Theosophy
in Spirituality ·
Paul Ross2 said:If anyone finds this interesting, feel free to read—if not, just skip it; it might be a bit tedious:An arithmetic ratio—which is essentially comparing two measurable quantities—is a specific instance of a standard logical operation. That operation involves:
1° Perceiving the functional relationship or the hierarchy of values between two objects.
2° Distinguishing or comparing those values, whether qualitatively or quantitatively.
If the ratio serves as a quantitative measure, it takes the form and properties of an arithmetic or algebraic fraction, $a/b$.
A proportion is simply the equivalence of two ratios—an analogous relationship between two comparisons, namely $a/b = c/d$.
A geometric proportion can consist of any number of terms:
a) A discontinuous proportion where $a/b = c/d = e/f = \dots$
b) A continuous proportion where $a/b = b/c = c/d \dots$
The ratio $a/b$ could be compound $(a+b)/b$, divided $(a-b)/b$, or inverted $a/(a-b)$.
From these, one can construct ten distinct types of proportions:
1° Multiple proportions.
2° Superparticular (the whole plus a part of the whole, divided by the whole), for example:
- Sesquialtera $(x + 1/2x)/x$ (a fifth)
- Sesquitertial $(x + 1/3x)/x$ (a fourth)
3° Superbipartite (the whole plus two parts of the whole, divided by the whole)—$5/3$.
Supertripartite (the whole plus three parts of the whole, divided by the whole)—$7/4$.
4° Multiple superparticular, such as:
Double sesquialtera $(2x+1/2x)/x$
Triple sesquialtera $(3x+1/2x)/x$
5° Multiple superbipartite.
6° - 10° Their respective inversions, such as submultiple...
The Pythagoreans back in Sicily had already established the three most significant types of proportions that can be expressed algebraically as:
$b-a = d-c$ — arithmetic proportion ($1, 2, 3$)
$a/b = c/d$ — continuous geometric ($1, 2, 4$)
$1/b - 1/a = 1/d - 1/c$ — harmonic ($2, 3, 6$)
These three proportions can be represented as:
$(c-b)/(b-a) = c/c = 1$
$(c-b)/(b-a) = c/b$
$(c-b)/(b-a) = c/a$
Alongside these, there remain seven others:
$(b-a)/(c-b) = c/a$ ($3, 5, 6$)
$(c-a)/(c-b) = c/a$ ($6, 7, 9$)
$(b-a)/(c-b) = b/a$ ($2, 4, 5$)
$(c-a)/(b-a) = b/a$ ($4, 6, 7$)
$(b-a)/(c-b) = c/b$ ($1, 4, 6$)
$(c-a)/(c-b) = b/a$ ($3, 5, 8$) — Fibonacci
$(c-a)/(b-a) = c/a$
---
The relationship between the means and the extremes:
Arithmetic mean: $(a+b)/2$
Geometric mean: $\sqrt{a \cdot b}$
Harmonic mean: $2ab / (a+b) = 2 / (1/a + 1/b)$
Up until this point, I suppose everything I’ve read from you regarding the intersection of music and mathematics has been focused on the numerical side of things. You know, numbers, ratios, quantitative data... you’ve mentioned before that math is a spiritual discipline just as much as music is, and that viewing mathematics through an exclusively numerical lens is quite limiting, which I think is a very fair point. I wonder, though, if you could perhaps elaborate on that more? Could you try to bridge that spiritual dimension of mathematics with the spiritual essence found in music?
Or maybe looking at it this way: take a solo by Charlie Parker, let's say this one again:
Is there a way to analyze this using mathematics? I mean, why does it resonate so deeply? To my ears, I hear these completely unpredictable rhythms; in those faster passages, he seems to be playing slightly ahead of the beat, then lagging behind, or sometimes he just 'swallows' a note, implying it without actually playing it. He finds this rhythmic unpredictability by varying his dynamics and accents, yet somehow, the whole thing remains perfectly in pocket... it feels more like a language to me. Parker isn't just playing notes; he’s speaking and singing. He’s conveying a message, a spiritual message, and I guess that’s why it sounds so incredible. What kind of mathematics would you even use to analyze something like that?
I'm genuinely curious, so I figured I'd ask. 🙂
And yeah, I realize this is a bit off-topic, but if we allow ourselves a slightly broader interpretation, I really believe it ties back into spirituality, and perhaps even into Theosophy.