Part 2 of 5

Part 2 – Numbers to Structure
Organizing the values to reveal a consistent system

Not a Scale – A System of Markers

Something different begins to emerge. The Puleo numbers don’t behave like a conventional musical scale. Instead of forming a complete set of notes, they appear at specific points within a larger structure. A helpful way to think of them is as reference points. They highlight where important relationships exist, rather than defining all of the relationships themselves.

This becomes clearer when looking at the intervals They come very close to multiples of eleven, but not exactly—indicating the structure is being approximated rather than fully expressed.

To bring the structure into focus, the values can be adjusted slightly. These adjustments stay within about 2 Hz, which is small enough that the ear cannot reliably distinguish the difference.

Both the notes and the intervals settle into consistent multiples of eleven.

Intervals show the distance between notes. The size of the step changes, but each step is still an interval. Although this process is purely numerical, it produces the same sequence of note relationships found in a musical scale.

The bold numbers are the frequencies that have changed.

Three candidates emerged for A^𝄰 /B^𝄬 : 462, 473, and 484.

This produced a fully consistent system.

Just Intonation (JI)

I began to look at different tunings, particularly Just tuning. I found a site by Kyle Gann, a composer and the author of seven books on American music, including books on microtonality, that explained Just Intonation, as well as many other pieces of information on music theory. I also found a page called the anatomy of an octave. This table gives ratios and decimal values for a host of numbers.
This chart shows the Ratios of a Just Intonation Scale. This table will be our reference table for interval names and ratios and the Note Name.

Semitone refers to the interval of the Equal Temperament Scale. Augmented fourth and diminished fifth are values for F ^𝄰 and there are several values for the 7^th. The Harmonic minor Seventh’s companion is 8/7 and is not a ratio within today’s music because our music is limited to factors no larger than the prime number 5. The ratios of 8/7 and 7/4 are ratios in the Lamdoma scale (discussed later) where the limit for factors is the prime number 7.

A key property of Just Intonation is symmetry through reciprocal ratios.

• The Perfect Fourth (4/3) pairs with the Perfect Fifth (3/2) – 4/3 x 3/2 = 2.
  • Four (fourth) + Five  (fifth) = 9.
• The Minor Third (6/5) pairs with the Major Sixth (5/3) – 6/5 x 5/3 = 2.
  • Three (third) + Six  (sixth) = 9.
• The Major Third (5/4) pairs with the Minor Sixth (8/5) – 5/4 x 8/5 = 2.
  • Three (third) + Six  (sixth) = 9.

The Perfect Fourth / Perfect Fifth along with The Thirds / Sixes paired together Major to Minor, Minor to Major. Numbers add to 9 while the product of the ratios yields 2.

We now take the Just Intonation reference table and add the Frequencies and decimal equivalent for the ratios for each note base on C=264 Hz. C=264 was chosen because that is the vale of C in the Light Scale.

F𝄰 is not listed in this table. The JI ratio is 45/32 with a frequency of 371.25.

Alternative Just Scale

Now I wanted to know how well the Light Scale compares to the Just Intonation Scale, When I tried to create a Just Scale using 264 as the fundamental, I ran into some limitations.

• The final frequency needed to be a whole number, because I wanted to keep the pattern of whole numbers with factors of 11 and intervals that were multiples of 11.
• I wanted to see if there were alternative ratios for a Just Scale.
• The ratios needed to be small, and the denominator (bottom number) needed to be multiples of 2, 3,4,6,8,11 because 264 is only divisible by these numbers. 264 is 11×24, = 11x2x12= 11x3x8= 11x4x6.
• There could be no ratio with a bottom number that had a factor of 5,7 or 9 because 264 is not divisible by these numbers.
• Each of the black note ratios in the above Just Scale had their denominator a number divided by 5 or 9 in the reference  table, in the Just Interval column. Therefore, I needed to find new fractions that would be in the range of the existing fractions and stay with my pattern of whole numbers, and multiples of 11.

Using the Anatomy of an Octave, I was able to find ratios that matched these criteria.
This became the Light Scale chromatic scale.

Notice that A^𝄰/B^𝄬 became 7/4, with new ratios for the other black notes (bolded).
These ratios are alternative ratios for a Just Intonation scale.

Notice that the white notes match the Just tuning for the diatonic scale. It seems like every way we look at the black notes we have a lot of flexibility with the intervals and with the ratios. Also note that pairing of the ratios is no longer an option. The consistency of whole numbers and intervals of eleven was too precise to ignore, so I retained the pattern and allowed the ratios to adjust accordingly. Now I have not forgotten about the elephant in the room, which is A=440. So, I next looked at the equal temperament tuning we have today for the middle C octave.

Continue to part 3