Discussion:
math(s) music question (cannot find answer)
Simon Roberts
2017-06-14 16:34:31 UTC
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Middle C = K 2^(0/12) Hz is C# = K 2^(1/12) Hz , and so on till octave is C = K 2^(12/12), Is this correct? What is K in Hz? I can find the answer to second question but not the first. Does it not work this way; fractions of 12 of a power of 2.

Are these twelves of a power of 2 found naturally, like a horn, drum skin, wind pipe, or dare I say some natural variable of strings (thickness or weight OR tension)
Peter Percival
2017-06-14 16:55:57 UTC
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Post by Simon Roberts
Middle C = K 2^(0/12) Hz is C# = K 2^(1/12) Hz , and so on till
octave is C = K 2^(12/12), Is this correct? What is K in Hz? I can
find the answer to second question but not the first. Does it not
work this way; fractions of 12 of a power of 2.
Are these twelves of a power of 2 found naturally, like a horn, drum
skin, wind pipe, or dare I say some natural variable of strings
(thickness or weight OR tension)
https://en.wikipedia.org/wiki/Equal_temperamenthttps://en.wikipedia.org/wiki/Equal_temperament
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
Jan
2017-06-14 17:36:46 UTC
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Post by Simon Roberts
Middle C = K 2^(0/12) Hz is C# = K 2^(1/12) Hz , and so on till octave is C = K 2^(12/12), Is this correct? What is K in Hz? I can find the answer to second question but not the first. Does it not work this way; fractions of 12 of a power of 2.
Are these twelves of a power of 2 found naturally, like a horn, drum skin, wind pipe, or dare I say some natural variable of strings (thickness or weight OR tension)
K can vary (esp. in Baroque music) but modern "concert pitch" is 440 Hz for
A above the middle C.

This is not really a "natural" scale, it's a "tempered" one. In this case
it's "equally tempered", i.e. the octave (doubling the frequency, or
multiplying by 2) is divided into 12 equal frequency multipliers, i.e.,
2^(1/12) each.

What's found naturally are even multiples of some fixed basic frequency you
start with, but the scale you end up with this way will "sound good" only in
specific few keys near C major. So people twiddled with different tunings to
allow as many keys as possible to be playable without the instrument appearing
out of tune. Bach used some of those systems and when his organ
or harpsichord pieces are played on instruments tuned that way, a really
interesting surprises pop up now and then (like a VERY strange-sounding
chord out of blue - definitely an intentional effect). OTOH a Beethoven
sonata can sound simply out of tune when played that way, not very usable.

At some point people just decided to divide the octave equally. This makes
all intervals except the octave slightly out of tune but not much. On most
pieces this is inaudible but on the organ it can be VERY obvious: it sounds
like "grit" or "dirt" on chords, what some people probably assume is the
normal organ sound.

Anyway.

--
Jan
bert
2017-06-14 17:40:16 UTC
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Post by Simon Roberts
Does it not work this way; fractions of 12 of a power of 2.
Not quite. The octave consists of five full tones (two with
frequency ratios of 10/9, and three with frequency ratio 9/8,
both of which are approximations to 2^(1/6)) and two semitones,
of frequency ratios 16/15, a rather poorer approximation to
2^(1/12). Multiply them all together to get the octave ratio 2.
The tones and semitones are interleaved so that non-adjacent
notes can have "nice" frequency ratios like 4/3, 5/3 etc.
--
SteveGG
2017-06-14 18:22:06 UTC
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Simply : 12 tone scale, where each subsequent tone frequency is the
current tone frequency multiplied by the 12th root of 2 or
approximately 1.059463 . Hence 12 tones are a doubling of frequency or
the common octave. Actually doesn't matter where you start.

Interesting to speculate why 12 seems to be the magic number in our
music, and why the common scale is 8 notes spaced 2, 2, 1, 2, 2, 2, 1
bert
2017-06-14 18:40:20 UTC
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Post by SteveGG
Interesting to speculate why 12 seems
to be the magic number in our music . . .
No need to speculate; it's because 2^19 is very
close to 3^12. Were it otherwise, none of it
would work out as neatly as it actually does.
--
Jim Burns
2017-06-14 19:32:59 UTC
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On Wednesday, 14 June 2017 20:22:11 UTC+2,
Post by SteveGG
Interesting to speculate why 12 seems
to be the magic number in our music . . .
No need to speculate; it's because 2^19 is very
close to 3^12. Were it otherwise, none of it
would work out as neatly as it actually does.
One way to find more magic numbers is to look at
the continued fraction series of
x = log(3)/log(2) = 1.58496250072...

This gives us a sequence of _best approximations_ p/q
of x, "best" meaning there is no rational number with a
smaller denominator which is closer than p/q to x.

So, best approximations of log(3)/log(2) are
1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53, ...

Twelve (approximately) equal ratios will bring us
from 1 to its octave 2, with a stop at 3/2
along the way, near to 2^(19/12).

We can do something similar to create an (approximately)
equal-tempered pentatonic scale, five (approximately)
equal ratios bringing us from 1 to 2 with a stop at 3/2,
near to 2^(8/5).

Or similarly, a 53-tone scale.

There are other choices for the magic number, not many,
but a few. Of course, what humans find appealing or
even usable restricts those few magic numbers further.
(And why must a scale have approximately equal
steps, anyway?)
SteveGG
2017-06-14 23:56:52 UTC
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Post by bert
No need to speculate; it's because 2^19 is very
close to 3^12.
You really should explain this in more detail. I don't see how 2^19 =
3^12 ( approx ) should lead to a 12 tone octave and the common
selection of the 8 scale notes. e.g. Why not a 9 tone "octive" based
on a frequency factor of 9th root of 2 = 1.08 ?
Simon Roberts
2017-06-15 12:13:47 UTC
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Post by Simon Roberts
Middle C = K 2^(0/12) Hz is C# = K 2^(1/12) Hz , and so on till octave is C = K 2^(12/12), Is this correct? What is K in Hz? I can find the answer to second question but not the first. Does it not work this way; fractions of 12 of a power of 2.
Are these twelves of a power of 2 found naturally, like a horn, drum skin, wind pipe, or dare I say some natural variable of strings (thickness or weight OR tension)
Harmonic Series (Horns)

https://en.wikipedia.org/wiki/Harmonic_series_(music)
Simon Roberts
2017-06-15 12:16:39 UTC
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Post by Simon Roberts
Post by Simon Roberts
Middle C = K 2^(0/12) Hz is C# = K 2^(1/12) Hz , and so on till octave is C = K 2^(12/12), Is this correct? What is K in Hz? I can find the answer to second question but not the first. Does it not work this way; fractions of 12 of a power of 2.
Are these twelves of a power of 2 found naturally, like a horn, drum skin, wind pipe, or dare I say some natural variable of strings (thickness or weight OR tension)
Harmonic Series (Horns)
https://en.wikipedia.org/wiki/Harmonic_series_(music)
just read. quite stupid.
Simon Roberts
2017-06-17 23:15:04 UTC