Originally posted on my old site Life In A Mikeycosm.

Q. Why are musical notes an octave apart considered to be the same note? -Charlotte V., Seattle, WA

Notes an octave apart are the same note because of the mechanics of vibration. Consider a piano string that is hit by a hammer and vibrates 1000 vibrations per second. So in 1/1000th of a second, it does this: Starts at center, then is hit by hammer. Snaps upwards. Hits the upper limit of its vibration, when the tension pulls it back towards the center. Crosses the center but keeps moving because of the momentum. Hits the downward limit of its vibration. Snaps back towards the center. Crosses the center on its way upward again, completing one cycle.

The precise timing of this motion is:
0 Seconds – position center – hit by hammer
1/4000 of a second: hits upper limit of motion
2/4000 of a second: crosses center moving downward
3/4000 of a second: hits downward limit
4/4000 (= 1/1000) of a second : crosses center again, completing cycle.

Now consider a string an octave higher. The definition of an “octave” is a doubling of frequency, so an octave above a 1000 vibrations per second is 2000 vibrations per second. In that same 1/1000th of a second, this second string vibrates twice. The timing for this string is like this:
0 Seconds – position center – hit by hammer
1/8000 of a second: hits upper limit of motion
2/8000 of a second: crosses center moving downward
3/8000 of a second: hits downward limit
4/8000 (= 2/4000) of a second : crosses center again, completing first cycle. 5/8000 of a second: hits upper limit of motion
6/8000 of a second: crosses center moving downward
7/8000 of a second: hits downward limit
8/8000 (= 1/1000) of a second : crosses center again, completing second cycle.

Now notice the zero crossings for that second string: 0 Seconds, 2/8000 of a second, 2/4000 of a second, 6/8000 of a second, 1/1000 of a second. Notice that for fully 50% of these center crossings, every other one, the first string is also crossing center. The “waveforms” of the first and second pitches could not coincide more frequently without the pitches being exactly the same (IE, both 1000 cycles per second.) Hence we say they are the same note, but in two different octaves.

Then you get into things where string#1 crosses zero every third time that string#2 does. This is what we call a fifth. If you hit a C on a piano, and then hit the G not in the same octave but an octave higher, this is exactly what’s happening.

The interesting thing about this is it means a C and the G an octave higher are actually more consonant than a C and the G in the same octave. Which is, from a physical standpoint, true. But generally, as long as the ratio of frequencies of two notes (frequency is directly related to the center crossings – in effect, the center crossing times define the frequency… crossing center 2000 times/second -once downwards, once upwards- equals frequency 1000 Hz) can be expressed in low integers, they are relatively consonant. The two strings given in the example above would generally be said to have frequency ratio 1:2. (1000 cycles per second:2000 cycles per second). A C and the G an octave up are 1:3. A C and a G in the same octave are 1:1.5, which is more clearly written as 2:3. (Obviously we’re getting into a little algebra here.) A 2:3 frequency is not quite as consonant as a 1:2 or 1:3, but it’s still pretty consonant. 91:129, for example, would NOT be very consonant.

1:4 (String#1 crosses center once for every fourth time string #2 does): 2 octaves
1:5 (String#1 crosses center once for every fifth time string #2 does): 2 octaves + a third
1:6 (String#1 crosses center once for every sixth time string #2 does): 2 octaves + a fifth
1:7 (String#1 crosses center once for every seventh time string #2 does): 2 octaves + a … fuck, I don’t remember, I think it’s somewhere around a seventh. This is the launching point for a lot of interesting music theory. If this interests you look up “well temperament”, “just intonation” or “mean tone” on the web for several years’ worth of fascinating reading.
1:8 (String#1 crosses center once for every eighth time string #2 does): 3 octaves
1:9 (String#1 crosses center once for every ninth time string #2 does): 3 octaves + a pling
1:10 (String#1 crosses center once for every tenth time string #2 does): 3 octaves + a durgen
1:11 (String#1 crosses center once for every eleventh time string #2 does): 3 octaves + a quackledaff

I made those last three up.

“If it’s music, it’s got to vibrate. If it don’t vibrate, it ain’t music.” – Wadada Leo Smith

[NOTE: I DON’T WANT TO HEAR FROM ANY ACOUSTICIANS CORRECTING THIS EXPLANATION. IT’S A SIMPLIFIED INTRODUCTION. IF YOU TAKE ISSUE WITH ANYTHING SAID HERE, YOU’RE TOO PEDANTIC. TAKE A CHILL PILL.]