Thanks
Looking forward to trying those out.
Not sure what could be causing the discrepancy between 1.6KHz and 2KHz. Perhaps the deep null is just at a different position at 1.6KHz that you managed to miss in the specific angles you modelled?
No the deep null is at A100 for 2kHz.
I'll make the A100 deep null / cancellation axis for 1.6kHz later today and upload it as well.
This is really fascinating. Sorry to ask you to explain a bit more, just wanting to try to understood a bit better:
- Do the brains of people not exposed to predominantly Western tonalities during their development also quantize to perfect octaves and fifths?
- What do you mean by "tune" in the context of e.g. "tune that minor seventh as an augmented sixth" or "out of tune major third"? The latter to my simple brain would mean that a note or all the notes in the chord were not tuned to concert pitch, but I assume you mean something different here?
- In terms of the Turkish rast for example, what is the difference e.g. between the Turkish Cb and the B in standard Western tuning?
- What do you mean by "transparent" in the context of "Extended Pythagorean tuning is transparent"?
- And finally, what do you mean by "high quality" in the context of that last paragraph of yours I quoted? Are you talking about tuning already-recorded music here, or are you talking about how musical instruments should be tuned (or something else)?
Anyway I'm going to read up on extended Pythagorean tuning and just intonation so no doubt that will clear up some of these questions...
EDIT:
@Thomas savage maybe these last couple of posts could be moved to a new thread?
Ah nice that you find it fascinating
- I believe the evidence is very much in favor that we are all born with this way of quantizing the musical interval space. Though not everybody agrees with me here. Though you can imagine someone from a tribe never in contact with the western world being presented two things, an audio recording of a piece by Bach, and an audio recording of a letter by Bach in German. He will be able to understand musically the piece / its intervals etc, though some aspects of it are culturally and it doesn't say whether he'll like it or not etc. The audio recording of the German letter though will be complete gibberish to him, he'll have to learn the German language first (which isn't a blank slate either, a language. Here Chomsky wrote a few things about universal grammar. But the basic "grammar" of music is what makes the intervals, we do not have to learn it.)
- By "in tune" I mean the ratio of the interval. For instance in 12-tone equal temperament the octave is tuned as 2/1 ratio, 1200 cents. for instance 100Hz an octave above is 200Hz. And an octave above 200Hz is 400Hz. In 12-tone equal temperament the perfect fifth is tune as 700 cents. This is about 2 cents short of 3/2 which is about 702 cents. As you can see all the intervals come from a chain of perfect fifths and octaves, here is the western major scale in fifths: F-C-G-D-A-E-B and in 12-tone equal temperament every fifth has a 2 cent error so they add up. The augmented fourth F-B is about 12 cents too low in equal temperament. A major third is about 8 cents too low in equal temperament. (though as a twist of nature the correct tuning for the major third of about 408 cents can in many circumstances appear to sound harsher than an equal tempered major third of 400 cents. This is due to the fifth harmonic / overtone of about 386 cents. And this is one of the main reasons why people have been led to a wrong path regarding tuning in the past and many still today.
- The difference between Cb and B is in relation to the G.
G - Cb is the interval of a diminished fourth. G - B is the interval of a major third, A - Cb is the interval of a diminished third, A - B is the interval of a major second, Cb - C is the interval of an augmented prime, B - C is the interval of a minor second. These are different intervals as far as our brain is concerned (even if we tune them exactly the same on an equal tempered instrument).
In monophonic music we can indicate the interval by tuning difference alone. This is what makes up the Turkish / Arab / Persian "quartertones", augmented seconds and diminished thirds. (often the difference is exaggerated a bit in tuning)
- With "transparant" I mean that the "haze" of floating intervals of equal temperament is removed (for fixed pitch instruments / electronic music etc). Take for instance the equal tempered perfect fifth at 700 cents and the Pythagorean perfect fifth of 702 cents. The third harmonic overtone is 3/1 and is 1902 cents which is 1200 cents + 702 cents. If the perfect fifth for instance C - G is tuned to 700 cents there is a 2 cents discrpancy between the third harmonic of C and the second harmonic of G which will slowly go in and out of phase. This produces a slightly hazy soundquality to the music. It holds for other intervals as well, and not even just because of the harmonics but that's another story don't want to make it too long.
- With high quality I mean that if you're using a subpar sampled piano for instance with correct Pythagorean tuning then any timbre deviation and any tuning deviation will be heard much more clearly which will often lead to errors in the sampled set jumping forward which were not so noticeable when playing that sampled piano in 12-tone equal temperament.
- As for reading up on this. I can strongly advice against it. The internet is a mess regarding this subject. What I'm sharing above is in part because of original research I've done for the past 12-years. Most of what's on the internet is simply completely wrong and there's no easy way to tell truth apart from error for you I'm afraid. Getting into microtuning leads most people completely astray and their music suffers a very great deal, both because they spend too much time on weird mathematical constructions behind tuning and mostly because the result will almost always sound very much out of tune.
Only thing I can recommend is to experiment with very simple western music which is well understood and tune it Pythagorean according to the correct written enharmonics. You can only do this if you've studied music theory and know how to spell correctly. And even this is somewhat bothersome as most instruments inc VST's etc are not made to be easily retuned.
And anything going beyond this I can only strongly strongly advice against as you'll probably end up making things more out of tune that 12-tone equal temperament
(edit: not that I question your individual musical hearing or anything like that. But it's a snake pit for everybody also the very best. Even many scientists who are famous in other areas, like Helmholtz, Mersenne, Kepler, etc etc wrote completely wrong things regarding tuning. They fell into the traps and never got out. And you can't even read about this because nobody knows what's right. This is a very hard part of the brain to study, it is not like sound and the auditory cortex etc but something else alltogether.)
edit: to make it extra clear. Only worth tuning very well understood "common practice" music to Pythagorean based on standard common practice harmonic theory and spelling.
Please don't attempt to do this with jazz/blues etc or late romantic period music etc. Especially jazz/blues is basically a mess in theory and spelling. With 12-tone equal temperament this doesn't matter much as when you spell a note wrong (for instance a G# instead of an Ab) it doesn't matter for the tuning. But for Pythagorean tuning this difference is audible. And when our brain interprets a note as a Ab and you tune it as a G# it is out of tune, and can be quite audible. Not very nice for your audience.