The easy answer to the second challenge might be to do a set from 0 to .999 repeating. Since it is a surprising fact that .999 repeating ends up equal to one, you'd cover the whole range as he challenges you to do; but it is also accepted that .999 etc. is 'less than one' for the purposes of the number line.
Although I suppose he could argue that doesn't satisfy since in some sense the value equals one in one way. But his solution around 8 minutes and 20 seconds talks about infinite series that converge on one in a similar way, so I think it's a plausible answer.
Having watched the video (fun!) here are a few professional comments about tuning:
Modern piano tuning is as described in the video, with "pure" Pythagorean intervals detuned in favor of evenness, which makes playing pieces in different keys more or less doable.
(They didn't do quite this back in Bach's day, which is why his keyboard music titled "The Well-Tempered Klavier" is, well, well-tempered. If you had transposed those selections and then played them in other keys on an instrument tuned to that temperament, they would have sounded noticeably different, because the intervals would not have been the same.)
We bowed-string players mostly tune to Pythagorean ratios (there are occasional strategic exceptions). My four cello strings are kept tuned in three perfect fifths, where the upper string of each pair has a frequency 1.5, or 3/2, that of the lower.
Pure intervals work better for flexible-pitched, sustaining instruments, I would suggest, while de-tuned intervals work better for fixed-pitch, percussive instruments such as pianos and fretted guitars.
But, since Pythagoras's comma is in play, that creates mismatches. In first position, an E on my D string that is tuned against the A string is about a millimeter away from the "same" E tuned against my C string.
So, we adjust.
But imagine the problem this creates for an entire orchestra? How does everyone know how to adjust in the same way?
The answer is a modified hybrid.
Set yourself a tonic note for the key you are in. That's scale degree 1. Next, add to it the Pythagorean pure-tuned notes one and two fifths above and below your tonic. That's four more notes, which gives you additional scale degrees 5 and 2 going up, and 4 and flat-7 going down.
Throw away the flat-7. That leaves us with 1,2,4, and 5.
Played as a chord, these four notes sound curiously dense and empty. IN C Major, they would be C, D, F, and G. However, they serve an admirable tuning purpose:
Since the farthest note (2) is no more than 2 iterations away from the tonic, the mismatch between Pythagorean purity and fixed=pitch adjustment is minimized. These four pure pitches form the anchor to orchestral tuning.
We simply tune those four precisely, and then adjust all other pitches to "fit" against those four.
Then, if we need to modulate to a new key, we can simply migrate the drone in our heads to the fresh key.
And that's the pitch adjustment culture that makes symphony orchestras possible.
5 comments:
The easy answer to the second challenge might be to do a set from 0 to .999 repeating. Since it is a surprising fact that .999 repeating ends up equal to one, you'd cover the whole range as he challenges you to do; but it is also accepted that .999 etc. is 'less than one' for the purposes of the number line.
Although I suppose he could argue that doesn't satisfy since in some sense the value equals one in one way. But his solution around 8 minutes and 20 seconds talks about infinite series that converge on one in a similar way, so I think it's a plausible answer.
Of course, finding an 'easy' solution misses his more interesting point about how rare harmonious intervals prove to be.
And of course, some intervals are _effectively_ not harmonious because it takes so long for a beat to form.
I will give it a whirl!
Piercello
Having watched the video (fun!) here are a few professional comments about tuning:
Modern piano tuning is as described in the video, with "pure" Pythagorean intervals detuned in favor of evenness, which makes playing pieces in different keys more or less doable.
(They didn't do quite this back in Bach's day, which is why his keyboard music titled "The Well-Tempered Klavier" is, well, well-tempered. If you had transposed those selections and then played them in other keys on an instrument tuned to that temperament, they would have sounded noticeably different, because the intervals would not have been the same.)
We bowed-string players mostly tune to Pythagorean ratios (there are occasional strategic exceptions). My four cello strings are kept tuned in three perfect fifths, where the upper string of each pair has a frequency 1.5, or 3/2, that of the lower.
Pure intervals work better for flexible-pitched, sustaining instruments, I would suggest, while de-tuned intervals work better for fixed-pitch, percussive instruments such as pianos and fretted guitars.
But, since Pythagoras's comma is in play, that creates mismatches. In first position, an E on my D string that is tuned against the A string is about a millimeter away from the "same" E tuned against my C string.
So, we adjust.
But imagine the problem this creates for an entire orchestra? How does everyone know how to adjust in the same way?
The answer is a modified hybrid.
Set yourself a tonic note for the key you are in. That's scale degree 1. Next, add to it the Pythagorean pure-tuned notes one and two fifths above and below your tonic. That's four more notes, which gives you additional scale degrees 5 and 2 going up, and 4 and flat-7 going down.
Throw away the flat-7. That leaves us with 1,2,4, and 5.
Played as a chord, these four notes sound curiously dense and empty. IN C Major, they would be C, D, F, and G. However, they serve an admirable tuning purpose:
Since the farthest note (2) is no more than 2 iterations away from the tonic, the mismatch between Pythagorean purity and fixed=pitch adjustment is minimized. These four pure pitches form the anchor to orchestral tuning.
We simply tune those four precisely, and then adjust all other pitches to "fit" against those four.
Then, if we need to modulate to a new key, we can simply migrate the drone in our heads to the fresh key.
And that's the pitch adjustment culture that makes symphony orchestras possible.
-Piercello
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