Maths vs. Music - Tuning and harmony

[Thumpalumpacus]

Alex, as much as I admire and appreciate the work you've done on this, I have to confess that I just don't understand Japanese.

In seriousness, I can read, mathematically, the difference in your second factors -- 1.259 vs 1.25, for instance, obviously a significant difference, because when you're cycling hundreds of times a second, those 9/1000s add up fairly quickly and throw the harmony out of whack.

I guess that as someone more fluent in the language of music than the language of mathematics, I experience music as qualia rather than abstraction?

[Alex K]

Yeah, forget the sine formula and all that, I just wanted to answer the question how these things are represented digitally - the important part is, the ratio is exactly 1.25 in one case which is a nice geometric ratio, and 1.2599... in the other.

Talking about subjective qualia, you can actually see visually when looking at the wave form that the pure one is more harmonious and even than the equal one

[bennyboy]

I found a VST synth that allows custom scale tunings, and found a just intonation scale.  I didn't set the scale frequencies myself, but I think they're right because I got the effect I expected. Here are two cadences repeated with a simple saw/square tooth sound, first with just intonation(i.e. all notes tuned to low-integer ratios based on the tonic), and second with equal-tempered tuning (freq. * 2^n/12).  You can really hear the "chorus" effect-- it's almost hard to believe that I didn't add an actual chorus effect, isn't it?

Also, sorry about the big LoL banner, but AF's soundcloud support doesn't currently allow for resizing.

[Thumpalumpacus]

Nintendo called, it wants its composer back.

[bennyboy]

Nintendo called, it wants its composer back.

Haha just be glad I didn't use sine waves.  An organized system of bee-farts is still slightly better than an organized system of dog whistles, no?

(And no offense meant to Alex, who really got the ball rolling on this. But after experimentation I think it's easier to hear the differences with a slightly more complex sound. Sine waves together usually sound like single pitches with strange overtones, at least to my ear, rather than real harmonies).

[Alex K]

benny, you're absolutely right. I think the more overtones you have in your sound( which always have exact geometric ratios), the more explicitely they will clash with other notes of the triad.

I used sine to the third power in my example though, which at least adds twothe second overtone

My piano allows pure and equal tunings, maybe I'll record some sample pieces in both later and post em.

[bennyboy]

Nintendo called, it wants its composer back.

You bastard!  You got the idea stuck in my head, and I just spent the last hour of my life fantasizing about life in the 80s. But yeah. . . I could have rocked Nintendo, I guess.

[Alex K]

Ok, so for good measure I recorded some nonsense (albeit after a few glasses), do not judge for artistic artistry.

First a C major chord
then some nonsense piece that safely stays with major C,F,G chords
then some vaguely jazzy nonsense also starting in C major but with sixth and seventh, ninth chords
Last, the circle of fifths from C down to F#

1. With equal tuning

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

As expected, everything sounds ok (tuning-wise, not the playing )

2. With a tuning which ensures pure C major thirds and fifths

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

The C major triad sounds a bit unfamiliar, but purer in the second case, and the F and G major triads sound good; the simple piece therefore sounds quite clear, the piece with jazzy chords starts to go wrong, and the circle of fifths just steers towards catastrophe at F#.

[bennyboy]

When art meets science! Love it.

Yeah, right off the first C-chord you can hear that the just-tempered one has that choir-style "power major" sound, and leaves the equal-tuned on sounding distinctly flat.

To be honest, I find it very hard to hear any sourness in the jazz. I think it's because I'm so used to hearing jazz played on out-of-tune pianos anyway, and sourness is a proper part of the jazz palette anyway. I did notice an event at 1:16 in the first sample and 1:38 in the second (why the different timings?). I wouldn't say either sounded "bad," but the second was definitely a more jarring moment. In a way I kind of enjoyed that little slap to the face.

btw you better not be placebo-ing me into thinking I hear stuff, when you just played the same thing twice. Then we can't be friends!

[Thumpalumpacus]

The C major triad sounds a bit unfamiliar, but purer in the second case, and the F and G major triads sound good; the simple piece therefore sounds quite clear, the piece with jazzy chords starts to go wrong, and the circle of fifths just steers towards catastrophe at F#.

No doubt the cycle of fifths breaks down at the b5 due to its position at the midpoint of the cycle.