Saturday, July 1, 2017

Pappas Music Post: The 17-Note Scale

This is what Theoni Pappas writes on page 182 of her Magic of Mathematics:

"During the 6th century B.C., Pythagoras and the Pythagoreans were the first to associate music and mathematics. The Pythagoreans believed that numbers, in some way, governed all things."

I've already written about Pythagorean music. We've seen that this music is 3-limit, based on perfect fourths, fifths, and octaves. Later on, Ptolemy and Didymus extended 3-limit to 5-limit music, so that we could add major and minor thirds. The next natural step, then, would be for us to extend 5-limit to 7-limit music. But what would this sound like?

Meanwhile, Pappas writes about another ancient scale:

"The Persian scale divided the octave into either 17 or 22 notes."

In yesterday's post, I wrote about the 22-note Indian scale, and most likely the Persian 22-note scale would have sounded similar. So today, we will look at the 17-note scale.

In my research of the 17-note Persian scale, I'm finding conflicting information. And so instead of trying to look up the scale, let's actually look at 17EDO and try to figure it out ourselves:

Degree      Cents
0               0
1               71
2               141
3               212
4               282
5               353
6               424
7               494
8               565
9               635
10             706
11             776
12             847
13             918
14             988
15             1059
16             1129
17             1200

By the way, what's so frustrating is that, while I don't have any instrument that can actually play this sort of microtonal scale, I might have had one at one time.

Back in my August 29th post, I wrote about the old computers I had back in the 1980's, when I was a young child. I used those computers to program in BASIC.

Now one command that my computer recognized was the SOUND command. This command took two parameters -- one for the pitch, the other for its length. Each parameter could fall in the range from 1 to 255. It was easy to figure out the length parameter -- one unit of time was about 6/100 second, and so the maximum possible length of the sound was about 15 seconds.

But the big mystery is the relationship between the first parameter and the note that was played. For example, I never knew what parameter corresponded to middle C -- nor did I even attempt to determine what it was.

A few years later, I upgraded my computer. In addition to the SOUND command, there was now a PLAY command that could play actual notes, like C, D, E, and so on. With PLAY, I never tried to used SOUND to create music, as PLAY was so much easier to use.

But we can attempt to use PLAY to reconstruct how SOUND might have worked. According to the instruction manual, PLAY had a range of five octaves. Since SOUND could play 255 different notes, this suggests that there were 51 notes per octave, or 51EDO. If we were to assume that Note 1 was a C, then Notes 52, 103, 154, and 205 were also C's. According to the manual, the second octave contained Middle C, so this implies that Note 52 was Middle C.

Of course, if Note 52 was C, then we know that Note 53 couldn't be C#, nor could Note 54 be D, since the next highest C wasn't until Note 103. With 51 notes per octave, this implies that a semitone was about four notes, so if Note 52 was C, then C# would be around Note 56.

But then what would Notes 53, 54, and 55 be? Back then, it never occurred to me that there could be microtones between C and C#. I probably would have thought that Note 53 was also a C, and so was Note 54, and then maybe Note 55 was when it turned into C#. I had believed that musical notes naturally divided themselves into 12 notes per octave.

Now, of course, I realized that I could have played microtonal music in 51EDO on my old computer, but I no longer own that computer. It's possible, of course, that the computer might have been set to 48EDO instead of 51EDO, as 12EDO is a subset of 48EDO. Then the range would be slightly more than five octaves, but at least the notes of 12EDO could be exactly played.

On the other hand, if my computer played true 51EDO, then 17EDO is a subset of 51EDO. So I could possibly have played Persian tunes on my computer -- tunes which would have been unplayable on my 12EDO keyboard or guitar. Imagine the possibilities!

Anyway, let's return to the 17EDO scale. We see that the 10th note of the scale is 706 cents, which is not far from a perfect fifth (702 cents), and so the 7th note of the scale approximates a perfect fourth.

But unfortunately, our major and minor thirds are missing. We want our major third to be 386 cents, but the closest available note is Note 5, which is 33 cents too low. Meanwhile, we want our minor third to be 316 cents, but again, neither Note 4 nor Note 5 is particularly close. In fact, Note 5 is almost exactly halfway between a minor third and a major third.

The solution is to skip the fifth harmonic and go straight to the seventh harmonic. So we can now consider ratios involving the number 7.

The first interval we will consider is 7/4. As it's based on the seventh harmonic, we can call this note the name "harmonic seventh." It's also known as "septimal minor seventh." Since we can convert cents to ratios by using exponents, it follows that we can convert ratios to cents by using logarithms:

log_2 (7/4) = log(7/4) / log(2/1) = 0.8074

that is, a harmonic seventh is 0.8074 octave. Since an octave is 1200 cents:

(0.8074)(1200) = 969

that is, a harmonic seventh is 969 cents. The closest note of 17EDO is Note 14, which is 19 cents too low, but this is better than the just major or minor thirds of 17EDO. Moreover, it's better

So we then create a scale by replacing all of the 5-limit intervals with their septimal equivalents:

Degree      Ratio        Cents      Name
0                1/1           0             perfect unison
3                9/8, 8/7    212         (septimal) major second
4                7/6           282         septimal minor third
6                9/7           424         septimal major third
7                4/3           494         perfect fourth
10              3/2           706         perfect fifth
11              14/9         776         septimal minor sixth
13              12/7         918         septimal major sixth
14              7/4, 16/9  988         (septimal) minor seventh
17              2/1           1200       perfect octave

We see that there are two new thirds -- the septimal minor third (7/6) and the septimal major third (9/7), while the ordinary major and minor thirds are unavailable. Meanwhile, the septimal minor seventh (7/4) is the same as the Pythagorean minor seventh (16/9). The same thing happens in 12EDO, which is why we often use the note Bb to represent the harmonic seventh.

Unfortunately, there are many intervals that are missing ratios. It's not quite as simple as the 22-note Indian scale, where we already had a list of all the correct ratios. Here are a few suggestions for the missing tones as given at the following website:

-- Note 1 is almost exactly 25/24, the just chromatic semitone. According to the link above, even though we can't use 5 in the ratio, we're allowed to use 25.
-- Note 2 is 243/224, a complex 7-limit semitone (analogous to 256/243 in 3-limit).
-- Note 5 is 49/40, a 7-limit third falling between the minor and major thirds. Hence we can call it a "neutral third," similar to the Chinese scale.
-- Note 8 is 25/18, which is a perfect fourth (4/3) raised by a chromatic semitone (25/24).

It's also possible to give these notes letter names:

Name       Cents
C              0
Db            71
C#            141
D              212
Eb            282
D#            353
E              424
F              494
Gb            565
F#            635
G              706
Ab            776
G#            847
A              918
Bb            988
A#            1059
B              1129
C              1200

Notice that C# and Db are different notes in 17EDO -- in fact, we can show that C# and Db are enharmonic only in 12EDO. It might be awkward to make C# higher than Db, but this is done in order to make the circle of fifths (or fourths) work out. In particular, we see that both F-Bb and F#-B are perfect fourths in this system, as they should be.

Here's a link to a YouTube video in which Persian music is played:

The musician plays a scale he calls "Abu-Ata." The notes of the Abu-Ata scale are given as:


where the note E* is not E, but is a lower note instead. Notice that had that note been a genuine E, the resulting scale would simply a D natural minor scale.

Anyway, the musician in the video refers to the E note as "E quarter flat." The term "quarter flat" suggests that it is 1/2 of a semitone (not 1/4 of a semitone), so it would be about 50 cents flat from where we expect E to be. Often this is rendered in ASCII as:


Let's rewrite the 17EDO scale from above, except we'll start it on D (as Abu-Ata starts on D):

Name       Cents
D              0
Eb            71
D#            141
E              212
F              282
Gb            353
F#            424
G             494
Ab           565
G#            635
A              706
Bb            776
A#            847
B              918
C              988
Db            1059
C#            1129
D              1200

We see that there really is a note between Eb and E here. Instead of D#, let's call it Ed (so in 17EDO, D# and Ed are enharmonic), and we'll rewrite the list with quarter flats:

Name       Cents
D              0
Eb            71
Ed            141
E              212
F              282
Gb            353
Gd            424
G             494
Ab           565
Ad            635
A              706
Bb            776
Bd            847
B              918
C              988
Db            1059
Dd            1129
D              1200

We can see why the name Ed (E quarter flat) is especially fitting -- its value of 141 cents really is about halfway between two semitones of 12EDO. On the other hand, all of the other notes of the Abu-Ata scale are within 24 cents of a 12EDO semitone.

So what exactly would the interval D-Ed represent? We see that 141 cents is about 243/224, but that seems a bit complex to include in a scale. Other nearby 7-limit intervals are 27/25 (at 133 cents) and 49/45 (at 147 cents).

On the other hand, what interval would D-F be? In 17EDO, D-F is a septimal minor third, but there's no reason that this can't be the just minor third instead. After all, there's no reason to assume that the 17EDO interval accurately captures the size of this interval.

This scale definitely merits more investigation. If only I still had that old computer that could possibly plat microtonal music!

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