4 ms·

> 'God's prank on musicians' (where He cruelly rigged the structure of reality such that (3/2)^n!=2^m for any integers n, m but IT'S SO CLOSE CMON MAN) What ex

> 'God's prank on musicians' (where He cruelly rigged the structure of reality such that (3/2)^n!=2^m for any integers n, m but IT'S SO CLOSE CMON MAN) What exactly is this about? Other than (3/2)^12 being 129.7 which is somewhat close to 2^7 I don't see much other being close. Also, musician's mathematics are so weird, talking about '11' while there are 12 notes, '5th' while no 5 is involved, ...

vaylian 2y ago
There seems to be something missing from the problem statement. If I can choose any integers n, m, then I will just choose n=m=0 which gives a trivial equation.
- 8note 2y ago
  That says the issue though. The only time you can tune a piano is when the frequency is 0 - that is, that there's no sound being played
  - recursive 2y ago
    Doesn't mean that. 2^0 is non-zero. I interpret it to mean that every note must have the same non-zero frequency. In other words, the ratio between any two notes is 2^0 = 1.
331c8c71 2y ago
3:2 is the perfect fifth. 2:1 is the octave. Hence if you climb (or descend) by perfect fifths you never align with the notes you get if you climb (descend) by octaves. Look up "just intonation" and "equal temperament" if you are interested. Also "The Well-Tempered Clavier" is highly recommended for listening!
- heikkilevanto 2y ago
  Just a side note: "The Well-Tempered Clavier" is not at all the same as the "Equally Tuned Piano". Bach was certainly aware of the equal tuning, where all intervals were the same. It had been used on lutes for some centuries. But with harpsichords, and to some extent pipe organs, an equal temperament was not what Bach or his contemporaries wanted - it didn't have a single pure interval!
giraffe_lady 2y ago
They are more like ordinals than "math" numbers the way you're thinking. It's the fifth scale degree, not 5 of something. The 11/12 contradiction, or seven notes in an octave too, is just because it's essentially 1-indexed. Interestingly indian classical music is distantly related through the shared proto-indo-european cultural history but counts it differently. They call the octave the saptak, etymologically related to the "sept" root for seven in romance languages.
zahlman 2y ago
The author is mistaken in the details. 12 5ths are stacked in order to get back to the start note (7 octaves higher). They're called "5ths" because they're the fifth scale degree, i.e. 4 diatonic scale (https://en.wikipedia.org/wiki/Diatonic_scale https://en.wikipedia.org/wiki/Diatonic_scale) notes above the start point (because they start counting at 1), which is an interval of 3 tones and a semitone. With the notes "equally spaced" (according to the logarithm of frequency, because we perceive intervals according to pitch ratio), the semitone becomes equal to exactly half the tone (despite the name, that isn't guaranteed for other tuning systems!) - thus, the interval that we play on an "equal tempered" piano as a "perfect fifth" is 700 "cents" i.e. a frequency ratio of 2^(7/12). Meanwhile, per music theory going all the way back to the ancient Greeks, this interval ought to be exactly 3/2, which is slightly larger. Contra what was written in the article, these intervals are thus tuned flat from that Pythagorean ideal. The difference is about 2 cents (where a cent is defined as a frequency ratio of 2^(1/1200), effectively). Most people wouldn't be able to hear the difference (https://en.wikipedia.org/wiki/Just-noticeable_difference#Music_production_applications https://en.wikipedia.org/wiki/Just-noticeable_difference#Mus...). Which is to say: yes, 129.746... might not seem that close to 128, but 1.498... is close enough to 1.5 that it lets us have music that sounds right (if sterile, according to some) in every key. Other intervals possible in this tuning arrangement (i.e, powers of 2^(1/12)) can approximate other reasonably-consonant intervals with varying accuracy.
dehrmann 2y ago
Prank or gift? The fact that we get an approximation to 3/2 in one of the best bases ever (12) is sort of a miracle.

vaylian 2y ago

There seems to be something missing from the problem statement. If I can choose any integers n, m, then I will just choose n=m=0 which gives a trivial equation.

8note 2y ago
That says the issue though. The only time you can tune a piano is when the frequency is 0 - that is, that there's no sound being played
- recursive 2y ago
  Doesn't mean that. 2^0 is non-zero. I interpret it to mean that every note must have the same non-zero frequency. In other words, the ratio between any two notes is 2^0 = 1.

331c8c71 2y ago

3:2 is the perfect fifth. 2:1 is the octave. Hence if you climb (or descend) by perfect fifths you never align with the notes you get if you climb (descend) by octaves. Look up "just intonation" and "equal temperament" if you are interested. Also "The Well-Tempered Clavier" is highly recommended for listening!

heikkilevanto 2y ago
Just a side note: "The Well-Tempered Clavier" is not at all the same as the "Equally Tuned Piano". Bach was certainly aware of the equal tuning, where all intervals were the same. It had been used on lutes for some centuries. But with harpsichords, and to some extent pipe organs, an equal temperament was not what Bach or his contemporaries wanted - it didn't have a single pure interval!

giraffe_lady 2y ago

They are more like ordinals than "math" numbers the way you're thinking. It's the fifth scale degree, not 5 of something. The 11/12 contradiction, or seven notes in an octave too, is just because it's essentially 1-indexed. Interestingly indian classical music is distantly related through the shared proto-indo-european cultural history but counts it differently. They call the octave the saptak, etymologically related to the "sept" root for seven in romance languages.

zahlman 2y ago

The author is mistaken in the details. 12 5ths are stacked in order to get back to the start note (7 octaves higher). They're called "5ths" because they're the fifth scale degree, i.e. 4 diatonic scale (https://en.wikipedia.org/wiki/Diatonic_scale https://en.wikipedia.org/wiki/Diatonic_scale) notes above the start point (because they start counting at 1), which is an interval of 3 tones and a semitone. With the notes "equally spaced" (according to the logarithm of frequency, because we perceive intervals according to pitch ratio), the semitone becomes equal to exactly half the tone (despite the name, that isn't guaranteed for other tuning systems!) - thus, the interval that we play on an "equal tempered" piano as a "perfect fifth" is 700 "cents" i.e. a frequency ratio of 2^(7/12). Meanwhile, per music theory going all the way back to the ancient Greeks, this interval ought to be exactly 3/2, which is slightly larger. Contra what was written in the article, these intervals are thus tuned flat from that Pythagorean ideal. The difference is about 2 cents (where a cent is defined as a frequency ratio of 2^(1/1200), effectively). Most people wouldn't be able to hear the difference (https://en.wikipedia.org/wiki/Just-noticeable_difference#Music_production_applications https://en.wikipedia.org/wiki/Just-noticeable_difference#Mus...). Which is to say: yes, 129.746... might not seem that close to 128, but 1.498... is close enough to 1.5 that it lets us have music that sounds right (if sterile, according to some) in every key. Other intervals possible in this tuning arrangement (i.e, powers of 2^(1/12)) can approximate other reasonably-consonant intervals with varying accuracy.

dehrmann 2y ago

Prank or gift? The fact that we get an approximation to 3/2 in one of the best bases ever (12) is sort of a miracle.