This week I learned a little more about how sounds and intervals work - I've always known that pitches were related and octaves especially so, but never realized it was so simple and so physical - an octave is specifically a 2:1 ratio in frequency. You double the frequency of a pitch to get the pitch an octave higher. A4 on a piano is 440Hz, and A5 is 880Hz. This is further divided into 12 semitones, corresponding to adjacent keys on a piano. Each of those is further divided into 100 cents. Given a base frequency in Hertz and a number of cents to increase along the logarithmic scale, you can calculate the final frequency quite easily:
For example, to add a single semitone from A4 up to B♭4, you increase by a twelfth of of what doubling it does: 440 * 2^(1/12). This works out to approximately 466.164, and that floating point imprecision is why intervals are defined as ratios calculated from a set standard instead of fixed numeric values. You don't get another whole number until you go up or down a full octave, to A3 - 220Hz or A5 - 880Hz.
This may be "common" knowledge, but it was new to me. Neat.
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This system tuning called twelve-tone equal temperament which was introduced around 18th century by Bach! Another fun fact: double bass tuning is opposite as violin which is E, A, D, G.
Totally related - the circle of fifths relates intervals to key signatures. A fifth is an interval with a 3:2 ratio, which is also 7 semitones. An octave has 12 semitones, so 12 keys in a given octave. Octaves start at C, so that's kinda like 0, and has no sharps or flat in the key signature. To get the next key in the circle, which has one sharp, you increase a 5th, or 7 semitones, which by counting on a piano gets you to G. To get through all the key signatures incrementally, one accidental at a time, you keep going up by perfect fifths. Once you come all the way back to C, you'll have hit all 12 keys, encompassing all possible key signatures.
That's to go clockwise - to go counter-clockwise, go up by a perfect fourth every time, which is 5 semitones.
Fun fact: a cello is tuned to C, G, D, and A, which is the first four parts of the circle of fifths.
Violins are tuned a fifth higher (G, D, A, & E), so you can count the string you are on for an easy way to tell how many sharps are in the key signature - A = third string = three sharps.
Hah, awesome! I actually played cello for a few years in elementary school, and I guess at that age they don't really go heavy on the theory. I never put this together even after learning all this stuff in high school.
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This week I learned a little more about how sounds and intervals work - I've always known that pitches were related and octaves especially so, but never realized it was so simple and so physical - an octave is specifically a 2:1 ratio in frequency. You double the frequency of a pitch to get the pitch an octave higher. A4 on a piano is 440Hz, and A5 is 880Hz. This is further divided into 12 semitones, corresponding to adjacent keys on a piano. Each of those is further divided into 100 cents. Given a base frequency in Hertz and a number of cents to increase along the logarithmic scale, you can calculate the final frequency quite easily:
For example, to add a single semitone from A4 up to B♭4, you increase by a twelfth of of what doubling it does:
440 * 2^(1/12). This works out to approximately 466.164, and that floating point imprecision is why intervals are defined as ratios calculated from a set standard instead of fixed numeric values. You don't get another whole number until you go up or down a full octave, to A3 - 220Hz or A5 - 880Hz.This may be "common" knowledge, but it was new to me. Neat.
Awesome! That’s sounds (pun intended) really cool!
This system tuning called twelve-tone equal temperament which was introduced around 18th century by Bach! Another fun fact: double bass tuning is opposite as violin which is E, A, D, G.
Is this somewhat related to the circle of fifths?
Not exactly - the circle of fifths is to do with chords, rather than individual pitches.
If you add a fifth to a given chord, you get the next key signature, so
Totally related - the circle of fifths relates intervals to key signatures. A fifth is an interval with a 3:2 ratio, which is also 7 semitones. An octave has 12 semitones, so 12 keys in a given octave. Octaves start at C, so that's kinda like 0, and has no sharps or flat in the key signature. To get the next key in the circle, which has one sharp, you increase a 5th, or 7 semitones, which by counting on a piano gets you to G. To get through all the key signatures incrementally, one accidental at a time, you keep going up by perfect fifths. Once you come all the way back to C, you'll have hit all 12 keys, encompassing all possible key signatures.
That's to go clockwise - to go counter-clockwise, go up by a perfect fourth every time, which is 5 semitones.
Fun fact: a cello is tuned to C, G, D, and A, which is the first four parts of the circle of fifths.
Violins are tuned a fifth higher (G, D, A, & E), so you can count the string you are on for an easy way to tell how many sharps are in the key signature - A = third string = three sharps.
Hah, awesome! I actually played cello for a few years in elementary school, and I guess at that age they don't really go heavy on the theory. I never put this together even after learning all this stuff in high school.