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Equal temperament vs other tunings


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#31 kenm

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Posted 25 March 2019 - 17:44

Lots of oddities in this, but I've just lost my reply for the second time. In brief:

Their Fig 1 has sets of 2, 3, 4, 5, 6, and 7 notes.not just 5 and 7.

Their Fig. 2 shows the spectrum that nothing acoustic has ever produced (a computer could generate it).

They don't consider freely vibrating sound sources (bells, drums etc.)

They confuse harmonics with partials.

I guess they don't have a lot of experience of physics.


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#32 kenm

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Posted 25 March 2019 - 17:50

kenm your suggestion of analysing chords in the WTC and looking for any that are avoided, sounds like a project my youngest would really enjoy - I might pass the idea on to him!

I guess he programs computers  If he wants elaboration point him at me.

 

Sorry about my unattached last: it was a response to Elemimele's last


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#33 elemimele

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Posted 26 March 2019 - 13:17

In fairness to them, of course one can argue that fig 1 contains sets of 2 notes in that a pentatonic scale contains sets of 2 notes, but all their pentatonic scales have 5 notes marked with dots. The dots aren't clear on the black background of the black notes. They could have made the figure more obvious.

I think fig 2 is intended to illustrate the exact frequences of harmonics for a neurophysiological reader, rather than an illustration of a real note.

I'll forgive them for ignoring drums, as drums can contain any partial whatsoever. Bells are a trickier issue. Part of it goes over to psychology again. Yes, a bell can contain partials that are not true harmonics, because they have non-integer divisors of the fundamental frequency. The critical point is that anyone can hear these partials as different notes, and sing them separately to the main note of the bell. On the other hand, a mix of harmonics will blend, and the listener hears a single note. This is, of course, what happens to a plucked "perfect" string, freely vibrating, or to a driven vibration in an air column (approximately!). Harmonics aren't just physically different to partials (integer divisors), they are also psychologically different (the human brain merges them and assumes they came from one source). This goes to the heart of the study, which depends on two questions: is this harmonic-merging something that we get from Western culture, and, extending this: is it fundamental to the way we, whatever our culture, create scales?

Meanwhile on Bach, and favoured intervals/unfavoured. It would be a fascinating thing to do, and quite tricksy to interpret. The missing half of the equation would be why Bach used, or didn't use, particular chords. We have several unknowns: his tuning, his attitude to which intervals are most important, and his intention when deploying a particular chord (i.e. sometimes he might deliberately use a badly-tuned, jarring chord, for its psychological effect, just as a chef will use sharp, sour ingredients to create a balanced meal). I think one could learn a lot by looking at your proposed study, and then going back to see the context where the little-used chords crop up.


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#34 kenm

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Posted 28 March 2019 - 10:23

In fairness to them, of course one can argue that fig 1 contains sets of 2 notes in that a pentatonic scale contains sets of 2 notes, but all their pentatonic scales have 5 notes marked with dots. The dots aren't clear on the black background of the black notes. They could have made the figure more obvious.

I think fig 2 is intended to illustrate the exact frequences of harmonics for a neurophysiological reader, rather than an illustration of a real note.

I'll forgive them for ignoring drums, as drums can contain any partial whatsoever. Bells are a trickier issue. Part of it goes over to psychology again. Yes, a bell can contain partials that are not true harmonics, because they have non-integer divisors of the fundamental frequency. The critical point is that anyone can hear these partials as different notes, and sing them separately to the main note of the bell. On the other hand, a mix of harmonics will blend, and the listener hears a single note. This is, of course, what happens to a plucked "perfect" string, freely vibrating, or to a driven vibration in an air column (approximately!). Harmonics aren't just physically different to partials (integer divisors), they are also psychologically different (the human brain merges them and assumes they came from one source). This goes to the heart of the study, which depends on two questions: is this harmonic-merging something that we get from Western culture, and, extending this: is it fundamental to the way we, whatever our culture, create scales?

Meanwhile on Bach, and favoured intervals/unfavoured. It would be a fascinating thing to do, and quite tricksy to interpret. The missing half of the equation would be why Bach used, or didn't use, particular chords. We have several unknowns: his tuning, his attitude to which intervals are most important, and his intention when deploying a particular chord (i.e. sometimes he might deliberately use a badly-tuned, jarring chord, for its psychological effect, just as a chef will use sharp, sour ingredients to create a balanced meal). I think one could learn a lot by looking at your proposed study, and then going back to see the context where the little-used chords crop up.

I guess my counts show that I saw no dots on black keys.  Enlarging them to the maximum of which this software is capable makes it just possible to see the red on the black.

Your nomenclature is mixed, but I assume that when you refer to a mix of harmonics you mean a mix of partials that are at at frequencies that have a harmonic relationship, in which case your sentence is correct but misleading: a mix of partials that approximate to harmonic frequencies, such as those of a real string, struck or plucked, are also heard as a single note.  Real strings are manufactured to be as flexible as possible e.g. by winding a long wire onto a fine load-bearing one, but they still have some stiffness, the effect of which is to make the frequencies of upper partials sharp relative to the corresponding harmonic frequency based on that of the lowest partial. A good piano tuner will set up a chromatic octave in the middle of the range, and then tune each pitch class (all the notes with the same name) by eliminating beats between the notes of an octave dyad. This makes the frequency of the first partial of the upper not equal to that of the second partial of the lower one.  On a baby grand, an electronic tuner shows the procedure to amount to about 20 cents from the lowest to the highest notes on the instrument. This does not impair the perception of consonance.  The longer the strings, the less the effect.

We should bear in mind that harmonics are abstract components of a mathematical model.  Partials are also part of the model, but relate closely to the real world in measuring equipment, inside the cochlea and presumably in the vestibulocochlear nerve.  Pitch becomes real inside the brain, but I know of no evidence to support the concept of harmonic frequencies as a distinct class* there, since the sensation of pitch is induced by non-harmonic mixes also.

* Don't divide a continuum into categories. Lawyers and Parliamentary draughtsmen do that; scientists make measurements.

Some drums (e.g. timpani, tom-toms, bongos) produce a recognisable pitch.

I agree with your comments on the Bach's use of discordances.  The interpretation of the measurements needs to be considered as part of the design of the experiment, but I don't think your add-on changes it, since it will presumably be looking at actual examples of the dyads (simultaneous or melodically adjacent pairs of absolute pitches) that are least used in the complete 48.  There is a great deal of discussion of temperaments in both the historical record and present-day research, so the measurements need to be put into each candidate tuning and some measure of total discordancy calculated.  I don't expect it to settle the question, merely to provide some more evidence for an argument that will continue.


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#35 Gordon Shumway

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Posted 11 April 2019 - 14:45

Something I could whinge about is so-called Pythagoras, because "he" is so misunderstood.

In fact we don't know much about him. It is doubtful if we have anything of what he ever said. The most likely candidate for anything genuine is a small quote about "not taking outside the circle what belongs in the circle" (i.e. don't blab secrets) in Plato's Laches, but I can't remember the details. Not long ago I was watching a telly prog in which Alice Roberts confidently talked of Pythagoras inventing music theory, but you should notice that they never get past discussing the perfect fifth when asserting that.

 

No, there were "Pythagoreans", self-appointed followers of Pythagoras. Partly they were scientific and partly they were a vegetarian religious sect. They didn't eat anything with souls. They didn't eat beans, because they thought the flatulence was escaping souls.

 

The JI major third, known by Ptolemy, had a ratio of 5/4, but Pythagoreans only worked with the ratios 3/2 and 2. You can manipulate 3/2 and 2 to get a chromatic octave, but that process was only finalised in the 15th century or thereabouts. You can't manipulate 3/2 and 2 to get 5/4, so the early Pythagoreans got as far as inventing the major third of 81/64. I would say it was so sharp that no-one ever used it for practical purposes, but I'm probably wrong - there was the problem of building church organs, and Christians were fond of Pythagoreans, and I think it's common knowledge that organs could only play in a few keys because of how out of tune they were.

 

Sorry to moan, but Alice Roberts gets my goat!

 

If you read Ancient Greek Music by Martin West, you'll find some oddities. The ancient Greek scale was constructed from, perhaps, approximately the notes EFA and a fourth note. Where did that fourth note go, you ask - it can only be G, surely? Well, no, there were three variants, one had G, one had F# and one placed it between E and F! Tell us how that fits Pythagoras, Alice! It's a good example of how music was first invented for the ear, and then the theory came later.


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