On dominant substitutions

The author looks at a variety of subsitutions for V7 and IVm6 within tonal harmony and why such procedures work. Meanwhile he explores some fundamental issues of music theory.

The asymmetry of the Harmonic Series remains one of the most interesting fundamental features of tuning science. Its implications for music theory are equally impressive. For the Harmonic Series has a reciprocal, usually called the Subharmonic Series. Take the pitch “C” and it yields a C major triad in  harmonics and an F minor triad in sub-harmonics (note, not a C minor!). The functions I and IVm thus form a special ambivalent relation that wants to waver back and forth if not influenced by counter-balancing forces. I sometimes call it the “see-saw” relation. The two sides form almost parallel universes.

The fact of asymmetry generates an interminable debate over the status of the reciprocals. Are they equal (polarity theory) or not (turbidity theory)? Both sides have long-standing and valid arguments. The polarity supporter affirms that they are, after all, reciprocals with all that this relation implies. The turbidity theorist counters that a minor chord clouds the medial consonances, and therefore it has less status than the major chord. Moreover, what we actually do hear forms harmonic series components – the sub-harmonics don’t even exist in nature! Countering this argument, the polarity theorist draws up a monochord, whose simple arithmetical division then generates or actualizes aspects of the sub-harmonic series. Indeed, the two realms have a strange inter-relation with each other. For (to give one instance) we can also find a minor triad in the harmonic series – harmonics 10, 12 and 15.

Rather than take sides in a rigid manner, I sometimes find the one a useful option here, the other there – without any ultimate resolution of the whole issue. For example, I defend the value and priority of the harmonic axis ( 1 5 ) which includes the dominant but not the sub-dominant. Here I’m guilty of turbidity theory because I have markedly differentiated their status.

On the other hand, the polarity theory also has its attractions. It explains the primary tendency of triad chords within tonal harmony. Major chords tend to act as dominants ( V ) and resolve to the tonic major chord ( I ), while minor chords tend to act as sub-dominants ( IVm ) and resolve in the other direction to the tonic minor chord ( Im ). These movements feel like floating downstream on the river. A reverse move from I  to V  feels rather like (delightfully) swimming upstream against the current. Consequently, one can argue that the primary or fundamental chord progression in the major key uses the chords I  IVm V  I. In the minor key it is Im  V  IVm  Im. Note the presence of the “see-saw” relation in the major (I  IVm) and the minor (Im  V). The same chords are used (IVm  and  V), only their direction has reversed between the major and minor modes. This jewel and mainstay of functional harmony rests squarely on  polarity theory.

The tendencies within simple triads become further accentuated when they are thickened into seventh chords. Thus the basic progression can be filled out in the major as I  IVm  V7  I and in the minor as Im  V  IVm6  Im. That is so to say, the IVm6 works in the minor mode in a mirror-like way analogous tot the way V7 works in the major mode. The polarity theorist accepts all of this without qualms and gives IVm6 and V7 equal status. The turbidity theorist says no, no. I leave the reader to decide – with a warning that the decision is not straightforward or simple!

This paper assumes a polarity position and treats IVm6 and V7 as married with children. It seems reasonable to me because this model allows one to approach the major and minor modes as alternative partners. Of course, the turbidity purist will object. Some music theorists even avoid the very use of the function IVm6 (that is,  4  b6  1  2) at all, consistently using the name II half-diminished seventh or IIob7 (elements 2  4  b6  1). But this practice is somewhat of a mystery to me, since IVm6 and V7 form an asymmetrical pair of hamonies pecuriarly bound to each other – one is the reciprocal of the other.

Moreover, they express a certain tolerance for each other. V7 prefers to resolve to I but it also accepts Im, (due to the see-saw relation) with no change of direction. Similarly, IVm6 prefers Im but also takes I without complaint. These functions have strong primary bonds with each other that are not quite found in (for example) IV and Vm. These two functions form a second tier in the hierarchical sequence whose next members are bVIIm and II. But now we approach the realm of extended dominants (and sub-dominants, the polarity theorist reminds us). We will leave that area for now.

The aim of this short paper is simple enough: look at a variety of functions that work well as replacements for V7 and IVm6 in our primary progression. In doing so we hope to get some idea about why they work.

Why does V7 behave the way that it does? One can rightfully argue that the presence of the diminished triad as VIIo (the elements 7  2  4) dominates it. Similarly, IIo (elements 2  4  b6) marks IVm6. This symmetrical dissonant triad is certainly a strong pointer. However, one can also make a variant that is not at all diminished, yet points just as strongly – V dominant seventh triad (5  7  4). Thus the most important source is the tritone interval (7  4 in the major and 2  b6 in the minor). The tritone leads the way because 7  4 (an expansion level seven harmony) wants to resolve into 1  3 (expansion level five). In other words, a relativity dissonant chord wants to collapse into a relatively consonant chord. Similarly in the minor mode 2  b6 seeks b3  5 and hence the minor tonic. Yet it can quite happily also resolve into 3  5 and the major mode. Moreover, 7  4 can also become 1  b3 and adapt itself to the minor mode. In the tritone one sees the potent source for that tolerance between the modes so evident in V7 and IVm6.

In the small collection of harmonies that follow the tritone forms a constant theme. We can place this interval into a number of interesting contexts.

The diatonic scale itself is bounded by the tritone on the circle of fifths, an E7 expansion. Thus by looking at subset hamonies of this master set (subsets that include the tritone) we generate a concise family of hamonies that serve as substitutions for V7 and IVm6.

The first and a most subtle one reserves the types. Thus V7 becomes IIm6, an introverted or recessive resolution (much used by Brahms). It works because it embodies only one changed element; that is, 5 7 2 4 becomes 2 4  6  7. The diminished chord remains. This shift of only one tone nevertheless imbues the harmony with a very different atmosphere. Yet it functions just like V7. Analogously in the minor mode IVm6 becomes bVII7 as 4 b6 1  2 becomes b7 2 4 b6. This pair forms one of the most harmonious of all substitutions. They also admirably illustrate the rule of thumb: a good substitution changes so little as possible.

Here is another harmony that changes little. Like the V7 it embodies the symmetrical diminished triad but is itself asymmetrical. We can call it VIIo add flat-two or VIIob2, the elements 7 1  2  4. It also works in the minor mode as IIob2, i.e. 2 b3 4 b6. Looking at its reciprocal we have IIo4 (i.e. 2 4 5  b6) in the minor mode and VIIo4 in the major (7  2  3  4). This pair holds much more dissonance than the last pair, yet they still function quite well as substitutes. In fact, any chord that keeps the diminished triad applies.

If one wants to maximize dissonance within an austere texture, try this asymmetrical pair which has absolutely no diminished triad subset. I call this type the five sharp-four triad. It gives IV5#4 (i.e. 4  7  1) in the major and bVI5#4 (elements b6  2  b3) in the minor. Its reciprocal is the five flat-two-type. Here we have V5b2 (i.e. 5  b6  2) in the minor and III5b2 (i.e. 3  4  7) in the major. These two harmony types combine to form an interesting superset symmetrical tetrad harmony with even more bite in its dissonance. Such complex structures are cumbersome to name but simple enough to specify. In the major mode we see 1  3  4  7, in the minor 2  b3  5  b6.

The following asymmetrical pair of tetrads also make supersets of the above triad pair. Perhaps we can consider these types more consonant than the above because they also contain a simple triad subset. The first type is called a major triad plus sharp-four. Thus we get IV#4 (elements 4  6  7  1) in the major and bVI#4 (elements b6  1  2  b3) in the minor. Its reciprocal is the minor triad add flat-two type. Thus we get Vmb2 (5  b6  b7  2) in the minor and IIImb2 (elements 3 4 5 7) in the major. Such chords suit a functional harmony that tolerates high degrees of dissonance.

A total change of character embodies the wholetone (w) scalar tetrad. This symmetrical harmony was much valued by the impressionists. In the major mode it goes 4 5 6 7. In the minor b6 b7 1 2. We could call them IVw and bVIw. A cousin of the diminished triad, this harmony works quite well as a substitute for V7 and IVm6. Only one element is changed. It also embodies much less dissonance than the foregoing two pairs.

Sometimes the substitution is more a scale than a chord. A good example can be found in the symmetrical Dominant Scalar Pentad, another harmony that retains the diminished chord. In the major mode it appears as VIIob2,4 or elements 7  1  2  3  4. In the minor mode it is IIob2,4 which is 2  b3  4  5  b6. It acts as a shortened version of the diatonic scale.

Sometimes the harmony is not so much a substitution for V7 and IVm6 as it is a thickening of the harmony. The most prominent example is the symmetrical pentad harmony that contains both of them. In the major mode the elements 5  7  2  4  6 means V7,2 but it also hides IIm6,4 as a secondary identity. In the minor mode the pattern 4  b6  b7  1  2 is defined as IVm6,4 but secondarily bVII7,2. The original harmony has incorporated the recessive substitution.

This harmony raises issues concerning function. Instead of naming the major mode harmony V7,2 and IIm6,4 it is perhaps more expedient or convenient to consider it a superset of the diminished triad, i.e.as VIIob7,b6. Then in the minor mode we have IIob7,b6. The usefulness of this approach means that we cannot just brush aside those theorists who support the half-diminished seventh chord ( IIob7). Inevitably we return to the dispute between polarity and turbidity theory. The polarity supporter reminds us that the reciprocal relation between V7 and IVm6 is unavoidable. The turbidity theorist undermines the importance of the root in the minor chord. Are these issues even resolvable?

It is time to pause, take stock, and gain a broader perspective. In the search for dominant substitutes I have presented a good sample of tetrads with the odd triad and pentad as well. The sample was never meant to be exhaustive, but it could be done. The reader may already see a certain pattern in the material. All of the harmonies form subsets of one scale, and any subset will work – although one must admit yhat some work better than others. At any rate we can simply compute all of the subsets for our diatonic mode. This is not so difficult for such a small master set, by using the algebra of subsets and supersets. In the early 1970’s I found an even easier method – by looking at the geometric patterns displayed on the circle of fifths. Every pattern type forms a chord type that can be modulated into the appropriate function.

The master scale in question is, of course, the Locrian mode. In the major key it appears on VII (elements 7 1 2 3 4 5 6), in the minor key on II (elements 2 b3 4 5 b6 b7 1). Since the base of the Locrian scale sits in the diminished triad, why not rationalize the whole assortment by using this base? Throw out both IVm6 and V7! In this treatment the major mode is serviced by VIIob7,b2,4,b6 while the minor mode has IIob7,b2,4,b6. From this standpoint one can see that the permutations are not very complicated. However, this approach still misses the sets that don’t contain the diminished triad. Perhaps such an approach will never become popular, since most theorists are content with IIob7 yet they balk at calling V7 a VIIob6. It reflects a turbidity stand. One can get away with gerrymandering a minor chord with its ambiguous root, but a major chord is not subject to such vagaries. It gets better treatment. The polarity theorist also feels uncomfortable about renaming V7, but insists that the minor triad deserves to be handled in a more dignified manner.

Thus far this paper has considered only subsets of an E7 expansion. Such an approach is understandable since V7 and IVm6 are themselves E7 harmonies. However, good substitutions can also be found in wider expansions of the circle of fifths. The search is a bit more complicated. No longer can we simply find a harmony with the requisite tritone and assume that it will act as a dominant substitute. In wider expansions the tritone is no longer “top dog” as far as dissonance is concerned. Consequently it tends to get swallowed up within greater dissonance and lose its effectiveness. Wide expansion dominant substitutes tend to be rather isolated special cases in  a sea of dissonant possibilities. In the rest of this paper I will look at only a small number of outstanding harmonies that uniquely exemplify the features of a good dominant substitute.

At E10 sits a remarkable harmony, perhaps the most special of all dominant substitutes. It is called the Diminished Triad, specifically the VIIo tetrad made up of the elements 7  2  4  b6. It forms the union of the two fundamental tritone functions 7  4 and 2  b6, as well as the two foundational diminished triad functions 7  2  4 and 2  4  b6. Consequently it no longer has any preference in its resolution. I and Im are equally acceptable. It also differs by only one element from V7, IVm6, IIm6 and bVII7. The dominant V7 (5  7  2  4) becomes 7  2  4  b6. The sub-dominant IVm6 (4  b6  1  2) becomes 7  2  4  b6. In  these two harmonies the change occurs by only one diatonic semitone. The dominant substitute IIm6 (2  4  6  7) becomes 7  2  4  b6. And the sub-dominant substitute  bVII7 (b7  2  4  b6) becomes 7  2 4 b6. Here the change of element involves only a chromatic semitone.

We should also note that the line of fifths that holds the diminished tetrad is bounded by b6 and 7 – the augmented second interval (#2) which is also called the septimal or dark minor third. When we plot the combination of V7 and IVm6 on the circle of fifths it also forms an E10 expansion bopunded by b6 and 7. Thus these harmonies are bonded with each other in various ways. Individually they express an E7 character but together they become E10 – the full diminished level of expansion. It favors the tonality as a whole rather than preferring I or Im.

This “holistic” personality is revealed through its symmetrical nature: it has axis symmetry about the harmonic axis (1  5). It breaks down into minor thirds with one sub-minor third in the symmetrical position between b6 and 7. It covers almost the whole chromatic tonality (E12) like an umbrella, happy to resolve either way. Thus it differs from the E7 diminished triad which has point symmetry around 7 2 4 and 2 4 b6. The axis symmetry puts the harmony in the company of the Ogdoad expansion (E8) and the Scalar expansion (E6). Indeed, the primary scalar tetrachord of E6 (elements 5 6 b7 1) has a peculiar affinity with the E10 “middle-eastern” tetrachord (elements 5 b6 7 1). These harmonies form cousins of the diminished tetrad.

Moreover, the diminished tetrad is quite rich in  its subsets. Not only does it have two sorts of minor third intervals (diatonic and septimal), it also has differing diminished triads. The much covered functions 7 2 4 and 2 4 b6 can best be named the symmetrical diatonic diminished triad. In addition we see an asymmetrical pair best named the two types of septimal diminished triad: 7 2 b6 and 7 4 b6. These complex harmonies will remain prominent later in the E11 expansion. Only note here that the diminished tetrad embodies this diversity of diminished triads within itself.

The special tetrad also highlights the differences between tuning systems. For 12-et cannot distinguish between diatonic and chromatic semitones. Moreover, its “#2” is really only b3, so that the “pointed” nature of the harmony as a whole is altered. As far as 12-et is concerned all the diminished triads are the same. The harmony becomes reductionist – three diminished tetrads instead of twelve. Consequently, the VIIo tetrad uses exactly the same pitches as IIo (for hat matter also Ivo and bVIo). This functional ambiguity of 12-et is a blessing or a curse depending upon your attitude, but it is unavoidable. On the other hand, 19-et and 31-et banish reductionism and harbor no functional ambiguity at all. For example, the IIo diminished tetrad uses elements 2  4  b6  b1 and refers itself not to the I(m) tonality but rather to the bIII(m) tonality. Thus the E10 diminished expansion shows where 12-et parts company with the prime systems 19-et and 31-et.

The diminished tetrad is such an important dominant substitute that superset chords containing it have a good shot at acceptance. Most prominent among this group is an asymmetrical pair that combines the diminished tetrad with V7 and IVm6. These pentad harmonies are not so much substitutions as thickenings of the pattern. They also form important subsets of the diminished scale. Thus V7 becomes V7,b2 (elements 5 b6 7 2 4) and IVm6 becomes IVm6,#4 (elements 4  b6 7 1 2). By virtue of the diminished tetrad subset these harmonies should have no resolution preference, but by virtue of the inherent V7 and IVm6 they should show at least a little bias. Just how much proves difficult to decide.

If we expand W10 just one more step to E11 we reach the Septimal expansion bounded by the augmented sixth (#6) interval between b2 and 7. Here we find another asymmetrical pair of harmonies with special characteristics. In fact these two are so prominent that they rival the diminished tetrad in importance. V7 becomes bII-7 while IVm6 becomes VIIm+6. The root of the harmony is transposed into the so-called “neapolitan” function.

Take the bII-7 (elements b2  4  b6  7) first. Two elements have been shifted from V7 but the tritone function 7  4 endures. The axis of the harmony (5  2) has been transposed a tritone in the sub-dominant direction tot axis b2  b6. As a result the harmony now incorporates one of the asymmetrical diminished triads (7  4  b6) that we saw above. Its mood or character changes a lot. Because the seventh chord uses a #6b rather than a b7, it forms a different flavor of seventh chord best named the septimal dominant seventh chord. It is to be distinguished from the E7 diatonic dominant seventh chord. The E11 chord is in fact more

consonant since it closely resembles the Harmonic Series norm. This matter has resulted in much controversy.  Some theorists contend that it need not act like a dominant at all – it has no necessity for resolution. I disagree here and contend that the seventh harmonic is in practical terms best treated as a dissonance. Some theorists think that we should replace V7 with V-7 in our classical cadence. But this view is totally misguided. The function is quite clear: V-7 normatively resolves to #IV, not to I. The neapolitan function rules.

bII-7 has much in common with V7. Both harmonies are tolerant but show a little bias toward the major resolution. They don’t have the “omni” character of the diminished tetrad. Moreover, both bII-7 and V7 emerge out of the diminished tetrad by only one change of element: VIIo tetrad (7  2  4  b6) becomes bII-7 (b2  4  b6  7) by a change of only a chromatic semitone. Hence bII-7 makes an excellent substitution for V7. Perhaps it is preferable to state that bII-7 and V7 make good substitutions for the VIIo tetrad. Well, now I tread upon controversial ground.

The function bII-7 already had usage and theoretical justification during the meantone era. The so-called German Sixth chord treats the standard extended dominant resolution II7  V by replacing II7 (2  #4  6  1) with the tritone-removed function bVI-7 (b6  1  b3  #4). Using the standard traditional meantone tuning (a line of fifths from Eb to G#) this works in the much-used keys of G and D. This progression reminds us that the dominant substitutes of this paper can be put to the wider service of extended dominants (and sub-dominants of course).

In the recoprocal situation IVm6 becomes VIIm+6 (7  2  #4  b6). Here the minor leaning tritone function 2  b6 forms the anchor while the axis IVm6 (4  1) is transposed a tritone on the dominant side to the axis 7  #4. The resultant harmony makes another flavor of the m6 chord – a version quite close to the Sub-harmonic Series norm. In some ways this chord is the turbidity theorist’s nightmare. Should or could we call it a name relating to its asymmetrical diminished triad b6  7  2? Moreover, it actualizes a good segment of te Sub-harmonic Series – something that is not supposed to exist in nature! The polarity theorist triumphantly retorts that we should call it the reciprocal neapolitan function. How about the sub-neapolitan? For various reasons this function was not valued to the extent of the bII-7. Some called its character “spooky”. Here again we detect the influence of turbidity theory. The sonority of the Harmonic Series has a certain sweetness to which one quickly gets accustomed. The Sub-harmonic Series is a different cookie that many feel to be strangely spicy, an acquired taste.

None of this means much of anything to one working in 12-et, which cannot make these harmonies. Instead the bII-7 is trivialized into bII7. The neapolitan function becomes just another diatonic dominant seventh chord. That being said, all of the 12-et advanced (chromatic) harmony can be approached and explained by reference to tritone-removed seventh chords. Indeed, the whole system is ruled by the reductionist tritone and its ramifications. On the other hand, in 31-et the bII-7 forms the jewel in the crown, one of the many strengths of the tuning  system. In 19-et the bII-7 becomes strangely dissonant and forms the gateway to its antipodal harmony. It exaggerates the difference between the diatonic and septimal seventh chords. Thus the Harmonic Series seventh chord (and its reciprocal) bring out the differences between 12, 19 and 31-et.

Coming back to our VIIm+6, the reciprocal also shares in the close relations to the VIIo tetrad that bII-7 exhibits. The diminished tetrad 7  2  4  b6 becomes VIIm+6 (7  2  #4  b6) by a single change of only one chromatic semiton e. Whether or not one likes its mood, it surely makes an excellent substitute for IVm6.

The harmonic functions V7,  bII-7, IVm6 and VIIm+6 form such an influential family that supersets containing them have priority as dominant substitutes. The most prominent example is the symmetrical E11 diminished scale. In the major mode the elements 5  b6  b7  7  b2  2  3  4 contain as subsets both V7 and bII-7. These harmonies dominate the scale. In the minor mode the elements 4  #4  b6  6  7  1  2  b3 contain both IVm6 and VIIm+6. The diminished scale thus displays that same tolerant bias seen in its illustrious subset chords.

We have seen that V7 and IVm6 possess a recessive alternative (IIm6 and bVII7) in which the chord types exchange places. The same situation exists for bII-7 and VIIm+6. I present these harmonies with great pleasure because (as far as I know) they are completely ignored by the standard music theory texts. Perhaps this oversight arises due to the overall fixation on 12-et, where septimal harmonies lose meaning.

Starting with the major mode, we generated bII-7 from V7 by holding the tritone 7  4 and shifting the axis 5  2 in the sub-dominant direction to the axis b2  b6. Why not try using the enharmonic alternative on the dominant side (#1  #5)? We get #1  4  #5  7.  Unfortunately this is an extreme dissonance which fails totally as a dominant substitute. One example of failure among many such possibilities, this harmony is even difficult to name (perhaps “dissonant major triad plus seven”). This exercise confirms that the tritone shift works in only one direction. Nevertheless, a harmony on the dominant side is entirely possible. Move the axis 5  2 up a chromatic semitone to #5  #2 and now we have #Vm+6 (elements #5  7  #2  4). In the reciprocal harmony that prefers the minor resolution, begin with IVm6, keep 2  b6, and shift the axis (4  1) down a chromatic semitone to achieve bIV-7 (elements b4  b6  b1  2).

These two harmonies are perhaps the most “distant” of this paper – yet they still work as dominant substitutes. We can appreciate their remoteness by compairing them to that “grand central station” of dominant substitutes – the diminished tetrad. Now two elements must shift instead of one. In the #Vm+6 the 7  2  4  b6 becomes #5  7  #2  4 through a chromatic semitone up and an enharmonic down (from b6 to #5). In the reciprocal harmony bIV-7 the 7  2  4  b6 becomes b4  b6  b1  2 through a chromatic semitone down and an enharmonic up (7 to b1). Now that ’s unusual! In fact I have not been able to find a dominant substitute that is more ”far out” than this pair.

We can confirm that the set is complete by applying the above procedure to the IIm6 and bVII7. Just as V7 generated bII-7 and #Vm+6 and similarly IVm6 generated VIIm+6 and bIV-7, so IIm6 generates #Vm+6 and bII-7 while bVII7 generates bIV-7 and VIIm+6. The same players reappear.

To summarize: the major mode has the services of V7, IIm6, bII-7 and #Vm+6. The minor mode holds IVm6, bVII7, VIIm+6 and bIV-7. That pretty well covers the normative seventh chords.

The reader may be forgiven for thinking that our project is now complete. But no, sometimes an excellent dominant substitute sits among the class of altered dominant chords. An outstanding case can be found in the seventh chord with the flattened fifth. A point-symmetrical E11 harmony and subset of the whole-tone scale, this function was called the “French Sixth chord” during the baroque era (function bVI-7#4). Later during the 19th century it became the “Tristan chord” after Wagner’s brilliant usage. It has a strong feature in common with the diminished tetrad – two tritones instead of one. Because of this feature the major mode has two positions for the 7  4, the minor two positions for 2  b6. So it has four reolution positions rather than two.

The harmony has another special and peculiar characteristic – an alternative functional name. Take the extremely useful function V7b5 (5  7  b2  4). With equal accuracy we can call it bII-7#4 ( b2  4  5  7). In fact it forms a sort of  “halfway house” between V7 and bII-7. Check: V7 (5  7  2  4), our 7b5 (5  7  b2  4) and bII-7 (b2  4  b6  7). Thus it is firmly bonded with the seventh chords. Staying with the major mode, the other resolution position is perhaps secondary in value: VII7b5 (7  #2  4  6) which equals IV-7#4 (4  6  7  #2). The minor mode has its primary defender in II7b5 (2  #4  b6  1) which can be co-named bVI-7#4 (b6  1  2  #4). Its secondary resolution is bVII7b5 (b7  2  b4  b6) which equals bIV-7#4 (b4  b6  b7  2). Thus the dominant substitute has four positions with a total of eight names. Extraordinary!

This profusion of functions arises because the same chord can be taken in two ways: as 1  3  b5  b7 and as 1  3  #4  #6. The former harmony has point symmetry around 4, the latter around 7. In the strong V7b5/bII-7#4 function the harmony has point symmetry around the tonic, a special trait. This function may have elevated status above the other three, but all four functions serve well as dominant substitutes.

Now the turbidity theorist waves the flag and smells victory. For we do find it convenient to consider these chords alterations of V7 and not IVm6. Why is this? The turbidity supporter explains that the V7 is just more stable than the IVm6, thus it makes a better anchor for pinning down these unruly altered chords. It innately holds a higher status. Although I cannot fault the argument at all, I see the polarity theorist calmly offering the reminder that II7b5/bVI-7#4 also serves as a halfway house between IVm6 and VIIm+6. Check: IVm6 (4  b6  1  2), our harmony (2  #4  b6  1) and VIIm+6 (7  2  #4  b6). Thus we cannot say that the IVm6 is absent any more than V7 in  the alteration. Moreover, we can also observe that VII7b5/IV-7#4 is halfway between IIm6 and #Vm+6. Check: IIm6 (2  4  6  7), our harmony (7  #2  4  6) and #Vm+6 (#5  7  #2  4). Finally, the function bVII7b5/bIV-7#4 sits between bVII7 and bIV-7. Check: bVII7 (b7  2  4  b6), our harmony (b7  2  b4  b6) and bIV-7 (b4  b6  b1  2). Thus the four functions of this harmony integrate beautifully into the set of diatonic and septimal seventh chords. Because we accept the validity of this set, the polarity theorist argues, the two sides need to be given equal weight. After all, the polarity position is nothing if not elegant and balanced. You decide.

The value of the 7b5 harmony arises out of its relation with the diatonic and septimal norms. It’s a delicate balance whose alteration is fraught with dangers. For example, suppose we convert V7b5 into V7#4, an operation that requires the change of only one element. Now the connection is broken, the dissonance  becomes more acute, and the tritone loses some effectiveness. Nevertheless, it still functions as a dominant substitute, if now demoted to “B grade”. V7#4 is a point-symmetrical E9 (augmented level) harmony, another whole-tone scalar subset, and it also has two tritones and therefore four resolution positions. These are: V7#4 (5  7  #1  4), IV7#4 (4  6  7  b3), bVII7#4 (b7  2  3  b6) and bVI7#4 (b6  1  2  b5). My point here is that one can easily proliferate a whole raft of second-rate dominant substitutes, but none will measure up to the 7b5.

The 7b5 harmony also has a prominent place in 12-et harmony. It forms the central plank in a way of modeling tonality (really a quasi-tonality) called Dual Modality. This theory has been proposed to help make sense of some very advanced chromatic harmony (notably Scriabin). The 7b5 type proves reductionist – six members instead of twelve, like the tritone interval. Therefore, take a stand for a special type of bi-tonality in which the key centers relate by a tritone.  We can do this because in 12-et V7b5 = bII7b5. One can reasonably argue that this form of dual tonality constitutes the only true bi-tonality in the system. Other forms tend to coalesce around one center, but the tritone relation retains its enigma. Proponents of this sort of harmony like to emphasize tritone-rich chords as wel – it shows almost an obsession with tritones. At any rate the end result is a vague and ambiguous tonality that verges on atonality. If not well handled it lapses into functional confusion. Such ambivalence is not present in 19-et and 31-et, where the V7b5 may well have two names but only one normative resolution. Functional ambiguity has no place here.

While on the subject of 12-et, reductionism and the 7b5, I cannot resist a mention of the so-called “Petroushka chord” (after Stravinsky, but Scriabin also used it). It is undoubtedly the most esoteric harmony in  the whole system because it boasts the only reductionist harmony (six members like the tritone and 7b5) that is asymmetrical. Every other reductionist harmony is symmetrical. It forms a complex hexad, but its construction is simple enough. Combine two major triads that relate by a tritone, e.g. V and bII. We get the elements 5  b6  7  b2  2  4. Its subsets include V7, V7b5 and the VIIo tetrad. We can call it V7,b2,b5. This harmony has three tritones no less, and hence six resolution positions. Since it is asymmetrical, its reciprocal is also under consideration, although as for as I know no one has bothered to name it. Here we combine IVm and VIIm to make the elements 4  #4  b6  7  1  2. It includes IVm6 and VIIo tetrad. Let’s call it IVm6,#4,#1. It also has three tritones and six resolution positions. But I will not write out the many resolution positions of this pair here – they are after all only mediocre dominant substitutions. For they are just too complex. I have included them in order to illustrate the kind of structures that animate Dual Modality. Needless to say, in 31-et the harmonies lose these esoteric qualities and simply revert back into E11 complexities.

Many theorists treat the augmented triad as a dominant substitute, namely as Vaug (5  7  #2). Others prefer to change the spelling to 5  7  b3 and claim it for the minor mode. But I detect some fuzzy thinking here. For the second spelling not Vaug at all, rather bIIIaug. The augmented triad is also subject to much abuse due to its reductionist character in 12-et. Moreover,  the harmony has no tritone. Are we wise to call it a dominant substitute? I think not. Once we make the conditions for inclusion so broad that we also accept many harmonies without a tritone, the very concept of a dominant substitution is undermined. I also feel uneasy about it because the augmented triad exhibits a certain “directionlessness” totally at odds to the tritone’s pointedness toward a particular resolution. This triad is ready to go anywhere rather than to one specific destination. I would use the rule of thumb that in general augmented harmonies (which are, after all, not such a large sub-group of types) make poor dominant substitutes.

An interesting exception involves the case where we take the Vaug and combine it with the tritone to make V7#5 (5  7  #2  4). For sure, this harmony does not quite show the value of V7b5, but it nevertheless beautifully expresses an augmented structure ”tamed” or given direction. In the minor mode it appears as bVII7#5 (b7  2  #4  b6). The harmony also demonstrates asymmetry. Its reciprocal, interestingly enough, yields the related -7#5 type. I refer to bII-7#5 (b2  4  6  7) in the major and bIV-7#5 (b4  b6  1  2) in the minor. The two reciprocal types show this difference: 7#5 (e.g. 1  3  #5  b7) and its partner -7#5 (e.g. 1  3  #5  #6).

Moreover, this group of four resolutions also displays that peculiar integration with the seventh chord norms that we saw in the 7b5 group. V7#5 sits midway between V7 and #Vm+6. Check: V7 (5  7  2  4), V7#5 (5  7  #2  4) and #Vm+6 (#5  7  #2  4). Meanwhile in the reciprocal, bIV-7#5 mediates between IVm6 and bIV-7. Check: IVm6 (4  b6  1  2), bIV-7#5 (b4  b6  1  2) and bIV-7 (b4  b6  b1  2). In the other group the bII-7#5 compromises between IIm6 and bII-7. Check: IIm6 (2  4  6  7), bII-7#5  (b2  4  6  7) and bII-7 (b2  4  b6  7). Finally, bVII7#5 comes between bVII7 and VIIm+6. Check: bVII7 (b7  2  4  b6), bVII7#5 (b7  2  #4  b6) and VIIm+6 (7  2  #4  b6). Thus these harmonies have some status above alternative augmented complexities.

This harmony is also an E11 expansion. The reader may have noticed that all of my examples in the latter part of this essay (after the central diminished tetrad) have been E11 harmonies. Consequently, one can compute all of the possible types as subsets in the usual manner. To be sure, the variety is much wider than that found in E7. However, one also sees a lot more “B grade” or even “C grade” material. I have chosen only the highlights in the interest of brevity.

The harmonies of this paper form a tight and inter-related group centered in the diatonic and septimal seventh chords. We can list them together under two headings: a bias for the major resolution and a bias for the minor. The VIIo tetrad is, of course, omitted from this list because it forms the exception. I have also left out the secondary E7 substitutes. Functions with double names are separated by the set.

These harmonies define the superior dominant substitutes. I leave the reader to ponder this question: do these two columns have equal weight?

Siemen Terpstra, November 2010, Amsterdam