Marcus,
OK.
The answer is "not by a single number or at least not very usefully by a single number", I agree. Even the single note values that I gave you could be reduced. Four more octaves of reduction if we stay in our range of hearing (by 1/2 value every octave). There is no scientific reason to stop there, and we know this goes further. In fact, it can be taken to its' Natural smallest unit, right?
"Is that a Planck in your pants, or are you just glad to be talking about RESONANCE?"
Yet, when one begins to approach mastery in making chords, this "single #" becomes second nature to us. Unmistakable. Automatic. Fundamental.
What is happening here? Certainly something as "predictable" as chord progression deserves its' own math. The Calculus of Gravity. Mere lack of "existance" didn't slow Newton down in communicating his vision, and he left us to ponder the results of this new vision.
When I drop into conversations between people who are studying the physical world, they seem to all be talking about this. Yet, they do not know how to derive this new math. So let us not wander from this topic without some experimentation.
By using RATIOs like 5/4, 4/3, 2/1, it allows the model to be folded up and taken with and communicated with other musicians. If we were using math instead of chords, we would use the decimal numbers that these ratios represent: 1.25, 1.33, 1.5, etc.
First, a word on the Root, or tonic note. This is the General Relativity of music scales. Simply put, the rule starts from where YOU start. All other particles will wait until there is a tonic note to allow themselves to be defined; until then they ALL have resonant potential. This note fixes the value in time (root), and creates the "perspective" needed to determine resonance.
Step One = f (to be determined by perspective of user)
Step Two = 1.125f
Step Three= 1.25f
Step Four = 1.33f
Step Five = 1.5f
Step Six = 1.66f
Step Seven= 1.875f
Step Eight= 2f
By the way, because in music, we call the values in 1f and 2f the same, it is intuitive to think of these values in angular format, so that you are returning to same start point
. Also note that there are SEVEN major notes, the 8th is just 2x the 1st.
What two groups come immediately to mind when seeing the steps of the scale laid out in "math"? Five #'s are "fast terminating", and whole # fractions of EIGHT. (hence the name "octave") The other two are non-terminating, and whole # fractions of THREE (and do not resonate with the 1/8 values).
Because step Eight has special rules already (it is neutral in relativity to tonic...it is the same, AND it is different), you should be able to see a "hidden" value at step Eight. It is 3/3, to finish the sequence of double end value. In decimals, this is .9999
Step Eight then, is quite a chameleon. It can be 8/8, 3/3, .9999, 2f, and, for the NEXT scale, the next 1f.
How to make a Triad. (chord) Pick starting note (tonic), figure value at step Three (mediant), and step Five (dominant). From previous example: G392, B494, and D588. I used D294 (1 octave up @ 1/2 value) because my finger can reach it! This is not necessary on the piano, it was invented and perfected during the time that the "equidistant" scale was being cooked up around 1690. You have to study the waveform of these 3 notes to fully understand why they resonate. The first note can never be overcome be the others because of the natural speed limit of C. They can only add to its' value (amplitude/pitch) without fundamentally changing the tonic. So the "sort of answer" we both agreed on is something like "a G chord with a base 392 is 392.4597846(yadda yadda)" Only the value to the right of the decimal point would change.
Great! you say, but where does this go?
Let's look at the Triad of light values that we used earlier.
Red 719
Green 539
Blue 453
Now, we too shall jump ship from the Kepler Galley to the Frigate of Andre Werkmeister. The equidistant scale uses 12 equal half-steps of 1.05946 (12sqrt2). In this format, the # 13 (intervals) is used as the neutral value of 2f in 12 half-steps, instead of 8 intervals for 7 whole steps. We will also divide the 7 "whole" colors into 12 even steps.
We know that these 3 values create the Triad we call "white light". From understanding waveforms and resonance, we must assume then, that this triad starts with a value that is white light. It's too bad that Science does not have a value for this. With some logical extensions, we can go 13 half-steps smaller from Red, and end at 359 Planck lengths (365nm), which is the first resonant value of UV in the next octave past visible light. As said before, we can not just dis-allow the values of IR and UV in our search for better understanding. This is saying that the 1/2 value of red, while in proximity, is indistinguishable from red, yet because of Planck speed, can not be caught or changed by the other resonant values. We call this note red because we only perceive ONE octave of color, none are repeated in higher or lower form. If we saw one more color up into UV, we would have no red (as is the case for many other eye bearing life forms). A curved eye (all eyes share common evolution)will always have the same limitations because of the angular format of the wave. Isn't it fascinating that our totally dissimilar ears have the same limitation spread out over 10 octaves?
To shorcut the rest of the story, by taking the natural commonality of vibration, and simply extending the 13 interval scale from music up to the e-7 "row of values" of light, all known colors can be produced and explained by triads of notes (chords).
The commonality of vibration is what I call your system of natural values. Take the "/1sec" out of the equation to simplify, and your left with "planck" values. This is what the chart is made from. Not frequency, not wavelength, yet both are there, under a Planck.
Take a longer look at
https://www.physicsforums.com/showthread.php?s=&threadid=13481
LPF