User:Eufalesio/Telicity

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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.

Feel free to change anything after ARTICLE START. I left out loads of cool bits, but I can't be writing articles all day now can I?

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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 can be derived.

n-m telicity in any equal division of n 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]

When m is equal to 2, the result is an n-2 telic edo, if it's not equal to 2, it's an edonoi.

Multitelicity

If said produced comma is also smaller than k halves 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]

Multitelicity is not the same as having many telicities. For example, 12edo is 3-2 telic and 5-2 telic, but only multitelic in the 3-2.

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, with the corresponding :

• 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 is not telic because its comma, the countercomp comma, is larger than half an edostep. (19.845*2 > 29.268). The next non-telic convergent is 111202edo.

Of those, 12, 53, 665 are multitelic, because they have a k-strength 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, and 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 offer astoundingly great approximations of intervals within their telic subgroups.

Non-telic edos with convergent fifths, or semiconvergent fifths (which are never telic), are also incredibly good and offer comparably great intervals, specially if the edo is big enough. Edos with semiconvergent fifths include 7, 17, 29, 94, 147, 200, 253, 359, 971...

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 never strictly proper, or improper scales (I hypothesize this)

Table of P-2 telic edos

3-2 telic	1	2	5	12 2-strong	53 3-strong	306	665 11-strong	15601	31867	79335	190537 28-strong
5-2 telic		3 4-strong	28	59	146 2-strong	643 3-strong	4004	8651	12655	21306 2-strong	97879 9-strong
7-2 telic		5 2-strong	26 2-strong	109 2-strong	571 2-strong	2694 15-strong	91313 2-strong	453601 4-strong
11-2 telic		2 3-strong	13	37 13-strong	986	1935	4856	16503 12-strong

WIP

See also

• Consistent circle