Oddbean

In Pythagorean tuning, we try to force all frequency ratios to be powers of 3/2. In music, 3/2 is the 'perfect fifth': the sweetest of intervals except for the octave. If we start with some frequency and go up and down by powers of 3/2, we create the 'circle of fifths' shown here. It's almost a 12-pointed star, with one point for each note in the 12-tone equal-tempered scale. Almost - but not quite! When we go up 12 perfect fifths, we get a note that's almost but not quite 2⁷ times the frequency we started with. In other words, it's almost but not quite 7 octaves higher. So there's a glitch. Here I've stuck that glitch at the opposite from the note labeled 1. The spot directly opposite 1 is called the 'tritone', or sometimes 'diabolus in musica' - the devil in music. 😈 The size of the glitch is called the 'Pythagorean comma'. It's (3/2)¹² / 2⁷ ≈ 129.74633789 / 128 ≈ 1.01364326477 https://en.wikipedia.org/wiki/Pythagorean_tuning https://en.wikipedia.org/wiki/Pythagorean_comma https://media.mathstodon.xyz/media_attachments/files/111/181/039/976/449/841/original/6423e662f54d0985.jpg

@b4c50e1b worth mentioning that perceptually, we hear this 3/2 interval as halfway between the root and its double one register higher (the octave). i.e., a "perfect fifth" in music theory is the "perceptual midpoint" between a note having frequency f and a note one octave higher, having frequency 2f.

@b4c50e1b Starting from ignorance on my part except for making junk guitars for a couple of years, it amuses me how simple things quickly are made complicated. I put only 4 marks on my fretboards ... halfway (Octave), 2/3 (perfect 5th), 3/4 (perfect 4th). The 4th mark equates to minor 3rd ... where the 3rd fret on a fretted guitar would be. Thus the dreaded tritone sits happily between the perfect 4th and perfect 5th.

@b4c50e1b Hey John, do you really find the fifth to be the sweetest of intervals subjectively (within the octave)? I ask because whenever I noodled as a kid, I always felt the fourth was the smoothest of the intervals. Then, years later I did some math / physics. Looking to measure the “interference power” of overtones in an interval relationship, a simple model of resonance damping had me look at the measure: \sum_{j=2}^{\infty} | (j \alpha - [j \alpha]) / j | with \alpha the frequency ratio of the interval, and [] the rounding operator. Basically, this is looking at the overtones series of the second note and measuring a distance from the overtones of the first to calculate destructive interference effects. It has model support from effects like power dissipation in beating and such. It’s not a full dissonance metric, which would need other terms for amplitude pain thresholds, blackboard frequencies, and the like - it’s purely a “harmonic interference” kind of dissonance metric. Anyway, when I plotted it, I was shocked by what I saw (see image). The strongest minima outside the unison and octave were the just perfect fourth and just major sixth, followed by the just major fifth which is very close to the Bohlen-Pierce fourth and just minor sixth. You can also see prominent views of the just minor third and BP seventh. My subjective preferences have validated this metric for my own tastes, but that’s obviously not a blind evaluation. However, since then, I’ve studied the history of the intervals across cultures, and there have been many different views on the consonance and dissonance of the different ratios. Because of my own interest in experimental noise music, I’ve found a lot of use of the maxima in this chart too which really do have a fair amount of harshness. It’s interesting to locate other famous dissonances, like the tritones and the wolf interval on the chart, as well as charting the dissonance introduced by well tempering. https://kolektiva.social/system/media_attachments/files/111/183/543/721/334/955/original/2afea3ed5e742681.png