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 mnumerator/ndenominator 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 multiple telicities in different primes. For example, 12edo is 3-2 telic and 5-2 telic, but only multitelic in the 3-2. To be more precise, 12edo is 5-2 telic, 2-strong 3-2 telic.

Pseudotelicity

If the p2-p1-telic circle in an edN reaches other (normally higher) primes [p3] with less than 25% error, then the edN is p3;p2-p1-pseudotelic. To cite some examples:

In 53edo, since the primes 5 and 7 connect to the telic chain of fifths with less than 25% error, then it is 7-5/3-2 pseudotelic. The prime 5 itself can be stacked four times, but not the 7, so it is more precisely 7-5⁴;3-2 pseudotelic.

In 159edo, since the prime 11, stacked thrice, connects to the telic chain of fifths with less than 25% error, so it is 11³-3-2 pseudotelic.

Examples

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

The commas that arise from these edos are the following, with the corresponding:

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.

A naïve way find if an edo is p-2 telic, multiply the relative error of that prime by the edo. If the error is less than 50%, it is 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...

Adding telic edos together results in semiconvergents when the smaller one is added to the bigger one or multiples thereof, like 94 = 53 + 41; 17 = 12 + 5. However, adding the smaller convergent more than once, or subtracting it, also results in edos with good fifths and possibly great properties. For this reason, the term "Continued Fraction Deviant" [CFD] may be used to extend the notion of semiconvergents, into adding telic edos together more generally.

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, non-telic P-2 convergent, and semiconvergent edos

3-2 telic 2 5 12 [24] 53 [159]

3-k

306 [612] 665 [7315]

11-k

15601 [78005] 31867 79335 190537

28-k

5-2 telic 3 [12]

4-k

28 59 [118] 146

2-k

643

3-k

4004 8651 12655 21306

2-k

97879

9-k

7-2 telic 5 [10]

2-k

26 [130]

2-k

109

2-k

571

2-k

2964

15-k

91313

2-k

453601

4-k

11-2 telic 2

3-k

13 [26]

2-k

37

13-k

986 1935 4856 16503

12-k

13-2 telic 3 7 10 [50, 80, 130, 270]

11-k

227 [908]

11-k

5458

4-k

54353

74-k

Table of 3-convergency
CFD-2 CFD-1 Convergent CFD+1 CFD+2 CFD+3
1
(1) 2 3 4 (5)
(1) 3 5 7 9 11
6 8 10 (12) (14) 16
(2) (7) 12 17 22 27
14 19 24 29 34 39
26 31 36 (41) 46 51
(17) (29) 41 (53) 65 77
(12) 53 94 135 176
(24) (65) 106 147 188 229
(77) 118 159 200 241 282
130 171 212 253 294 335
183 224 265 (306) 347 388
(200) (253) 306 359 412 465
506 559 612 (665) 718 771
(53) (359) 665 971 1277 1583
718 1024 1330 1636 1942 2248
1383 1689 1995 2301 2607 3003

1edo and 0edo excluded. Bolded telic edos are notable. Edos in brackets are notable telic edo multiples, due to high consistency or accuracy of their intervals. If italicized, the edo is a multiple of the telic edo but beyond its k-strength range.

Of these, the only edos (that I know so far) that are telic in many primes are 3edo (13-5-2 telic), 5edo (7-3-2 telic), 12edo (5-3-2 telic), and 26edo (11-7-2 telic).

WIP

See also

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