Alternative Tuning and How We Hear
Alternative Tuning and How We Hear
The following chart shows the scale. Each note's frequency is obtained by multiplying the fraction times the fundamental frequency. The first number of each fraction is the number of the harmonic. The second number is chosen to shift the frequency down into a single octave.
The chart ends at 22 simply because 23 can't be easily mapped into the 12-tone scale. This is purely arbitrary. Perhaps A could be used for this purpose, but it would be out of sequence. Or G and Ab could be shifted up to make room, but it seems more natural for G to stay put due to its relation as a fifth with C.
One could argue that stopping at 16 might be best, but as we'll see, having some extra notes might be valuable for composition.
The octave folding is shown as numbered Cycles to retain the information about where in the harmonic sequence each note originated.
| approximate 12-tone note | First Cycle | Second Cycle | Third Cycle | Fourth Cycle | Fifth Cycle | frequency (Hz) | pitch (cents) |
| B | - | - | - | 15/8 | - | 490.548 | 1088 |
| A# Bb | - | - | 7/4 | 14/8 | - | 457.845 | 0969 |
| A | - | - | - | - | - | - | - |
| G# Ab | - | - | - | 13/8 | - | 425.142 | 0841 |
| G | - | 3/2 | 6/4 | 12/8 | - | 392.438 | 0702 |
| F# Gb | - | - | - | 11/8 | 22/16 | 359.735 | 0551 |
| F | - | - | - | - | 21/16 | 343.384 | 0471 |
| E | - | - | 5/4 | 10/8 | 20/16 | 327.032 | 0386 |
| D# Eb | - | - | - | - | 19/16 | 310.680 | 0298 |
| D | - | - | - | 9/8 | 18/16 | 294.329 | 0204 |
| C# Db | - | - | - | - | 17/16 | 277.977 | 0105 |
| C | 1/1 | 2/2 | 4/4 | 8/8 | 16/16 | 261.626 | 0000 |
It may be valuable to use a modal version of this scale with the 3 used as the tonic.
Compositions tend to gravitate towards a drone note as the basic chord structure. On the other hand, there is quite a lot of potential for melodic complexity.
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.
As you can see, the plot is pretty sparse (compared to a typical just tuned scale).
Notes have several alternate meanings in the scale.
For instance, 1,2,4,8, and 16 are all the same pitch because of octave folding, yet they can function differently in a melody.
For example the sequence 6-7-8 might be used initially and later on it might return as 12-14-16.
The 7 might be treated as a passing tone, where as the 14 might be part of a chord structure.
I think the notion of octave interchangeability can be challenged by this scale, since the function of a note depends on which Cycle it belongs to.
MELODIC
A simple starting point is to assume that a melodic phrase should step through the scale without skipping any notes.
This is not nearly as restrictive as it first seems.
The low notes are large intervals.
As we proceed up the scale, the intervals become smaller.
CHORDS
Chords based on the fundamental can be built by "stacking seconds."
This is somewhat analogous to "stacking thirds" in the 12-tone scale.
Because of the strong harmonic structure of the scale, seconds can be stacked very far - half way through the Fourth Cycle - if you acquire a taste for this scale.
It seems to be necessary to lead into the higher notes by providing a fairly low root note, otherwise the chord sounds rather dissonant.
Some of this may be due to lack of familiarity with this set of notes.
Or perhaps explicitly affirming the root helps the listener match the harmony template.
By redefining the Cycle the notes are considered to be in, the stacking of seconds can be redefined as stacking of thirds.
The 11th harmonic is particularly obnoxious at first, although if presented with the 7th and 9th, it is acceptable. There is no analogy in the 12-tone scale for this note, so for listeners steeped in Western tonality, it is incomprehensible. Perhaps it is initially perceived as the rather discordant tritone. I have found it becomes more acceptable with experience.
The basic chords are: the tonic (4,5,6), the dominant seventh (12,15,18,21). There is no subdominant. The tonic with a major 7th (8,10,12,15) or minor 7th (4,5,6,7) is available. The tonic with minor 7th is so smooth, that it demands use. And stacked thirds like (4,5,6,7,8,9,10,11,12) sound acceptable.
A mode based on the fifth (3) can be created. Here, the tonic is (12,15,18) and the subdominant is (4,5,6). The problem is that there is no dominant!
Inversions of chords are also useful. In particular, an inversion with the third in the base (5,8,12) sounds very smooth (unlike its 12-tone analog).
An excellent description of the various intervals available in this scale was written by David Canright.
This document is ©2003 by Bill Grundmann.
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