The Conventional Theory of Chords is Just a Special Case

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Recording our two quartertone kalimbas together has reinforced our long held suspicion that the conventional theory of chords is just a special case.  So according to conventional theory two notes are said to be consonant  when their fundamental frequencies are related by some simple integral ratio, and certainly this is true in the sense that two notes which are so related do sound good together.   However when we play our two “quartertone” kalimbas together,  though they often sound consonant,  in fact no two notes are related by a simple integral ratio.
This is so because though  we like to call them “quartertone” instruments,  for many reasons the in-between notes are in reality only roughly half-way between their neighbors.   Not to mention that even the “chromatically” tuned keys (every other key) are only approximately 1/2 note apart, in part  because of the limitations of Mitsuko’s ears (she’s the tuner in our family, and she’s damn good,  but still tuning is not her main occupation),  but also because since the keys were hammered out by unspecialized, ham fingered, unskilled me,  they are all physically very different (unlike machine made keys or keys cut from sheet metal, they vary in thickness, width, shape, taper…..) so each and every one produces a different stew of miscellaneous frequencies.   Sometimes it is even true that the fundamental note goes up between two keys when some of the other harmonics go in the opposite direction.   In this situation what is a poor tuner to do?  obviously Mitsuko has no choice but to take a deep breath and do the best she can.
Well when irregularity like this exists even among the keys of each instrument, clearly it’s highly unlikely that any note from one of them would have a fundamental frequency related by an integral ratio to some note from the second.
And yet when we play them together they sound sweet and often they even produce a quasi-chordal feeling.
So if this is true there must be ways for two notes to be in harmony (to sound harmonious?) even without having what is conventionally held to be the necessary mathematical relationship between their frequencies.
which means the conventional theory is just a special case.
q.e.d.
What we have come to believe is that the brain of the listener is somehow able to perceive harmony even when the mathematical relationship is not there.
To take an analogy from another sense,  conventional color theory insists that different colors are perceived  to be different because they have different amounts of the three primary colors (a.k.a. “frequencies”),  and that the retina can perceive this because it has three types of sensors which are differentially sensitive to these three different primaries.
Now it is certainly true that varying the amount of these primary colors does change the perceived resultant color, but here again the situation is similar to that which exists in the world of sound, in that there is strong evidence that the brain can do a quite good job of perceiving a full range of colors even in the absence of the primary colors which conventional theory insists should be there…..
So 50+  years ago I witnessed a demonstration by Edwin Land (the inventor of the polaroid camera),  involving two pictures of the same scene which he had taken through two different filters, one of which admitted only a narrow band of  reddish orange light and the other only a narrow band of yellowish orange light.  With my own eyes  I watched as he projected the two through the same two filters onto a single screen, got them properly superimposed,  and then voila! there was the original scene in full color.
Again this is something which I actually witnessed…..

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Playing our quartertone kalimbas ( Kalimba Family ) has reinforced our long held suspicion the conventional theory of chords is just a special case.

According to it two notes sound consonant only if their fundamental frequencies are related by the proper integral ratio.

But while it’s true such mathematically related notes do sound good together, it’s also true the sounds of our quartertone instruments blend together and sound sweet ( that is sound “consonant ) even though none of their “notes” are related by any such simple ratio.

Indeed there’s no way they could be so related because though we like to call them “quartertone” instruments, in reality the “quartertone” keys are only roughly half-way between their “chromatically tuned” neighbors.  And even those keys
( every other key ) are only approximately 1/2 note apart, since unlike most kalimba keys which are cut from uniformly machined sheet metal, ours are hammered out by hand from high carbon steel rods and vary in thickness, width, shape, and taper….. which means each and every one produces a different stew of miscellaneous frequencies.  ( Notation )
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Sometimes it’s even true the fundamental note goes up between two keys when some of the other harmonics go down.  In this situation what’s a poor tuner to do?  Obviously there’s no choice but to take a deep breath and do the best one can.
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Well when irregularity like this exists even among the keys of each instrument, clearly it’s highly unlikely any note from one would have a fundamental frequency related by an integral ratio to any note from the second.

Yet when played together they sound sweet and often even produce a quasi-chordal feeling.

So it appears two notes can sound consonant even without what is conventionally held to be the necessary mathematical relationship between their frequencies.

But doesn’t this mean the conventional theory is just a special case?

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The Brain Does the Work
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What we’ve come to believe is the brain of the listener is somehow able to perceive harmony even when the supposedly necessary mathematical relationships are not there.

To take an analogy from another sense, conventional color theory insists different colors are perceived to be different because they contain different amounts of three primary colors ( a.k.a. “frequencies” ), and the retina detects this because it has three types of sensors which are differentially sensitive to these three primaries.

And in an experimental situation varying the amount of the primaries does change the perceived color, but even as we’ve discovered in the world of consonant sounds, there’s strong evidence the brain can perceive a full range of colors even in the absence of the primary frequencies which conventional theory insists should be there…..

So 60 years ago Edwin Land ( the inventor of the Polaroid camera ) demonstrated if two pictures of the same scene taken through different filters, one of which admitted only a narrow band of reddish orange light and the other only a narrow band of yellowish orange light, were projected through the same two filters onto a single screen and properly superimposed, viewers perceived the original colors.
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To be sure it wasn’t brilliant technicolor, but the sky was blue, the leaves were green, and the dirt was brown, even though the only light projected on the screen was two different flavors of orange, even though the only light on the screen was from the “orange” range of the spectrum.
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Clearly in both the case of consonant tones and perceived colors, there’s much more involved than just a bunch of physical parameters.

…. or as we have ranted on our Notation page, these are both cases where the clear-thinking scientific worldview has forced its way into an area where it’s not really appropriate.

It’s enough to make one suspect the ancient Indians were closer to the truth to insist all this is just some sort of show, some playful exercise of power by forces beyond our ken who for reasons known only to themselves enjoy watching us dance through their dazzling displays of ever shifting Maya…….

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Posted by untravelledpath at 1:30 pm on October 18th, 2009. No comments... »