@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 

 @3132ba89 - that's fascinating!  I really like that graph and should someday think about various measures of dissonance or 'resonance' between frequencies.  (I'm already curious about celestial mechanics and how resonances sometimes stabilize orbits and sometimes destabilize them; I've never had time to really understand it.)   

I don't really myself find the fifth to be 'sweet'; I was just trying take the attitude of common practice harmony theory, roughly 1650-1850, where you often read things like:

    The perfect fifth and the perfect octave are considered perfect consonances. The unison is a consonance insofar as it can be considered an interval at all (many say it cannot).
    The major third and sixth, as well as the minor third, sixth, are considered to be imperfect consonances.
    The perfect fourth is dissonant in some contexts but consonant in others (see below). Specifically, the perfect fourth is dissonant when it is formed with the bass note of any sonority.

https://en.wikibooks.org/wiki/Music_Theory/Consonance_and_Dissonance

If I'd wanted to be more technical I would have said "the fifth was the only interval besides the octave to be considered a 'perfect consonance' in the common practice era."   But that would have killed off half my readership!