Maths vs. Music - Tuning and harmony

[Little lunch]

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.

Same here.
The way I, and probably everybody, detects one sound as being out of tune with another is the dissonant waves.

So I wonder if it would be easier to detect a sound that is getting closer to the original sound, if you know what I mean.
You would hear a fast wave length that gradually reduces.
I find it hard to put into words.

[bennyboy]

I find it hard to put into words.

"Wah wah"

There's your words.

By the way, I only got to about 25 seconds, so you guys both did a lot better than me! Although, actually, every few seconds, I could suddenly notice a difference. The click at the start of each new cycle is kind of distracting as well. Curious. I'm suspicious that limitations in my sound equipment (resonance etc.) might be doing something funny. I should try it with studio headphones later.

[bennyboy]

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

Brainfuck indeed.  Split-ear stuff like this can induce OBE I think.  However, I'd bet a million dollars people can hear the difference better if they play it back in monophony.  I'm curious whether we can detect interference in split-brain listening-- I think not.  But with this example, you can definitely feel the planets coming into alignment. The interesting thing is about halfway through, each increment made me feel "Aha! That's in tune!" but then the next increment made me feel "Aha! that last one was a little off. This one is clearly in tune."

[Little lunch]

Wah wahh wahhh wahhhhh.
Yep, that's it. :-)

[Thumpalumpacus]

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

Same here.
The way I, and probably everybody, detects one sound as being out of tune with another is the dissonant waves.

So I wonder if it would be easier to detect a sound that is getting closer to the original sound, if you know what I mean.
You would hear a fast wave length that gradually reduces.
I find it hard to put into words.

I think Benny has it right, that it's much, much easier to use the harmonic discrepancies in a dyad or chord than it is to identify a continuous change in a single note once it starts climbing the asymptotic hill, so to speak.

[Thumpalumpacus]

I find it hard to put into words.

"Wah wah"

There's your words.

By the way, I only got to about 25 seconds, so you guys both did a lot better than me! Although, actually, every few seconds, I could suddenly notice a difference. The click at the start of each new cycle is kind of distracting as well. Curious. I'm suspicious that limitations in my sound equipment (resonance etc.) might be doing something funny. I should try it with studio headphones later.

Yeah, I listened on my phone, and the discrete changes at the beginning were very noticeable. I'll listen at home on headphones and see what difference, if any, there may be.

[Alex K]

I can try to remove the clicks at each transition by matching the phases, didn't have the brainwill yesterday to attemot that...

[Thumpalumpacus]

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

Interesting, I didn't realize harmonic beating could be generated digitally.

[bennyboy]

I actually find the lower one more noticeable-- there's a very clear chorus-effect "wah wah."

[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

Interesting, I didn't realize harmonic beating could be generated digitally.

I use the software mathematica to define the waveform as a mathematical function, e.g. for the equal temperament A major triad

sin(2 pi * 440 * 1 * t)
+sin(2 pi * 440 * 2^(4/12) * t)
+sin(2 pi * 440 * 2^(7/12) * t)

corresponding to the fundamental and the 4th and 7th semitone, and then take values of this function in steps of 1/44100th of a second, normalize it such that the maximum and minimum are near +/- 32000 and write those values into a file as 16 bit integer numbers. Voilà your CD quality wav file.

For comparison, the triad with pure third and fifth would be
sin(2 pi * 440 * 1 * t)
+sin(2 pi * 440 * 5/4 * t)
+sin(2 pi * 440 * 3/2 * t)

the "pure" ratios 5/4 and 3/2 are numerically close to, but not quite equal to the corresponding steps 2^4/12 and 2^7/12 in equal temperament.