to incorporate Chapter 6 of Eugenia Cheng's Beyond Infinity.
This is what Theoni Pappas writes on page 183 of her Magic of Mathematics:
"Scientists have also observed oscillations from the sun which they surmise produce vibrations that are in various periods. Are musical scales needed to produce music?"
I'm not sure how to answer that question, but I do have another scale to discuss here. But it's not a scale that appears in Pappas -- the 19-note scale.
So far, we've seen that the 17EDO scale incorporates septimal intervals and the 7th harmonic, but it's difficult to play intervals involving the 5th harmonic in that scale. The simplest scale that allows us to play both the 5th and 7th harmonic tolerably is 19EDO.
Let's take a look at the notes of 19EDO:
Regarding 3-limit intervals, a just perfect fifth is 702 cents, and so the 11th note, 695, is close enough to the fifth. Two perfect fifths add up to an octave plus a tone of 189, which is in between the major and minor tones (hence is considered to be a "meantone").
Regarding 5-limit intervals, a just major third is 386 cents, and so the sixth note, 379, is close enough to the third (certainly more accurate than in 12EDO). The just minor third is 316 cents, and look -- the fifth note is almost exactly just!
Regarding 7-limit intervals, a harmonic seventh is 969 cents, and so the 15th note, is close enough to the seventh (certainly more accurate than in 12EDO). This is not quite as accurate as 17EDO, but 19EDO makes up for it with its superior thirds.
Let's try to fill in some ratios -- which is much easier, since we have both 5- and 7-limit available:
Degree Ratio Name
0 1/1 Perfect unison
1 25/24, 21/20 (Septimal) chromatic semitone
2 15/14, 16/15 (Septimal) diatonic semitone
3 9/8, 10/9 Major second
4 7/6, 8/7 Septimal major second/minor third
5 6/5, 25/21 (Tempered) minor third
6 5/4 Major third
7 32/25, 9/7 Diminished fourth, septimal major third
8 4/3 Perfect fourth
9 25/18, 7/5 Augmented fourth, septimal diminished fifth
10 36/25, 10/7 Diminished fifth, septimal augmented fourth
11 3/2 Perfect fifth
12 25/16, 14/9 Augmented fifth, septimal minor sixth
13 8/5 Minor sixth
14 5/3, 42/25 (Tempered) major sixth
15 7/4, 12/7 Septimal major sixth/minor seventh
16 9/5, 16/9 Minor seventh
17 15/8, 28/15 (Septimal) major seventh
18 48/25, 40/21 (Septimal) diminished octave
19 2/1 Perfect octave
As usual, the following site provided the ratios:
These notes can be given names, as follows (from the same link above):
Degree Cents Solfege Color
C 0 do white unison
C# 63 di blue 2nd
Db 126 ra green 2nd
D 189 re yellow 2nd
D# 253 ri/ma red 2nd/blue 3rd
Eb 316 me green 3rd
E 379 mi yellow 3rd
E#/Fb 442 mo red 3rd
F 505 fa white 4th
F# 568 fi blue 5th
Gb 632 se red 4th
G 695 sol white 5th
G# 758 lo blue 6th
Ab 821 le green 6th
A 884 la yellow 6th
A# 947 li/ta red 6th/blue 7th
Bb 1011 te green 7th
B 1074 ti yellow 7th
B#/Cb 1137 da red 7th
C 1200 do white octave
Here are a few things that we notice here:
-- C# is now lower than Db. (C# is higher than Db in 17EDO, and the same note in 12EDO.) The reason for this is the circle of fifths. If the fifth is higher than 700 cents then C# is higher than Db, if the fifth is lower than 700 cents then C# is lower than Db, and if the fifth is exactly 700 cents then C# and Db are enharmonic.
-- On the other hand, we have two additional notes (as compared to 17EDO). One of these is E#/Fb, and the other is B#/Cb. Notice that E# is enharmonic to Fb only in 19EDO. (In 12EDO, E# is enharmonic to F, and in 17EDO, E# is enharmonic to Gb.)
-- The septimal (7/4) and Pythagorean (16/9) minor sevenths are equivalent in 12EDO and 17EDO, but not in 19EDO. Instead, the harmonic seventh equals the septimal major sixth (12/7).
-- The names "septimal diminished fifth" and "septimal augmented fourth" appear to be in the wrong order, but they aren't. The septimal diminished fifth (7/5) is made up of a just (6/5) and a septimal (7/6) minor third, and two minor thirds add up to a diminished fifth, not an augmented fourth,
-- With the colors, "white" means perfect, "green" means minor, and "yellow" means major. In addition, we now have "red" for septimal major and "blue" for septimal minor. The color "blue" was chosen because septimal minor intervals sound "bluesy."
In fact, here is a link to an actual 19EDO keyboard that can be played right on the computer:
I prefer using the QWERTY option to the mouse option, because then we can play chords:
C major triad: Z-C-B (plays as C-E-G)
D major triad: X-G-N (plays as D-F#-A)
Db major triad: E-V-U (plays as Db-F-Ab)
D minor triad: X-V-N (plays as D-F-A)
and so on. Now in 12EDO, we would add a Bb to a C major triad to form a dominant seventh. But let's try playing the following chords:
Z-C-B-I (plays as C-E-G-Bb)
Z-C-B-J (plays as C-E-G-A#)
To me, C-E-G-A# sounds much more consonant. This is because the A# functions as the harmonic seventh, so the chord sounds like a 4-5-6-7 chord. In 17EDO, Bb is the harmonic seventh rather than A#, and in 12EDO, A# and Bb are enharmonic. We notice the same difference with:
D dominant seventh: X-G-N-< (sounds as D-F#-A-C)
D harmonic seventh: X-G-N-K (sounds as D-F#-A-B#)
One more harmonic seventh is playable in 19EDO: D# harmonic seventh, which is D-Y-J-<. This sounds as (D#-Fx-A#-Bx), where x is double-sharp. We'd like to get rid of those double-sharps, but we must remember 19EDO enharmonics: Fx is equal to Gb (not G) and Bx is equal to C (not C#), so the resulting chord is (D#-Gb-A#-C).
Here's a YouTube link to a guitarist who refretted his guitar to accommodate 19EDO:
The musician at this link states that 19EDO may have been used on Baroque keyboards. Some musicians believe that 19EDO is the future of music, since one can play harmonic seventh chords like C-E-G-A# mentioned above.
Oh, and let's answer the question from Pappas earlier:
"Are musical scales needed to produce music? If they were, how would birds sing?"
That definitely gives us something to think about....