to incorporate Chapter 8 of Eugenia Cheng's Beyond Infinity.
This is what Theoni Pappas writes on page 186 of her Magic of Mathematics:
"Two people can carry on a normal conversation from the two focal points [of the Rotunda in the United States capitol building], undisturbed by the noise level of the hall."
Pappas writes that the ceiling at the House of Representatives is a parabola -- and based on the info provided at this YouTube link, it really is a parabola.
According to the link, President John Quincy Adams took full advantage of this whispering spot.
Let's continue with our musical scales. By now, you should have figured out the pattern:
-- Our next prime is 13, so it should incorporate the 13th harmonic.
-- Our last two scales were 19EDO and 31EDO, so we combine these to form a 50-note scale.
It would take too long to write out all the notes of 50EDO. But since an octave is 1200 cents, each step of 50EDO must be 1200/50 = 24 cents. So all the multiples of 24 cents are steps of 50EDO
Let's check out how accurate all the harmonics are in 50EDO:
-- 3/2: The perfect fifth is 702 cents, so we choose Note 29 (696 cents).
-- 5/4: The major third is 386 cents, so we choose Note 16 (384 cents).
-- 7/4: The harmonic seventh is 969 cents, so we choose Note 40 (960 cents).
-- 9/8: The meantone is Note 8 (192 cents).
-- 11/8: The harmonic eleventh, is 551 cents, so we choose Note 23 (552 cents).
-- 13/8: And the newest harmonic, the 13th, is 841 cents, so we choose Note 35 (840 cents).
As we can see, the 11th and 13th harmonics are very accurate. The seventh is the furthest away.
Here is the link to our Xenharmonic site with regards to 50EDO:
By the time we reach 50EDO, most musicians would say that there are too many notes. I couldn't find any YouTube links to 50EDO music. But I do see a poster on the dozenal (base 12) website, Leopold Plumtree, who is apparently a huge fan of 50EDO:
Dividing the octave into a dozen parts gives us a 1:2 semitone:whole tone ratio (of their logs). Using the next pair of Fibonacci numbers (including 2 again) would be 2:3 which results in 19 divisions per octave (very close to 1/3 comma meantone) and next up would be 3:5 giving 31 divisions (with a fifth a tad wider than that of 1/4 comma meantone). [Keep on going and we approach the golden meantone].
But, if we go a little further and use 5:8 for the diatonic semitone:whole tone ratio, we get 50 divisions per octave, so the diatonic semitone is a tenth of an octave.
The 50-edo major scale in a decimal octave is...
0.00, 0.16, 0.32, 0.42, 0.58, 0.74, 0.90, 1.00
The major triad is 0.00, 0.32, <0.26>, 0.58, the 0.26 being the minor third between the major third and fifth.
Compare to the 5-limit Just major triad...
0.000, 0.322, <0.263>, 0.585
And we get a lovely fit for the thirds; the major third being out by about 2 millioctaves and the minor third by about 3 millioctaves.
As others have noted, 50-edo is also close to the least squares *meantone mapping of the major triad compared to its Just counterpart (or minor triad...same intervals in a different order), which results in a fifth of 0.580137 octave.
So unfortunately, we can't hear what 50EDO music actually sounds like -- nor for that matter what 13-limit music actually sounds like.
But one thing I've noticed is that, while the third harmonic is actually a (perfect) fifth and the fifth harmonic is actually a (major) third, the seventh harmonic really is a seventh. Likewise, the 11th harmonic actually is a sort of "eleventh" (octave+fourth) while the 13th harmonic actually is a sort of "thirteenth" (octave+sixth).
In fact, we actually have eleventh and thirteenth chords in addition to seventh and ninth chords. But one thing I notice about 11th chords in music (especially jazz) is that the 11th is often sharpened:
We already know that jazz musicians often play a harmonic seventh for the Bb. And so I often wonder whether the F# is intended to be a harmonic 11th, 11/8 -- since F# is the closest note to 11/8 in a 12EDO scale.
So perhaps in 31EDO and 50EDO, we could try to play a true harmonic 11th chord -- as well as a true harmonic 13th chord in 50EDO.