}

Perhaps our listener encounters three consecutive whole steps in a melody, and correctly identifies an $ \alpha $ tritone. Thinking this must indicate the augmented fourth in a major scale, she therefore concludes that tonic must lie one half-step above the $ \alpha $-tritone's highest pitch, just as it would in the major mode. In this case, the strategy would predict tonic to be B$ \flat$ based on the $ \alpha $-division of the lower tritone, and anticipation of a $ \delta $-division above. This represents a failure of this heuristic for melodic minor, caused by misinterpreting the tritone between $ \hat{\flat3}$ and $ \hat{\natural6}$ as being between $ \hat{4}$ and $ \hat{7}$.

However, with an interval vector of <254442>, the melodic minor scale contains not one, but two tritones a whole-tone apart, resulting in two different possible tritone decompositions! If the melodic minor scale is rotated, we find another decomposition, which also happens to include an $ \alpha $-subunit, as well as a $ \beta_{1}$ (H|WT|H|WT):

Figure 4: Melodic Minor as $ (\alpha ,\beta _{1})$
\begin{figure}\begin{center} \begin{tabular}{p{.7cm}\vert p{.7cm}\vert p{.7cm}\v... ...umn{5}{c}{} & \multicolumn{2}{c}{F} \\ \end{tabular}\end{center}\end{figure}

In this case, the listener would in fact infer the correct tonic pitch, even though the scale as a whole could not be considered diatonic. What is important to the successful identification of tonic in this case that the locally diatonic scalar region of the melodic minor detected by our listener perfectly corresponded with the equivalent location in the major mode. Equivalently, the tritone decomposition $ \alpha\beta_{1}$ splits the melodic minor at $ \hat{4}$ and $ \hat{7}$, the ``correct'' dominant-functioning augmented fourth. We can conclude that the (strict) heuristic of tracking $ \alpha $ and $ \delta $ tritone divisions is evidently successful for all major tonalities as well as at least some melodic-minor situations.

When the harmonic minor scale is encountered, however, the waters become somewhat murkier. The interval vector for the harmonic minor is <335442>, indicating again two possible tritone-decompositions. In order to create them, it will again be necessary to introduce two new species of tritone--this time in order to accomodate the step interval of an augmented second. These will be $ \gamma_{1}$ (--3--|H|WT) and $ \gamma_{2}$ (WT|H|--3--). Both tritone species contain only two intervening notes, and they therefore share an affinity with the $ \alpha $-division. The two harmonic minor tritone decompositions are:

Figure 5: Harmonic Minor as $ (\gamma _{1},\delta )$ and $ (\gamma _{2},\beta _{1})$
\begin{figure}\begin{center} \subfigure[]{ \begin{tabular}{p{1.05cm}\vert p{1.05... ...umn{5}{c}{} & \multicolumn{2}{c}{F} \\ \end{tabular}} \end{center} \end{figure}

While one tritone continues to be found at $ \hat{4}$-$ \hat{7}$, the other is now located at $ \hat{2}$- $ \hat{\flat6}$--a minor third away. Both the $ (\gamma _{1},\delta )$ and $ (\gamma _{2},\beta _{1})$ decompositions show an augmented second occurring once. However, only one of these two decompositions includes a tritone species from the diatonic set! Therefore, the chances that our listener using the strict $ \alpha $/$ \delta $ heuristic could encounter a conforming tritone are dramatically reduced. Alas, this $ \delta $-divided tritone sits between D and A$ \flat$, so our listener inevitably concludes that E$ \flat$ is the tonic pitch, and not the true minor tonic of C. Since the $ \gamma_{2}\beta_{1}$ decomposition cannot be interpreted at all by our listener, the tritone between F and B would be impossible to detect, and even the slightest hope of correct inference was lost from the outset.

Liberal $ \alpha $/$ \delta $-matching strategy

Happily, the strict-$ \alpha $/$ \delta $ heuristic might be somewhat unlikely to be actually employed in practice. After all, the value of the tritone as a uniquely powerful tool for position-finding in the diatonic set derives in part from the fact that only the two component notes plus a single additional tone is sufficient to form a minimal cue-cell for position finding. Furthermore, Brown and Butler provided experimental evidence that listeners can indeed use three note cue-cells with high accuracy to locate tonic. We could interpret the study participants' behavior as inferring the presence of either an $ \alpha $ or a $ \delta $ tritone from only three notes. They subsequently can deduce the location of the other, forming a complete image of the diatonic set.

Accordingly, imagine that our hypothetical listener switches to exactly such a strategy--a more liberal version of our $ \alpha $/$ \delta $ heuristic. Now, she will require only a tritone and one extra pitch, and accordingly infer the filling-in of the tritone to conform to either $ \alpha $ or $ \delta $. Then, a tonic can be identifed based on the assumption that the scale was diatonic.

In other words, the liberal $ \alpha $/$ \delta $-matching strategy will infer either an $ \alpha $- or $ \delta $-tritone to exist, depending on where the additional note lies. In order to determine the outcome of the liberal $ \alpha $/$ \delta $ heuristic strategy, it will be helpful to use pitch class set notation to emphasize similarities and differences of note content. The six species of tritones we have encountered are given in Table 1, alongside their step composition and pitch class set identification.


Table 1: Tritone Subdivision Species
Four-Note Species Five-Note Species
           
$ \alpha $ WT|WT|WT [0246] $ \delta $ H|WT|WT|H [01356]
$ \gamma_{1}$ --3--|H|WT [0346] $ \beta_{1}$ H|WT|H|WT [01346]
$ \gamma_{2}$ WT|H|--3-- [0236] $ \beta_{2}$ WT|H|WT|H [02356]


This presentation facilitates several observations. The relationships between the three-note and four-note species can be seen as related to their counterparts in the diatonic decomposition $ (\alpha ,\delta )$. Additionally, the kinship between $ \beta_{1}$/ $ \gamma_{1}$ and $ \beta_{2}$/ $ \gamma_{2}$ is more clearly highlighted. Finally, the inclusion of pitch class set information for each species of tritone allows the systematic exploration three-note cue cells. Now, possible third-note positions can be described numerically as a pitch classes from 1 to 5 in reference to either of the constituent pitches of the tritone.2The memberships are shown below.


Table 2: Third-note Membership
pc set $ \alpha $ $ \beta_{1}$ $ \beta_{2}$ $ \gamma_{1}$ $ \gamma_{2}$ $ \delta $  
               
1   X       X  
2 X   X   X    
3   X X X X X  
4 X X   X      
5     X     X  


Hence, we see that the liberal $ \alpha $/$ \delta $ heuristic will operate as follows: If the third pitch is 2 or 4, the enclosing tritone is assumed to be $ \alpha $, and if the third pitch is 1, 3, or 5, the enclosing tritone is assumed to be $ \delta $. Under this liberal strategy, it turns out that any of the $ \beta$ and $ \gamma$ species might be mistaken for either $ \alpha $ or $ \delta $, depending on the third context note.

Now, the interpretation of the melodic minor scale will be depend on both the identity of the tritone, and on the identity of the third note of the cue cell. By working out the algorithmic heuristic for each particular combination of tritone and context tone, we can determine exactly how our listener will respond to any given cue-cell. The results are summarized in Table 3.


Table 3: Melodic Minor Inferred Tonics.
$ \hat{4}$-$ \hat{7}$ $ \hat{\flat3}$- $ \hat{\natural6}$
  $ \alpha $ $ \beta_{1}$ $ \alpha $ $ \beta_{2}$
  WT|WT|WT H|WT|H|WT WT|WT|WT WT|H|WT|H
1   C    
2 C   B$ \flat$ E
3   C   B$ \flat$
4 C   B$ \flat$  
5       B$ \flat$


As might be expected, if the tritone is $ \hat{4}$-$ \hat{7}$, our listener correctly infers the tonic note as C, and if the tritone is $ \hat{\flat3}$- $ \hat{\natural6}$, she mistakenly will select B$ \flat$ as the presumptive tonic. Notice, however, that with the liberalization of the heuristic, it is now possible to mistake $ \beta_{2}$ tritone as an $ \alpha $. It is because the melodic minor scale lacks the diatonic tritone subdivision's complementarity relationship that an enharmonic error can now arise.

If we were to assume for argument's sake that the tritone and the third tone is chosen randomly, the probability for correct identification of tonic would be 1/2, the incorrect identification of B$ \flat$ would be 7/16, and of E would be 1/16. Given the low possibility of enharmonic error, this is essentially an equivalent result to that obtained with the strict heuristic approach.


Table 4: Harmonic Minor Inferred Tonics.
$ \hat{4}$-$ \hat{7}$ $ \hat{2}$- $ \hat{\flat6}$
  $ \gamma_{2}$ $ \beta_{1}$ $ \gamma_{1}$ $ \delta $
  (WT|H|--3--) (H|WT|H|WT) (--3--|H|WT) (H|WT|WT|H)
1   C   E$ \flat$
2 C      
3 G$ \flat$ C A E$ \flat$
4   G$ \flat$ E$ \flat$  
5       E$ \flat$


When considering the harmonic minor on the other hand, the liberal heuristic fares rather better than its strict counterpart (Table 4). The situation becomes quickly more complex, largely due to the proliferation of possibilities when the context note is in location 3. Every tritone species involved in the two possible decompositions includes this possibility, so potential tonics accordingly proliferate. In musical terms, this is due to the enharmonic symmetry of the diminished seventh chord, which could potentially enharmonically resolve to any of four different locations. Using the same statistical assumptions as before, application of the liberal heuristic to the harmonic minor results in correct identification of C with probability 5/16, incorrect identification of Eb with probability 7/16, of G$ \flat$ 3/16, and of A 1/16. Although the strategy is still more likely than not to err, there is still a substantial improvement in that C is now a possibility. In this case, the conservative approach actually is less successful than the more permissive one!

Conclusion

If we imagine the purpose of such a tritone-based algorithm for position-finding to be the correct identification of diatonic location, then we must treat each instance of malfunction for the harmonic and melodic minor to be a complete failure--the mechanism had no failsafe to detect the nondiatonicity of the scale. After all, if the purpose is to map the diatonic space, then attempting to do so when none is in fact present is somewhat akin to systematically deriving the political affiliations of pet turtles. However, if the tritone system is thought to be for tonic-identification, then the system seems to be quite effective, even for music that does not exclusively use the diatonic set. The derivative scales are close enough to the diatonic in structure, that the feature of the tritone might remain sufficiently undisturbed to produce the correct solution. If this is to be a useful strategy in the complex world of real compositions, then the ability to draw successful inference in less-than-ideal situations would be very desirable.

The problem of minor modality is a vexing one indeed, fraught as it is with different definitions over the course of Western musical history. For example, although the modern natural minor scale is identified with the Aeolean mode, the minor mode of Johann Sebastian Bach, was that of re mi fa--the Dorian. Indeed, if we were to apply either heuristic to the natural minor scale, complete failure results--in every instance, our listener misidentifies the relative major (E$ \flat$, in the case of C minor).

On the other hand, it is possible that this is a feature, not a bug. That this technique is predictive of modal affiliation invites reconsideration of the strategy not in terms of identifying the tonic per se, but rather in mapping the implicit diatonic space, irrespective of the presumptive tonic tone. If the listener is merely attempting to derive a reference diatonic set agnostic of the tonally privileged member, then we can consider the heuristic as applied to the natural minor scale as a resounding success.

Perhaps then, it bears observing that the melodic minor results split almost evenly between C and B$ \flat$--C's ``Dorian root.'' Similarly, the tonic-finding strategy applied to harmonic minor scales favors E$ \flat$--C's ``Aeolean root.''3If tritone-based position-finding strategies are in fact occurring during the tonal listening experience, then any such `mistakes' in identifying the surrounding diatonic field are likely to produce exactly the sort of self-contradictory information which seems to define the uneasy and tonally unstable character of the minor mode. Potentially, the impact position-finding has on the ineffable qualities of minor mode music outweighs that of, say, detecting individual semitone displacements.

There are several future avenues to pursue. One immediate demand is to show that there are no other viable tritone subdivisions, subject to certain constraints, such as Tymoczko's consecutive semitone constraint (Tymoczko, 1997). Another is to pursue these tritone species as helpful mnemonic ways to comprehend tonal music. Perhaps most importantly, theorizing in this direction will encourage exploration of melodically-driven systematic studies of scalar structure.

Bibliography

Brown, H. and Butler, D. (1981).
Diatonic Trichords as Minimal Cue-Cells.
In Theory Only, 5(6-7):39-55.

Browne, R. (1981).
Tonal Implications of the Diatonic Set.
In Theory Only, 5(6-7):3-21.

Gamer, C. (1967).
Some Combinational Resources of Equal-Tempered Systems.
Journal of Music Theory, 11:32-59.

Huron, D. (2008).
A Comparison of Average Pitch Height and Interval Size in Major- and Minor-key Themes : Evidence Consistent with Affect-related Pitch Prosody.
Empirical Musicology Review, 3(2):59-63.

Tymoczko, D. (1997).
The consecutive-semitone constraint on scalar structure: A link between impressionism and jazz.
Integral, 11:135-79.

About this document ...

Position Finding with Tritones in Nondiatonic Music

This document was generated using the LaTeX2HTML translator Version 2008 (1.71)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

Footnotes

... tritone1
In this paper, `tritone' will refer generically to the interval [06] in 12-space--including both the augmented fourth and the diminished (false) fifth.
... tritone.2
If the third note is encountered in pitch space beyond the sounded tritone, one needs only invert the tritone to include the third pitch.
... root.''3
This relationship is most immediately the consequence of shifting the second tritone from $ \hat{\flat3}$- $ \hat{\natural6}$ to $ \hat{2}$- $ \hat{\flat6}$.