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How is that if we take A3 at 220Hz then we double the frequency to 440Hz it still is A?

I have been reading about the frequencies and I understand that if we have pitch A3 and we decrease/increase the frequency where it's sharp/flat we still retain enough of it's quality to remain an A but if we decrease/increase even more then were onto a new distinct pitch such as B but if we double it, triple it, up or down were back to the A but an octave or up or down.

Or is it a pitch class and our ears hear similar qualities so we only need the 12 notes (natural, sharps, flats) and the pitch is way different but has some same qualities as it's fellow A up and down octaves?

Can some help me understand this or point me to a good article or video on it.

And this is why I can't focus. Always trying to learn something new.

Frequency is continuous.when your piano gets out of tune the frequency will be something between two adjacent keys.
Try listening to the mooc course “reinventing the piano”, there is a full chapter on different methods for tuning the piano.

Earing is not a linear phenomenon. Rather, our response to frequencies is exponential. Here is a good primer on this:

https://www.animations.physics.unsw.edu.au/jw/frequency-pitch-sound.htm

Small world, that is where I did physics, so long ago this course on perception of sound did not exist.

Interesting, none the less.

It works the other way... it's BECAUSE the frequency is doubled, that they're the same note.


Sound is waves. When we increase frequency, we decrease wavelength.

When we double frequency, the wavelength is halved.




So in this graph, the red wave is 2x the frequency and 1/2x the wavelength when compared to the blue wave.

Because the increase factor is a nice round number like 2, instead of some random multiple like 1.732, you can see that 2 red waves neatly fit together in 1 blue wave.

This neat property, for some reason, makes the two sound waves sound similar to us, and therefore we assign them the same letter.

All two notes an octave apart have this property, from one A to the next A, from one G to the next G, etc.

Re-using the names of notes/pitches, such as A at 440hz and A again at 880hz reflects the fact that A880 is the pitch that is most harmonious with the A440 out of all pitches possible, except the same A440.

If you simultaneously strike two A 440 notes, at every possible point in time they will be at exactly the same place in their sine wave travel. They will never deviate from each other. This is the maximum amount of harmony possible, perfect harmony. Tuners and other music disciplines call this a Unison. And, the two pitches have exactly the same name since they are exactly the same note.

At the opposite end of the spectrum would be a note that nearly never gets in sync with the first note. If you simultaneously strike an A 440 along with another note that vibrates at, for example, 473hz, they would not get into harmony until 208,120(440 x 473) cycles later, which could be several seconds (minutes?). Unless, there is a common factor to both numbers, in which case they would reach and depart a state of harmony at that number of cycles. But, these two notes would sound poorly together as their vibrations would rarely get into sync. They might never get into sync in a real instrument, as the sustain could die out before the sync point was ever reached.

Between these two extremes are frequent and regular occurrences of note pairs that sound very much in harmony with each other. For instance, A440 and A880. If you strike these two notes at exactly the same time, they will deviate from each other a little bit. See Yao's graph above. But, at the conclusion of EVERY cycle of A440, the A880 will be in exactly the same spot. In other words, 440 times per second, the two notes will be in exact harmony. They will differ from the Unison only in that the A880 will go through an extra cycle for every one cycle of the A440. There is not a more harmonious relationship possible, other than the aforementioned Unison. Or, strike an A440 and an A1760 (2 x 880hz) and they too will be in perfect harmony 440 times over the course of one second, differing only in that the A1760 will go through 4 cycles for while the A440 is cycling once. This is the second most harmonious relationship possible. A440 and A3520 (2 x 1760hz) would be the third most harmonious relationship possible.

I don't know, but I suspect, that the naming convention that repeats note names as you go up the scale has to do with the occurrences of these second, third, fourth, etc., most harmonious possible relationships.

https://pages.mtu.edu/~suits/notefreqs.html

The sound spectrum is continuous. So selecting 12 notes in an octave is a choice in terms of practicality. The human ear can detect a difference in pitch of about 25 cents (sometime less), knowing that a tempered half tone is 100 cents. So a normal ear can detect a quarter of a half tone difference.

Our ear is sensitive to the way vibrations (sine waves) creates or not resonances. If one takes 2 sine waves with a frequency that is double they align exactly (except that one is just 2x faster)and thus do not create any disturbance (ie they are in unison). In practice for a given pitch real instruments have multiple harmonics which are not exactly the same at every frequency (which is due to the way the instrument is built) and is also different for each instrument, but they are close enough to give us a sense of unison. So if you take a simple chord and record its pattern of harmonics, the first one you will find in terms of strength going upward is the octave above (the the Fifth, then the third). That also explains why playing the pitch an octave above sounds very similar; not only is it the same sine wave but they also have several harmonics in common.

Once you start to move away from the unison, as said as low as 25 cents, the amount of disturbance (ie non alignment of the waves) starts to increase and the ear senses that there are 2 pitches which are not in tune.

>How is that if we take A3 at 220Hz then we double the frequency to 440Hz it still is A?

The octave is the least dissonant tone (after the unison) in our music. Calling it also A highlights this octave based tonal structure and makes it easy to recognise many of the patterns following from the theory. So it's a useful way to describe what we do with music.

It's much more than just a useful convention. Octave equivalence is a fundamental neurobiological phenomenon that is experienced even by other species. Children will naturally sing a tune along adults even though their voice is actually higher pitched. They do that because their brain is wired to hear equivalence between the notes of different octaves.

This was very helpful. Thanks everyone for the replies. I imagine if we gave every pitch its own name it would very confusing. I see now that A 440 and A 880 are two distinct pitches but with a lot of unison occurring.

@ralphiano that explanation and example helped greatly! This is what I was was looking for on Google and couldn’t find.

I will also check out links you all provided to learn more too.

I know this may have not been directly a piano question so I appreciate you all still taking the time to help me understand this.

I struggled with this for a long time. My father was into ham radio, TV repair, etc., and I used to tune my guitar with a sine wave generator. The octave (doubling) made sense but I never could understand the chromatic scale. Finally I stumbled onto the 12th root of 2 and I could at least describe it mathematically.

As I understand it, adding the piano to the orchestra was what made the tuning change from just intonation to even temperament. The original tuning (just intonation) sounds ever so slightly better but only for a single key. I guess the rest of the instruments in the orchestra can make tiny pitch adjustments but but the piano requires a tuning that works in any key or a long break between songs. So the math no longer works for why we use call some pitches 'perfect', they aren't mathematically perfect any more. I can just barely hear the difference between just intonation and even temperament tunings but I've always wondered if other people are bothered by not having those perfect, integer derived, ratios.

I can't back this up but I have been told that some violinists use just intonation when playing solos.