Yahoo Tuning Groups Ultimate Backup tuning The Conundrum of the Bent Tuning Forks

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. http://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: http://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.
http://dkeenan.com/Music/nonCT3_CT4s.gif

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

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:
DK> ... 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 ... 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 ... 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.

Very interesting post, Dave! The "valleys" that appear in the contour plot of 3HE also seem to arise from triads A:B:C that satisfy relations pA+qB+rC=0 where [p,q,r] are small integers; I wrote something about this to Dustin Schallert.

DK> 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 ... 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 ...

I saw that on XA2 but wasn't smart enough to do anything with it. However, I've just realised that my 4HE code should be able to find the extended ratios (it returns the most likely seed point as well as the 4HE for any specified tetrad).

0-1100-1700-2200 => 9:17:24:32
0-300-700-900 => 16:19:24:27
0-1812-2622-3212 => 7:20:32:45
0-1002-1444-1951 => 23:41:53:71

These all satisfy the rule you gave ...

Steve M.

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:
>

>
> 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.
*********************
An important point above different and combination tones exist on a continuum. It is relationship though that if one continues along the chain it will converge on an irrational and this is the only place that phi becomes useful possibly. This though admittedly is saying a recurrent series will converge on an irrational which is trivially true as something we know but is useful as an attactor. In my own experience working with whole numbers, it is the places before it converges that appear to be the most musically interesting. These often involve very high numbers. In the case of a tuning for Satie's "Vexation went into the millions. At this point one can say it is a territory where it is hard to define exactly what one is working with except as the method one used to get there. :)
these pre-convergent scales or chords can produce beats with spacing the chords so that they clash with the harmonics of lower tones also.. example spacing the 27-40-67 triad as 27- 80- 134 where the third and fifth harmonics clash with the higher tones ( this occurs in and out, yet predominantly over 55 minutes in my Beyond the Windows, Perhaps among the Podcorn)
>
> 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).

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.
****************
My experience here is that a tuning generated by recurrent sequences seems to work well and consistently despite different inversions in the music within itself. One can use the tuning and never touch upon the the CT chord. Sometimes with a large cluster of notes in such a tuning, the resultant beating does not appear in the numbers directly. Albeit i have not spent more than a few minutes with this situation.
>
> 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.
*****************
Such methods as whole numbers or phi are merely a way to get into the territory we might say and not the whole picture~
>
> 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.
*****************
if we take his example of
D = C + B - A and slightly detune it to cause beats, it is something i have noticed when i went through Kunst's Slendro measurements that they often subtracted 2 beats from making a perfect recurrence sequence (meta-slendro for example). This is interesting in that one could say have a sequence where A was quite a ways down a recurrent sequence can be used to generate a whole scale and not just a chord. the beating could be in tune with the actual sequence and beyond just being an unrelated artifact. It also points out that no math is needed at all once one can hear this as somehow the Indonesians seems to have done. numbers serving just as a tool for us and a way of reflecting on what it is others do.

Didn't Dudon use subtraction in his series also ?

> 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).
************
fun play with 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 visualization.
> http://dkeenan.com/Music/nonCT3_CT4s.gif
**************************
I have often heard these effects done by jazz keyboard players especially ( from Ellington to Sun Ra!) so this chart is quite useful .
>
> -- Dave Keenan
>

🔗Brofessor <kraiggrady@...>

🔗dkeenanuqnetau <d.keenan@...>

Thanks Steve,

I'm not sure if you've missed my point, or are helping to make it. :-) Certainly the fact that your software is forced to use such large whole numbers as 23:41:53:71 to get any kind of approximation to the 0-1002-1444-1951 (cents) chord, means I have done my job.

I designed that chord by starting with the saturated noble triad 0-1001.6-1424.1 (rounded to nearest 0.1 cents). Since all its intervals are noble, this triad is as far as possible from _all_ nearby ratios of whole numbers. Its lower interval is the limit of the fibbonacci-like sequence of mediants
4:7, 5:9, 9:16, 14:25, 23:41, 37:66 ...
Its upper interval the limit of
7:9, 11:14, 18:23, 29:37, 47:60 ...
Its outer interval the limit of
4:9, 7:16, 11:25, 18:41, 29:66 ...

I then calculated, in cents, the pitch of the 4th tone required to give frequencies D-C = B-A. This implies D = C+B-A and so in cents we have
d = lg2(2^(c/1200) + 2^(b/1200) - 1) * 1200
= 1936.2 cents

This did not result in a saturated noble tetrad (as far as I can tell it never does) and indeed the new outer interval was far too close to the ratio 1:3 (1902 cents), so I adjusted the top two notes to (a) move the outer interval further away from 1:3 without making any other intervals too close to any other simple ratios, and (b) to introduce a slow beat between the B-A and D-C difference tones.

So I expect that any extended ratio of whole numbers that your software finds for this, is only one of a number of possibilities that are all equally bad approximations. i.e. If you change the parameter in your software that determines the tradeoff between accuracy and complexity (size of numbers), you will get a different "best fit". Or if you change the software to also give a "second-best fit" then I think you will find that this is almost equally as bad an approximation as the best fit, or at least that accuracy only improves at the expense of complexity.

And yes, it is almost inevitable that the coefficients of any such extended ratio approximation will obey D-C = B-A. But the frequencies of the actual chord only _approximately_ obey it, because we like the sound of the approximately 2 Hz beat.

Harmonic entropy takes no account of combination tones, except, by accident, those of combination tone chords based on small whole numbers. I think it would be quite a stretch to suggest that the "harmonic template" hypothesised to exist in the human brain by the proponents of Harmonic Entropy, extends to the 71st harmonic, given that natural sounds having detectable levels of such high-numbered harmonics are essentailly non-existent. And even when a 71st partial does exist, it is as likely to be 71.3 times the fundamental due to the inharmonicity of some decaying tones.

I want to emphasise that there is nothing special about noble numbers with CT chords either. I merely used noble (and near-noble) numbers to prove that whole number ratios are not required, just as many have used whole-number ratio versions of these chords which shows that noble numbers are not required, and we have the non-whole-number non-noble version 0-1100-1700-2200 to show that neither noble numbers nor whole-numbers are required. And that many of these "paradoxical" tetrads are lurking unsuspected-by-most in 12-EDO, and many other tunings.

The point is that the entire explanation of the unusual beating properties of these chords is contained within the formula D-C ~= B-A where A, B, C and D are the actual frequencies in hertz (possibly fractional). _This_ is what "Paul Erlich figured out straight away", not any extended ratio. I see that I was a little ambiguous about that in the article.

Equivalently, the entire explanation is contained in the formula

d ~= lg2(2^(c/1200) + 2^(b/1200) - 2^(a/1200)) * 1200

where a, b, c and d are the pitches in cents.

🔗dkeenanuqnetau <d.keenan@...>

🔗martinsj013 <martinsj@...>

🔗kraiggrady <kraiggrady@...>

🔗martinsj013 <martinsj@...>

🔗dkeenanuqnetau <d.keenan@...>

I have updated my Combination Tone Tetrad (CT4) chart
http://dkeenan.com/Music/nonCT3_CT4s.gif
and created a second chart that zooms in on the higher octaves of the first chart where the lines were getting rather crowded.
http://dkeenan.com/Music/nonCT3_CT4z.gif

These charts are not only useful for finding CT4s of this specific type (D-C=B-A or D=C+B-A) but they can also be used to find combination tone _triads_ (CT3s) of the four major types (C=B+A, C=B+2A, C=2B-A, C=2B+A). Except that the easiest way to find them on these charts is to ignore B, and so they appear as D=C+A, D=C+2A, D=2C-A, D=2C+A.

While CT4s constitute 2D continua (every point on the chart above the diagonal), CT3s corresponds to 1D continua (the fat curves labelled D=C+A, D=C+2A, D=2C-A, D=2C+A). So while you might want to avoid these (and the other fat curves) when finding non-CT3 CT4s, you will use points _on_ them when finding CT3s.

I note that in the zoomed chart you can no longer use the D axis to look up the value of B, so I have labelled every fifth curve with its B value.

I remind you that these chords exist in all kinds of tunings. They do _not_ require the tuning to approximate small whole number ratios. I suspect that many 12-EDO chords that have been given "17-limit" or "19-limit" "explanations" in the past, may actually be examples of combination tone chords, that owe nothing to such high harmonics (which rarely exist at a significant level in any case).

However, I note that combination-tone beating effects are very subtle (at least they are for me) unless either (a) the timbre has few partials or (b) distortion is applied to the chord, or both.

But electric guitar output is routinely distorted, so it is a wonder that more research hasn't been done before now, into predicting the effects of distortion on particular chords, or finding chords that give a particular effect when distorted. That's one possible application of these charts. I mentioned that they can be applied to tunings other than 12-EDO, but when doing so, it would be more convenient if the horizontal and vertical gridlines (and the family of B-value curves) corresponded the notes of the tuning.

One aspect of this that I have avoided so far, for simplicity's sake, although it was mentioned by Paul Erlich in the XA2 "Conundrum of the Bent Tuning Forks" thread, is that distortion comes in different kinds that can accentuate certain combination tones and not others. You might have seen this in the book by Fastl & Zwicker that I linked to earlier.
http://books.google.com.au/books?id=eGcfn9ddRhcC&pg=PA277

Quadratic distortion versus cubic distortion. Also asymmetric versus symmetric or bipolar versus unipolar. Crudely speaking these pairs respectively emphasise combination tones |nA + mB| (n, m integers) where |n|+|m| is even versus odd. Where |n| is the absolute value of n.

Regards,
-- Dave Keenan