## Monday, July 3, 2017

### Pappas Music Post: The 31-Note Scale

This is what Theoni Pappas writes on page 184 of her Magic of Mathematics:

"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:

http://www.rome-tour.co.uk/saint_peters_basilica.htm

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:

Degree      Cents
0               0
1               39
2               77
3               116
4               155
5               194
6               232
7               271
8               310
9               348
10             387
11             426
12             465
13             503
14             542
15             581
16             619
17             658
18             697
19             735
20             774
21             813
22             852
23             890
24             929
25             968
26             1006
27             1045
28             1084
29             1123
30             1161
31             1200

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:

http://xenharmonic.wikispaces.com/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#             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
E               387        mi             yellow 3rd
E^             426        mo             red 3rd
Fv             465        fe               blue 4th
F               503        fa               white 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
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:

http://xenharmonic.wikispaces.com/31edo+solfege

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: