The Conundrum of the Bent Tuning Forks

[Originally posted as: https://groups.yahoo.com/neo/groups/TUNING/conversations/topics/105487 by dkeenanuqnetau (Dave Keenan), Dec 3, 2012. Archived here: https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_105487.html]

As a theorist, when people start talking about writing a "history" of the "paradigm" you've been working in, you know it's time to move on. :-) http://x31eq.com/paradigm.html

The work I did with Margo Schulter on metastable intervals, using noble numbers, is one direction beyond the obsession with small-whole-number frequency-ratios and their approximations. And when composer David Guillot turned up on the Xenharmonic Alliance II (XA2) facebook group, just over a month ago, with some interesting questions about chords and scales involving the golden ratio or phi ((sqrt(5)+1)/2, the least rational number), I leapt at the chance to try to answer his questions. This is a brief summary of some results of our email collaboration over the past month.

When David mentioned he was using sine waves for his experiments, I realised they had nothing to do with metastable intervals and everything to do with combination tones. https://books.google.com.au/books?id=eGcfn9ddRhcC&pg=PA277 I pointed out that deliberately applied electronic distortion allows even those of us with linear sound systems and linear ears, to experience the effects of combination tones. And David taught me that Ocarinas and Tuning Forks are acoustic instruments that come close to generating sine waves.

Kraig Grady has, in the past, often chastised us (quite rightly) for ignoring combination tones when computing the consonance of chords, or the utility of scales. But at the time, there was more than enough mathematical fun to be had in the dominant paradigm.

David Guillot presented me with various slowly beating triads of sine waves, to which I applied distortion by playing them through my daughter's guitar amp, to make the effect more obvious. His examples were of two kinds:

1. Triads that were 12-EDO approximations of small-whole-number extended ratios such as 8:9:17 or 8:19:27, in which case he gave the formulas he had worked out, that related these numbers, such as C=A+B and C-B=B-A where A:B:C is the extended ratio (with C>B>A), and

2. Triads using octave-reductions and octave-inversions of small powers of phi, from a phi-based scale by Heinz Bohlen that was designed with combination tones in mind.

I showed David that there was no requirement for either small whole numbers or phi-based ratios to produce these kinds of combination-tone chords, and that the _only_ things that mattered were those formulas he had worked out, as applied to the _frequencies_ of the notes, irrespective of whether they are whole numbers or fractions, rational or irrational. So if we express the chords in cents as 0-b-c, e.g. 0-200-1300, then the condition C=A+B becomes 2^(c/1200) = 2^0 + 2^(b/1200) and therefore c = lg2(1 + 2^(b/1200)) * 1200.

So for _any_ interval 0-b in cents, we can calculate the third pitch c in cents, that will give that specific kind (C=A+B) of combination tone triad (CT3).

The only thing special about phi (the golden ratio, approx 833 cents) here, is that it is the only interval that stacks with itself to make a CT3 (combination tone triad) of the kind C=A+B. Other kinds of CT3 have different self-stacking intervals. The C=2A+B kind has the octave as its only self-stacking interval. The C=A+2B kind has the silver ratio (sqrt(2)+1), approx 1526 cents) as its self-stacking interval, and the C=2B-A kind has only the degenerate case of the unison as self-stacking. But if you do not insist that the two atomic intervals must be equal, then there is nothing special about these values. I should mention that the silver ratio is not a noble number, and the golden ratio is the only noble number that appears in this context. I should also mention that CT chords are no respecters of octave equivalence or chord inversions. If you transpose one of the pitches in a CT chord by an octave, the resulting chord may or may not be a CT chord.

David then presented me with the conundrum of the tetrad 0-1100-1700-2200 (A3-G#4-D5-G5) for which he could find no obvious extended ratio, and no involvement of the golden or silver ratios. This tetrad, when made with near-sine-waves and gently distorted, has slow beating, but none of its subset triads do. i.e. take away any note and the beating stops.

I suggested to David that it might be fun to introduce this topic to the XA2 facebook group with the same tetrad conundrum he had presented to me. And I came up with two other examples, not in 12-EDO, which were hopefully even more difficult to match to any extended ratio of small whole numbers (or phi), because, in the past, some folks have assumed that the relative consonance of such chords derives from their proximity to whole number ratios, even when those whole numbers have gone way beyond 19, with few common factors, and for phi-related chords of this type, some folks have assumed it derives from the well known mathematical properties of phi.

David also composed a lovely little musical piece that made use of several of these CT4-beating chords whose triad subsets do not beat. You can hear the 3 tetrads, and David's "Music for bent tuning forks", in "The Conundrum of the Bent Tuning Forks" here: https://soundcloud.com/dmguillotine-1/the-conundrum-of-the-bent

Don't read any further if you want to try to solve it yourself.

As I predicted, Paul Erlich figured it out straight away, and the XA2 folk didn't even bother trying to match the chords (given in cents) to extended ratios. In fact they went in somewhat the opposite direction, with Mike Battaglia pointing out that "... you could probably replicate this effect even with JI if you tried hard enough". Smart guys.

The beating of these tetrads can be explained as the difference tone of the two low notes beating against the difference tone of the two high notes (and beating at the same rate between several other pairs of combination tones). i.e. It is because the chord is an approximation of A:B:C:D where B-A = D-C with no requirement that A, B, C or D are whole numbers, or rational, or noble, or any other condition. David challenged me to find others like this in 12-EDO.

In response to David's challenge, I have produced the following visualisation.

Any point on this graph, above the cyan diagonal, corresponds to a CT4 chord of the kind where D-C = B-A. So these chords form a two-dimensional continuum, unlike the discrete cases of JI chords. You can read off the cent values of the four pitches making up any such chord as follows. A is always 0 cents. Note that there are thin curves and fat curves. The thin curves are more or less parallel to each other and correspond to values of B, in 12-EDO steps. Follow the nearest such thin curve down and left to the vertical axis to read off the value of B. Follow the nearest vertical gridline down to the horizontal axis to read off the value of C and follow the nearest horizontal gridline left to the vertical axis to read off the value of D.

So B and D are read off the same axis (the vertical one). B is read via the sloping coloured thin curves, and D is read in the usual manner via the grey horizontal gridlines. Interpolate as required, for non 12-EDO values.

What are the fat curves for? Well part of the brief was that none of the triad subsets should exhibit beating, so the fat curves represent triad subsets that have CT3 beats of various kinds as indicated in the legend. The intention was to consider all CT3 (and hence CT2) possibilities involving combination tones up to the 3rd degree. That is, for two frequencies A and B, combination tones of the form

|nA + mB| where n and m are integers (positive, zero or negative) and |n|+|m| <= 3. These are by far the most audible, with typical forms of distortion, including those in our ears.

To find an example of such a tetrad in 12-EDO, look for a place where a thin coloured curve passes through, or nearly through, the intersection of a vertical and a horizontal gridline, but not near any fat coloured curve. Then read off the cent values as described above. An example is shown, along with the locations of the 3 tetrads from the "The Conundrum of the Bent Tuning Forks".

-- Dave Keenan