Maths vs. Music - Tuning and harmony

[Thumpalumpacus]

In guitarist circles it's said to sound more sonorous. My tuner is a fixed calibration a 440. My ears can't tell the difference.

[LostLocke]

There's a great article about that frequency here:
https://ask.audio/articles/music-theory-...om-fiction

Of course, I assume *some* people here will just wave it off.  

[Alex K]

There's a great article about that frequency here:
https://ask.audio/articles/music-theory-...om-fiction

Of course, I assume *some* people here will just wave it off.  

Man that is some dumb shit. But the "debunking" of the first two is a really stupid as well. They couldn't have tuned to 432 hz because they didn't have seconds yet? Duh, if 432 hz really had such special properties, they would have found it without explicitely referring to it as "432 hz" or "432 cycles per second".

[Alex K]

I'm going to go off track here a little.
I was looking at that chart with the colours corresponding to musical notes and was thinking, 'what about all the notes in between.'
The ones that middle eastern music plays like the notes between G and A flat, etc.
Then I remembered that there are infinite numbers between 1 and 2.
Are there infinite notes between G and A flat?
Can one note transcend or descend to another forever?

Yes. If you play the function

sin(2 pi * 440*(1+e^-t)*t)

as a sound (where t is in seconds), you get a note that quickly decends from a'' towards a' in a continuous fashion without ever reaching it. I might put it as a mp3 later l. Whether you call all that stuff "notes" is, I suppose, convention.

I realize my mathematical ansatz for the waveform of the descending tone was too naive. I need to refine it.

[Alex K]

For the specialists:

---

I thought with the

Sin(2pi * 440 * t * (1+E^-t))

I was implementing a sine wave that at each point in time had the frequency 440 * (1+E^-t) Hertz, but that's not true. The change in phase of the sine wave is dominated by the E^-t factor becoming smaller, and the result is a pitch that quickly becomes lower and then approaches 440 from below. This is easy to solve by adding a

Sin(2pi * 440 * t * (1+E^-IntegerPart[t] ))

because now within each second we have a constant note corresponding to the frequency 440 * (1+E^-IntegerPart[t] ) which does what we want, namely asymptotically approach 440 without ever reaching it, and the decrease in t now appears as a sudden phase change whenever IntegerPart jumps to a higher value, which corresponds to an audible clicking noise.

For my actual sample, i didn't use the exponentially decaying example because it so quickly approaches 440 that differneces soon becomes inaudible. Instead I used

Sin(2 pi * 440 * (1 + 1/IntegerPart[t+1])

which also starts with a'' 880 and approaches a' 440 asymptotically without ever reaching it, but not as quickly. Last but not least, I used sin^3 instead of sin in order to include some harmonics and make the sound more instrument-like:

---

https://dl.dropboxusercontent.com/u/4685...ngnote.mp3

How far in do you think you still hear differences in pitch?

p.s. as proof that it keeps descending even if you don't hear the individual steps any more, you can click from say 30s to 50s while playing.

[Alex K]

As an additional help (or brainfuck, ymmv), here I keep the 880 reference note on the right speaker. People with musical ears will now hear the left side increasingly approach but not quite reach the lower octave.

https://dl.dropboxusercontent.com/u/4685...erence.mp3

[Alex K]

And last but not least, a comparison of an A major triad in pure tuning with the exact ratios 5/4 for the third and 3/2 for the fifth, and in equal tuning with 2^4/12 and 2^7/12 for the third and fifth.

https://dl.dropboxusercontent.com/u/4685...arison.mp3

It plays the exact ratios for 2 seconds, then the equal temperament for 2 seconds, then repeat. Equal temperament really sounds wrong when contrasted directly with the exact ratios...

The effect is maybe more subtle, but still very noticeable when everything is taken one octave lower, i.e. based on 220 hz.

https://dl.dropboxusercontent.com/u/4685...rison2.mp3

[Thumpalumpacus]

For the specialists:

---

I thought with the

Sin(2pi * 440 * t * (1+E^-t))

I was implementing a sine wave that at each point in time had the frequency 440 * (1+E^-t) Hertz, but that's not true. The change in phase of the sine wave is dominated by the E^-t factor becoming smaller, and the result is a pitch that quickly becomes lower and then approaches 440 from below. This is easy to solve by adding a

Sin(2pi * 440 * t * (1+E^-IntegerPart[t] ))

because now within each second we have a constant note corresponding to the frequency 440 * (1+E^-IntegerPart[t] ) which does what we want, namely asymptotically approach 440 without ever reaching it, and the decrease in t now appears as a sudden phase change whenever IntegerPart jumps to a higher value, which corresponds to an audible clicking noise.

For my actual sample, i didn't use the exponentially decaying example because it so quickly approaches 440 that differneces soon becomes inaudible. Instead I used

Sin(2 pi * 440 * (1 + 1/IntegerPart[t+1])

which also starts with a'' 880 and approaches a' 440 asymptotically without ever reaching it, but not as quickly. Last but not least, I used sin^3 instead of sin in order to include some harmonics and make the sound more instrument-like:

---

https://dl.dropboxusercontent.com/u/4685...ngnote.mp3

How far in do you think you still hear differences in pitch?

p.s. as proof that it keeps descending even if you don't hear the individual steps any more, you can click from say 30s to 50s while playing.

Without a reference pitch handy, the last really noticeable change for me was around 32 seconds.

[bennyboy]

And last but not least, a comparison of an A major triad in pure tuning with the exact ratios 5/4 for the third and 3/2 for the fifth, and in equal tuning with 2^4/12 and 2^7/12 for the third and fifth.

https://dl.dropboxusercontent.com/u/4685...arison.mp3

It plays the exact ratios for 2 seconds, then the equal temperament for 2 seconds, then repeat. Equal temperament really sounds wrong when contrasted directly with the exact ratios...

The effect is maybe more subtle, but still very noticeable when everything is taken one octave lower, i.e. based on 220 hz.

https://dl.dropboxusercontent.com/u/4685...rison2.mp3

Well done.

if you really want to hear the difference, I recommend playing a simple cadence: I ii V I in major or i iv V i in minor.  The difference becomes very obvious then.

The problem is that you are (i think) using sin waves: it's such a pure sound that it sounds like single notes rather than chords.  I've considered writing a computer program that will let you choose how you derive each note by linking it to previous ones; for example, you could say, "Use the B that is the major 3rd of that G chord we used last bar," or "Use the B that is the perfect 5th above the e-minor (iii) chord that we modulated to" or whatever.  Ultimately, I'd like to do this with entire pieces of MIDI-recorded music, like say a Beethoven Sonata.  But the problem is that MIDI pitches are only adjustable in cents, making the MIDI system useless. So I'd have to collect the data using MIDI, and then do the math to adjust all the pitches and generate my own .wav files. So you could have the entire piece in a kind of "dynamic just tuning," something that I think nobody has attempted yet.

I think it's the kind of project that could land me a professorship somewhere, no?

[bennyboy]

https://dl.dropboxusercontent.com/u/4685...ngnote.mp3

How far in do you think you still hear differences in pitch?

The ability to hear differences in a single pitch probably isn't that high. However, the ability to hear the HARMONIC RELATIONSHIP BETWEEN pitches is epic. If you take a perfectly-tuned perfect fifth and detune it even a little bit, you can start to hear the "wah wha wahhhhh" from interference among the notes right away.


This means that our ability to tell the difference between pitches is almost infinitely better than our ability to tell the difference between different colors. That's why, for example, I can't use MIDI (which uses cents) in my tuning project. It's also why real audiophiles will never accept digital media as being better than a really good record player: the interference patterns lead to whistles, hums and buzzes that normally-gifted listeners don't notice. I, for one, can't hear the difference-- but I can totally understand how/why it's there.