Alternative Tuning and How We Hear

APPLICATIONS (continued)

JUST TUNED SCALE

Notice that as we perform this construction various assumptions and trade-offs need to be made. Often, a different and equally useful scale could be formed by making different decisions. Already we have assumed the need for octave equivalence and the need for fifths and fourths.

We'll start with C being 1/1. Following the definition of the just tuned fifth being a 3/2 ratio, we set G to 3/2, and F to 4/3.

The way to think about how to get F is to say that the interval of F to C must be the same as the interval from C to G.

F / C = C / G

F / (1/1) = (1/1) / (3/2)

F = 1 / (3/2) = 2 / 3

Now we'll add the third, E, to complete the major triad (C E G). The relative ratios for these notes should be (4 5 6). So that makes E be 5/4. This can be seen by rewriting C as 4/4, and rewriting G as 6/4. This is another point where you could make a different decision and head off into another direction to create another scale. Thirds are not necessary to form usable scales.

Our scale so far has:

We continue to fill out the rest of the table by considering what notes are needed for the key of G and F. We want to maintain the same relative ratios. All the notes from the key of G are 3/2 of the frequency of the corresponding note in the key of C. All the notes from the key of F are 4/3 of the frequency of the corresponding note in the key of C. These are shown in the last two pairs of columns. The ratios can be expressed in simplest terms, and for the ones that fall outside of the octave, we can multiply or divide by two to shift them back into the same octave.

This helps us define the ratios for D, A, Bb, and B. We can add these new notes to our table. (Let's skip Bb for now, since it's not really in the key of C). Here's how our table looks so far:

All that's missing now is a few accidentals. We can copy the combined scale from the key of C back into the other keys to begin to do this.

The first thing to notice is that we're at odds on what the proper ratio is for A and D. Our previous analysis said that A ought to be 5/3 - yet now the key of G would like to see the second note of the scale be 27/16. It's not as far off as it may appear. Converting these ratios to their common denominator, we get

 5/3  = 80/48
27/16 = 81/48

If we want to modulate to the key of G through its dominant seventh (the D7 chord), we'll need to drop the A. The chord (D F# C) resolving to (D G B) actually works fairly well. Again, these are decisions that could be changed to lead to alternative scales.

We'll solve the problem of the ratio for D the same way. It's also marked with '*' to show it's not available in that key.

Returning to our task, defining the accidentals, we find that we now have found the ratio for F#: 45/32! It looks kind of bizarre, but it's consistent. Doesn't sound too bad either. We've gone pretty far away from our starting point of C. We went up a fifth to the key of G, then up a third to get B - or the seventh in the key of C. Then we found what the seventh in the key of G would be. That's up another fifth. Finally, we need to lower the note by an octave to fit in our range. Numerically, here's all the transformations:

 up      up      up      down            
fifth   third   fifth   octave   combined
  3       5       3       1         45   
  -   x   -   x   -   x   -    =    --   
  2       4       2       2         32   

Now what about that Bb? It might be useful for modulating to the key of F. We could modulate to the key of F through its dominant seventh (the C7 chord). That says we want the Bb to be harmonic with (C E G). The purest ratio for this purpose makes Bb be 7/4.

But Bb is also needed to make the subdominant in the key of F (the Bb chord). This says we want Bb to be 16/9. Here's another point where you can take your choice. I'll pick 16/9 - the rationale being that it is analogous to the ratio for F in the G7 chord for the key of C.

The remaining notes are pretty far from C: C# Eb and Ab. Their proper values depend a lot how how you plan to modulate keys, and the interaction of the melody with the chords. Let's arbitrarily choose their ratios to allow creating the Cm and Fm chords. We can consult the ratios in the Am chord to do this. (1/6 1/5 1/4).

(A C E) = ( 5/3  2/1 10/4)
        = (10/6 10/5 10/4)
        = ( 6/6  6/5  6/4)

Finally we copy all of these new ratios back into the scales for the key of G and F. The ones that conflict with existing ratios are marked with '*'. And from our analysis of the key of F, we finally find a ratio for C# (and the key of G even matches it up with Ab). So here's our complete table:

The biggest trouble with this tuning is that in the key of G, you can't form a complete D chord. And in the key of F, you can't form a complete Bb chord. So you can't modulate away from C without constraints.

But there's enough flexibility available to get the modulation done from the key of C. Then the tuning can be dynamically switched to the new key with only three notes actually changing their tuning. If the chords and melody can be arranged to stay away from those changing notes long enough, the transition can be effectively hidden from the listener.

If you find that you need to use the notes that are being retuned across the modulation, a possible solution is to tweak the decisions used to build the tuning table.

MAPPING ONTO THE STANDARD 12-TONE SCALE

COMPOSITIONAL POSSIBILITIES

A nice way to visualize the harmonic relationships in this scale is to plot the cycle of fifths on a clock face, and draw lines joining each pair of notes that share a simple ratio. The dark lines are the stronger ratios: fifths, fourths and major thirds. The light lines are the weaker ratios: minor thirds and seconds.

MELODIC

CHORDS

This document is ©2003 by Bill Grundmann.

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