Fractions and Orbits

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Those fractions!

The ratios for all seven tunings are summarized in Table 6. Although some of the fractions look quite odd, the patterns still show only two distinct intervals between notes, which we are calling A and B. It’s time we looked at where the fractions came from.

We know that the seven fractions in Nis Gabri come directly from the Circle of Fifths, because we went through the steps above. Just as I wanted to find patterns in the intervals between notes, I also wanted to find any patterns between tunings. Playing around, I happened to multiply all the ratios of one tuning by the inverse of the second fraction. Boy, did I get a surprise!

Try it yourself. Take Nis Gabri as a starting point. Multiply each fraction by 8/9. To make a whole octave, you’ll have to drop off the starting 1/1 fraction and add a new 2/1 at the end to complete your octave.

  •    So 9/8 times 8/9 is 1/1, keeping similar notation.
  •    Next, 8/9 times 81/64 is 9/8, the same as we had.
  •    Once more, 8/9 times 729/512 is 81/64. And again, 8/9 times 3/2 is 4/3. Here’s a new fraction that doesn’t exist in Nis Gabri.
  •    8/9 times 27/16 is 3/2.
  •    8/9 times 243/128 is 27/16.
  •    Lastly, 8/9 times 2/1 is 16/9.
  •    We tack on a 2/1 to complete the octave, and what have we got?

We’ve got Mitum! We diddled the numbers so that a new scale started on the second note of our first scale. All we did was simplify fractions, dividing through by 9/8 (or multiplying through by 8/9). I couldn’t believe my eyes, so I repeated the whole process on Mitum, and got Kitmun. The pattern holds true for all seven tunings. You read up each row in Table 6, and when you get to Ishartum, just cycle back down to Nis Gabri and start again. Each tuning is related to each other tuning according to some underlying schema that I didn’t see yet. All I knew was the tunings turned into each other in the simplest manner possible.

Orbits, cycles and the missing notes

Think of a huge Hula Hoop that somehow got bent out of shape so that it’s no longer round. At one point on this hoop is attached a bracelet with seven charms. At another point nearby, there is another bracelet, which has some of the same charms and some new ones. As we progress around the hoop, we find seven bracelets, each of which has seven charms attached, but they are never the same seven – they seem to be picked from a pool, some often, some only once.

The bracelets, of course, represent the seven tunings, and the charms represent the ratios or fractions in each tuning. The rule for selecting the charms to use says start with the single interval 3/2 or 1.5, then pick seven notes. This determines the first set of charms. The other sets are determined by multiplying through by the inverse of any one ratio; this immediately jumps you to another bracelet and determines its charms.

But what does the Hula Hoop represent? What is the grand pattern that determines what the bracelets are, and even how many of them there are. To do this, we have to go back and fill in the missing notes.

Table 4 is repeated below.

Note Ratio Fraction Intervals
C 1.0000 1/1
C#
D 1.1250 9/8 9/8
D#
E 1.2656 81/64 9/8
F 1.4238 729/512 9/8
F#
G 1.5000 3/2 256/243
G#
A 1.6875 27/16 9/8
A#
B 1.8984 243/128 9/8
C’ 2.0000 2/1 256/243

Nis Gabri. Table 4.

We more or less arbitrarily skipped over the notes marked with a # sharp, so as to correspond with the “white notes” on a piano. And initially, there seemed to be no reason to go further. Eventually, though, I got thinking about the Circle of Fifths. There are twelve steps, and there are twelve notes in the modern chromatic scale – both the white and black notes. So I continued multiplying by 3/2, and finally calculated all the ratios and all the fractions, as shown in Table 7.

Note Ratio Fraction Intervals
C 1.0000 1/1
C# 1.0679 2187/2048 I
D 1.1250 9/8 J
D# 1.2014 19683/16384 I
E 1.2656 81/64 J
F 1.3515 177147/131072 I
F# 1.4238 729/512 J
G 1.5000 3/2 J
G# 1.6018 6561/4096 I
A 1.6875 27/16 J
A# 1.8020 59049/32768 I
B 1.8984 243/128 J
C’ 2.0000 2/1 J

The Twelve Notes of Nis Gabri. Table 7.

The fractions get ugly fast, as no amount of multiplying powers of three in the numerator and twos in the denominator will ever reduce to a simple fraction. (Oh, yeah. Did you notice? All the numbers in all the fractions in all the tunings are a power of two or three.)

But here I was anyway, again looking for patterns. Lo and behold, here’s another one! And this is the Queen of them all, hiding the lost Pythagorean Comma, no less.

The first interval, that between C and C#, is 2187/2048. Expressed as a decimal this is 1.0679 (rounded). Call this number I.

The next interval is between C# and D, and is 1.0535 (rounded). Expressed as a fraction, this is 256/243. Call this number J.

But the interval between D and D# is again I. In fact, all twelve of the intervals are either I or J. I occurs 5 times; J occurs 7 times. And as a little math check to make sure we haven’t made a mistake somewhere,

I5 * J7 = 2.0000

In other words, multiplying all the intervals together should equal one octave exactly, which they do.