Just Triadic Tuning

A just triadic tuning in which just M3 and P5 are possible is easy to construct.

$$\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox{... ...4}}\:x_t + \ensuremath{\underline{3/2}}\:x\hspace{-1pt}_f+ x_r.$$

This maneuver of adding a ``thirds'' dimension seems to magically solve all our problems. But as is always the case with tuning (and life), you can't get something for nothing; it is all about compromises.

One potential problem with the just triadic tuning is that, like regular diatonic tunings with irrational , its size is infinite. Mathematically, this means that just triadic tuning is a one-to-one mapping, i.e. any change to x results in a change of value for $\tau(\mbox{\boldmath$x$})$. To show this, let us look at $\tau(\mbox{\boldmath$x$})$(not its log),

$$\tau(\mbox{\boldmath$x$}) \ = \ \left( \frac{5}{4} \right)^{... ...5^{x_t} 3^{x\hspace{-1pt}_f} 2^{x_r - 2x_t - x\hspace{-1pt}_f}.$$

Since 2, 3, and 5 are prime, any change to their exponents will result in a different value of $\tau(\mbox{\boldmath$x$})$. Any change to x results in a change to these exponents, so $\tau(\mbox{\boldmath$x$})$ is one-to-one and the just triadic tuning has infinite size. Of course size is not a problem for implementation on electronic instruments.

A big problem with just triadic tuning is that much of the repertoire relies on enharmony between pitches separated by P1₁, like A₀ and A₁. This enharmony does not exist in just triadic tuning since

$$\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(\mbox{... ...\ensuremath{\underline{81/80}} = C_S \approx 17.9 \mbox{ mil}.$$

In other words, P1₁ is tuned to a syntonic comma.

Let us examine how this enharmony is required in a simple musical example. Consider the chord progression I-IV-ii-V-I, whose diatonic pitch classes are indicated in Table 3.4 for C major.

 

 

G C A D G

E A F B E

C F D G C

How shall we assign triadic pitch classes to this progression?

Harmonically, we would like to think of major chords as having a M3_-1 between root and third and minor chords as having a M3_-1 between third and fifth. We will assume that diminished triads are tuned as two stacked m3₁, and augmented triads are tuned as two stacked M3_-1. Examples of subscripting using these harmonic criteria appear in Table 3.5.

 

 

major minor dim. aug.

G₀ G₀

E_-1 E_-1

C₀ C₀ C₀ C₀

Melodically, it would be nice to keep the triadic pitch class the same if two consecutive notes have the same diatonic pitch class.

As listed in Table 3.6, there is a solution to the harmonic and melodic constraints we have discussed above, but unfortunately it results in a P1$_\mathit{1}$ slip, a phenomenon whereby all parts become transposed by a P1₁ (usually downward). This is often called a comma slip, but it seems more appropriate to discuss these phenomena without reference to a specific tuning.

 

 

G₀ C₀ A_-1 D_-1 G_-1

E_-1 A_-1 F₀ B_-2 E_-2

C₀ F₀ D_-1 G_-1 C_-1

It is possible to imagine music in which P1₁ enharmony is not relied upon and therefore slips cannot occur. For example, music restricted to the five interlocking triads listed below cannot slip.

F₀ A_-1 C₀ E_-1 G₀ B_-1 D₀

Unfortunately, this would only be useful for music in C major that stayed strictly within the key and did not use triads based on B or D. Little if any such music exists. Richer sets of triads can be accommodated, for instance the set of seven interlocking triads below.

D_-1 F₀ A_-1 C₀ E_-1 G₀/G_-2 B_-1

Still, this triad set would only be useful for music in A minor that stayed strictly within the key and did not use triads based on G, G, or B. Again, little if any such music exists.

Arbitrarily large ``slip-free'' triad sets can be formed. This is an important point, because many people assume that in order to avoid slippage, music must be limited to small triad sets such as the examples above. The problem with slip-free triad sets is not that they are inherently small (they are not), it is that they do not suit the needs of traditional Western music. It turns out that P1₁ enharmony is a critical aspect of Western music, because a single pitch must be able to have multiple harmonic functions.

For example, if a piece is to have a C₀, F₀, and G₀, it cannot have a D that is m3₁ below the F₀ (D_-1) and P5₀ above the P5₀ (D₀). This supertonic problem is one of the well-known consequences of slip-free triad sets [18, 61] [27].

Having decided that slip-free triad sets are not practical, how else can we avoid slippage? One option is to use tunings that enharmonically identify P1₁. Unfortunately, this is the same thing as returning to a diatonic representation of pitch and using a diatonic tuning.

Another option is to relax our harmonic and/or melodic constraints. Harmonically, we could allow M3₀ (and its co-conspirator, m3₀) back into the picture for occasional appearances. We could even allow P5_-1 to appear on rare occasions, even though it is best not to tamper with this all-important interval. Melodically, we could allow a repeated note to have two different subscripts. We could even allow melismatic intonation, the adjustment of the intonation of a note while it is sounding. This way, a note held between two chords could have its subscript change as its function changes [30], [17, 143]. A change in subscripting to a held or repeated note could be disconcerting, especially if it were in an outer voice.

P1₁ slippage aside, another problem is that it is not always easy to assign triadic pitch classes (subscripts) to the notes of a piece of music. The primary reason for this is that not all notes have a triadic (or even harmonic) function. For example, what is the harmonic function of an ``A'' passing tone between a C major and a G major chord, since it belongs to neither? Another example is the seventh of a dominant seventh chord. It certainly has harmonic function, but how should it be subscripted triadically? In a dominant seventh on G₀, is it F₀, functioning as a subdominant to C₀, or is it F₁, functioning as an m3₁ stacked on top of D₀?

So not only is it difficult to avoid P1₁ slips, it is also difficult to know how we would like to assign subscripts even ignoring them. In other words, we can't always get what we want, and we don't always even know what we want! In some sense this is good, though, because if it is unclear how to subscript a pitch, we can let the avoidance of slippage dictate the subscripting.

At some point, judgments about subscripting cannot be made without knowledge of the triadic tuning that will be applied to them. For instance, without knowing just how bad an M3₀ sounds, it is difficult to decide how much it should be avoided in favor of M3_-1.

When such painstaking judgments are being made, it becomes unclear that the abstraction of triadic pitch class is really offering anything, and in fact one should not just take recourse to ``raw'' intonation rather than tuning. For instance, instead of assigning a subscript to each note, an intonation instruction in the form of a deviation from Pythagorean tuning could be given.

Indeed, for the most demanding applications this raw intonation approach is probably the best. In many circumstances, it makes sense to use a diatonic tuning for most notes, and carefully choose the intonation only in parts of a piece where it is most important, for instance in slow sections or final, held chords.