Plot of consonance/dissonance of true intervals

[Z1202]

Hi all

I was interested in the relative consonance/dissonance of various intervals and their offsets from the equal temperament scale. I ended up with the following plot and thought I might share it with you. The regular vertical lines mark semitones, the leftmost line correspond to the unison, the rightmost one to two octaves. The black lines mark the true intervals, the higher is the line, the more consonant is the interval. You can easily see prominent peaks at unison, octave and the fifth (including +1 octave). I hope I didn't do any mistakes in the computations

It should be possible to see why consonant intervals are easier to hit (they have higher tolerance) than dissonant ones.

Also some interesting questions arise. E.g. there are two peaks next to the minor seventh. Which of them do we really hear as a minor seventh then?

The image probably is going to be corrupt in the scaled view. You need to download it and see in full zoom not to miss any lines.



Regards,
{Z}

[Z1202]

Here's an alternative plot if anyone's interested (this time interval dissonance is measured based on the undertone distance, so octave and unison e.g. are considered equally consonant)

[Endolith]

Can you post the calculations you used to generate these?

[llatham]

It should be possible to see why consonant intervals are easier to hit (they have higher tolerance) than dissonant ones.

I interpret this as showing the opposite - you better hit a consonant interval dead on, because it's more likely to sound "wrong" than those other intervals, whose tolerance is "wider".

But what are you doing here?

It looks like you're simply showing us how far "off" equal tempered intervals are from pure intervals (or vice versa).

Steve

[Endolith]

There's a lot of explanations of what this graph means in Tuning mailing lists and such, but I couldn't find a concise example of the equations used to plot them. This paper explains one method: http://www.nafindix.com/math/sensory.pdf

Their estimated dissonance function for two frequencies is pretty simple:

hv(s, v) = LCM(s/GCD(s, v), v/GCD(s, v))

I'll try to make some plots with this later.

[Ogg Vorbis]

http://img362.imageshack.us/img362/4106 ... vd8.th.png[/img][/URL]

How can one apply this data?

[Endolith]

Here's my attempt based on that dissonance function, in Processing:

http://www.endolith.com/processing/java/interval_lines/

This shows just one octave:

http://www.endolith.com/processing/java ... h_opacity/

You can see the fractalness more clearly with the colors, I think.

Also some interesting questions arise. E.g. there are two peaks next to the minor seventh. Which of them do we really hear as a minor seventh then?

That depends. There's actually more than one seventh. Equal temperament makes it out of tune no matter what, but the pure mathematical intervals are what you end up singing:

http://en.wikipedia.org/wiki/Harmonic_seventh

[Endolith]

More plots here: http://www.flickr.com/photos/56868697@N ... onsonance/