"Mathematical ideas have been twisting and turning music and sound waves for centuries. A walk around the interior of the dome in St. Peter's Cathedral in Rome will convince you that the curve of the dome's walls carries one's whispers to a listener on the opposite side."
This begins a new section in Pappas, "Mathematics & Sound." I've heard of the concept of a "whispering gallery" before. Pappas suggests that the whispering gallery at the Vatican consists of two parabolas. But actually, St. Peter's Basilica is an ellipse. Let's watch the following video:
A person who stands at one focus of the ellipse can listen to one who stands at the other focus. I assume that in this video, the camera or phone is at one focus, and what we are hearing is at the other.
I'm not quite sure why Pappas writes "parabola" instead of ellipse. At first, I thought it was because she fears that the readers are less familiar with ellipses than parabolas. But Pappas herself explains what an ellipse is on Pages 18-19, so that can't be the case.
Fortunately, parabolas do share with ellipses the whispering property. Ellipses have two foci but parabolas have only one focus, so, as Pappas writes, we need two parabolas to make it work.
Actually, further investigation reveals that there really is a parabola at St. Peter's. But it's not the whispering gallery. Apparently, Michelangelo designed the dome as a parabola. Here is a link to the interior of this parabola -- it's the last photo at the following link:
I hope this settles the debate about which conic sections are at St. Peter's Basilica.
Technically, we are done with musical scales in Pappas. But actually, we can go on forever with scales, since there's so much information from the Xenharmonic website.
The next limit to consider after the 7-limit isn't the 9-limit, since 9 is already included in 3 (so we already had 9/8 with Pythagoras). Instead, it's the 11-limit, as 11 is the next prime number.
Is the 11-limit worth playing in music? Well, we reduce our 11 by octaves to 11/2 and 11/4, and finally 11/8 is the interval to consider. This interval lies somewhere between the major third of 5/4 (or 10/8) and the perfect fifth of 3/2 (or 12/8). In fact it is 551 cents, and so in 12EDO, it's very slightly closer to the 600-cent tritone than the 500-cent perfect fourth.
Vi Hart -- yes, the same Hart who prefers tau to pi -- created a video about the harmonics. She intentionally stops at the 10th harmonic, since the 11th harmonic is essentially a tritone -- one of the two most dissonant intervals. (She skips the 17th harmonic for the same reason -- 17/16 is the other of our dissonant intervals, the semitone.)
But some musicians wish to include the 11th harmonic anyway. Intervals based on 11 are known as "undecimal," just as those based on 7 are "septimal."
Now we just need a scale to in which to play our undecimal intervals. As it turns out, some of our scales fit into a pattern: 5, 7, 12, and 19.
Moreover, in the 7EDO, we played a 5-note subset (the Chinese scale). In 12EDO, we can play both a 7-note subset (white keys, major scale) and a 5-note subset (black keys, pentatonic scale). A certain musician (Joseph Yasser) continued this pattern with 19EDO -- a 12-note subset (all the white keys and the flats, a chromatic scale) and a 7-note subset (all the sharps, a C# major scale).
And so the next scale in this pattern would be 31EDO, which would contain both a 19-note subset and a 7-note subset:
Let's find our harmonics:
-- 3/2: The perfect fifth is 702 cents, so we choose Note 18 (697 cents).
-- 5/4: The major third is 386 cents, so we choose Note 10 (387 cents).
-- 7/4: The harmonic seventh is 969 cents, so we choose Note 25 (968 cents).
-- 9/8: The meantone is Note 5 (194 cents).
-- 11/8: And the newest harmonic, the 11th, is 551 cents, so we choose Note 14 (542 cents).
All of these harmonics are improvements from 19EDO. In fact, these are all the best approximations for the harmonics in scales we've seen except for the perfect fifth (slightly better in 12EDO and 17EDO) and the 11th (slightly better in 22EDO, but that note was already assigned to 27/20 based on the Indian scale).
The Xenharmonic website is very detailed as to which ratios correspond to each note of 31EDO:
Degree Ratio Name
0 1/1 Perfect unison
1 55/54, 33/32 Diesis
2 22/21, 25/24 (Undecimal) chromatic semitone
3 15/14, 16/15 Diatonic semitone
4 12/11, 35/32 (Undecimal) neutral 2nd
5 9/8, 10/9 Major 2nd
6 8/7 Septimal supermajor 2nd
7 7/6 Septimal subminor 3rd
8 6/5 Minor 3rd
9 11/9 Undecimal neutral 3rd
10 5/4 Major 3rd
11 14/11, 9/7 (Undecimal or septimal) supermajor 3rd
12 21/16 Septimal sub 4th
13 4/3 Perfect 4th
14 11/8, 15/11 Undecimal super 4th
15 7/5 Septimal diminished 5th
16 10/7 Septimal augmented 4th
The chart abruptly ends here, since each new interval is an octave inversion of a previous interval. So Note 17 is an undecimal sub 5th, or 16/11, since (11/8)(16/11) = 2/1, the octave. We can invert any interval be changing "major" to "minor" (and vice versa), "super" to "sub," "augmented" to "diminished," and keeping "perfect" and "neutral" the same. Oh, and 2nds become 7ths, 3rd become 6th, and 4ths become 5ths.
We notice that these new undecimal intervals tend to be "neutral." This is because the 11th harmonic is almost exactly halfway between a perfect fourth and a tritone. So neutral intervals tend to differ from Pythagorean intervals by 33/32, which is an undecimal "quarter tone."
Notice that if we scroll down to Note 13 (perfect 4th/meantone) we can see how all of the previous scales fit into the 31EDO -- three-, five-, seven-, 12-, and 19-note scales.
To name these notes, we see that there are too many notes for a simple sharp-flat system. We could use either double-sharps and -flats, or half-sharps and -flats. The link above uses half accidentals, which are symbolized using up (^) and down (v) symbols:
Degree Cents Solfege Color
C 0 do white unison
C^ 39 di jade quarter-tone
C# 77 ro blue 2nd
Db 116 ra green 2nd
Dv 155 ru amber 2nd
D 194 re yellow 2nd
D^ 232 ri red 2nd
D# 271 ma blue 3rd
Eb 310 me green 3rd
Ev 348 mu jade 3rd
E 387 mi yellow 3rd
E^ 426 mo red 3rd
Fv 465 fe blue 4th
F 503 fa white 4th
F^ 542 fu jade 4th
F# 581 fi blue 5th
Gb 619 se red 4th
Gv 658 su amber 5th
G 697 sol white 5th
G^ 735 si red 5th
G# 774 lo blue 6th
Ab 813 le green 6th
Av 852 lu amber 6th
A 890 la yellow 6th
A^ 929 li red 6th
A# 968 ta blue 7th
Bb 1006 te green 7th
Bv 1045 tu jade 7th
B 1084 ti yellow 7th
B^ 1123 to red 7th
Cv 1161 da amber octave
C 1200 do white octave
The solfege system is explained at the following link:
The new undecimal intervals require two new colors, "amber" and "jade." The colors must appear in pairs since the inversion of an amber interval must be jade, just as the inversion of a yellow interval must be green. Notice that jade is visually similar to green -- and in fact, green intervals are converted to jade by raising them 55/54. Yellow intervals are converted to amber by lowering them 55/54.
We also see that all of the septimal intervals are distinguished in 31EDO. So the harmonic 7th is equal to neither the Pythagorean minor seventh nor the septimal supermajor sixth. But the harmonic 7th must still be played as A#, just as in 19EDO. The harmonic 11th is played as F^.
As usual, here is a song played in 31EDO: