User:Eufalesio/Telicity

This is a major rewrite of the "Telicity" article, as it was incredibly mathed up and very hard to parse and understand what is meant to be explained. So I stepped in. (Thanks to Aura for suggestions and help)

There's still math, but much less math. And also continued fractions are important.

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ARTICLE START

Telicity is a property of both equal temperaments and commas and how they relate to each other. An edo is p-2 telic when it tempers a comma in 2.p subgroup for a prime p, and that comma is smaller than half an edostep.

Commas and equal temperaments that demonstrate this property are referred to as as being telic. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be multitelic.

Telicity and Continued fractions

In order to understand how and why telicity is useful, one must first look at continued fractions to see how telicity is derived.

m-n telicity in any equal division of an equave satisfies the following:

The equal division of m is a denominator appearing in the continued fraction of logm(n).
The comma that arises from stacking m^{numerator}/n^{denominator} of the convergent is smaller than half an ed-m-step.

Mathematically, this is satisfied with the following:

[math]\displaystyle{ d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2} }[/math]

Multitelicity

If said produced comma is also smaller than k/2 of an ed-m-step, then the edm is k-strong m-n telic, which means that the comma is smaller than not only half of an ed-m-step, but also half/2 (a quarter), or half/3 (a sixth)... etc. Essentially not only the {denominator}ed-m is convergent, but also its multiples. This makes it multitelic.

Mathematically, this is expressed as the following:

[math]\displaystyle{ k\cdot d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2} }[/math]

Examples

Here is the series of convergents for log2(3):

1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53, 485/306, 1054/665, 24727/15601, 50508/31867, 125743/79335, 176251/111202...

The commas that arise from these edos are the following:

1edo: 4/3 [bixby, degenerate case]
2edo: 9/8 [antitonic, degenerate case]
5edo: 256/243 [blackwood]
12edo: 531441/524288 [compton]
41edo: [65 -41⟩ [countercomp]
53edo: [-84 53⟩ [mercator]
306edo: [485 -306⟩ [sasktel?]
665edo: [-1054 665⟩ [satanic?]

Of those, 41edo fails to be telic because the comma is larger than half an edostep. (19.845*2 > 29.268).

Of those, 12, 53, 665 are multitelic, because they have a k-strong value greater than one; being 2, 3, and 11 respectively, which means that 24, 106, 159, 1330, 1995, 2660, 3325, 3990, 4655, 5320, 5985, 6650, 7315. are also 3-2 telic.

Applications

Prime approximations

3-2 telic edos have record-breakingly accurate perfect fifths. As well as 5-2 telic edos having record-breakingly accurate ptolemaic major thirds, and so on. These telic edos can stack their optimized intervals extremely well with minimal error, being perfect for incredibly intricate modulations, and overall because they have ridiculously good approximations of intervals.

Non-telic edos with convergent fifths, or semiconvergent fifths (which are never telic), are also incredibly good.

MOS

MOS scales generated by a pure prime interval have strictly proper scales, with the softest hardness when they have a scale size that corresponds with a telic edo. For Pythagorean tuning these include 1L 1s (monowood), 2L 3s (pentic), 5L 7s (p-chromatic), 41L 12s. Scale sizes for edos that have semiconvergent or non-telic convergent generators may generate proper but not strictly proper, or improper scales (I hypothesize this)

Table of P-2 telic edos

WIP