User:Eufalesio/Telicity: Difference between revisions

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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)

{{Editable user page}}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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== 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.

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

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

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).

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<math>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/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'''.

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>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 ===

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

The commas that arise from these edos are the following:

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

* 1edo: [[4/3]] [<nowiki/>[[Bixby|bixby, degenerate case]]]

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* 5edo: [[256/243]] [<nowiki/>[[blackwood]]]

* 12edo: [[Pythagorean comma|531441/524288]] [<nowiki/>[[compton]]]

* 41edo: [[41-comma|[65 -41⟩]] [<nowiki/>[[Countercomp family|countercomp]]]

* ''41edo: [[41-comma|[65 -41⟩]] [''<nowiki/>''[[Countercomp family|countercomp]]]''

* 53edo: [[Mercator's comma|[-84 53⟩]] [<nowiki/>[[mercator]]]

* 306edo: [[Qian's small comma|[485 -306⟩]] [sasktel?]

* 665edo: [[Satanic comma|[-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, 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-strong value greater than one; being 2, 3, and 11 respectively, which means that [[24edo|24]], [[106edo|106]], [[159edo|159]], [[1330edo|1330]], [[1995edo|1995]], [[2660edo|2660]], [[3325edo|3325]], [[3990edo|3990]], [[4655edo|4655]], [[5320edo|5320]], [[5985edo|5985]], [[6650edo|6650]], [[7315edo|7315]]. are also 3-2 telic.

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=== Prime approximations ===

3-2 telic edos have record-breakingly accurate [[3/2|perfect fifths]]. As well as 5-2 telic edos having record-breakingly accurate [[5/4|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.

3-2 telic edos have record-breakingly accurate [[3/2|perfect fifths]]. As well as 5-2 telic edos having record-breakingly accurate [[5/4|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.

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 [[7edo|7]], [[17edo|17]], [[29edo|29]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[359edo|359]], [[971edo|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|1L 1s (monowood)]], [[2L 3s|2L 3s (pentic)]], [[5L 7s|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)''

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|1L 1s (monowood)]], [[2L 3s|2L 3s (pentic)]], [[5L 7s|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 ===

~~WIP~~

TBA

== See also ==

* [[Consistent circle]]

[[Category:EDO theory pages]]

[[Category:Terms]]

@@ Line 1: / Line 1: @@
-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)
+{{Editable user page}}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?
 ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
@@ Line 12: / Line 14: @@
 == 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.
+In order to understand how and why telicity is useful, one must first look at continued fractions to see how telicity can be derived.
-m-n telicity in any equal division of an equave satisfies the following:
+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).
@@ Line 22: / Line 24: @@
 <math>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/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'''.
+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>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 ===
@@ Line 35: / Line 41: @@
 * 1/[[1edo|1]], 2/1, 3/[[2edo|2]], 8/[[5edo|5]], 19/[[12edo|12]], 65/[[41edo|41]], 84/[[53edo|53]], 485/[[306edo|306]], 1054/[[665edo|665]], 24727/[[15601edo|15601]], 50508/[[31867edo|31867]], 125743/[[79335edo|79335]], 176251/[[111202edo|111202]]...
-The commas that arise from these edos are the following:
+The commas that arise from these edos are the following, with the corresponding :
 * 1edo: [[4/3]] [<nowiki/>[[Bixby|bixby, degenerate case]]]
@@ Line 41: / Line 47: @@
 * 5edo: [[256/243]] [<nowiki/>[[blackwood]]]
 * 12edo: [[Pythagorean comma|531441/524288]] [<nowiki/>[[compton]]]
-* 41edo: [[41-comma|[65 -41⟩]] [<nowiki/>[[Countercomp family|countercomp]]]
+* ''41edo: [[41-comma|[65 -41⟩]] [''<nowiki/>''[[Countercomp family|countercomp]]]''
 * 53edo: [[Mercator's comma|[-84 53⟩]] [<nowiki/>[[mercator]]]
 * 306edo: [[Qian's small comma|[485 -306⟩]] [sasktel?]
 * 665edo: [[Satanic comma|[-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, 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-strong value greater than one; being 2, 3, and 11 respectively, which means that [[24edo|24]], [[106edo|106]], [[159edo|159]], [[1330edo|1330]], [[1995edo|1995]], [[2660edo|2660]], [[3325edo|3325]], [[3990edo|3990]], [[4655edo|4655]], [[5320edo|5320]], [[5985edo|5985]], [[6650edo|6650]], [[7315edo|7315]]. are also 3-2 telic.
@@ Line 54: / Line 60: @@
 === Prime approximations ===
--2 telic edos have record-breakingly accurate [[3/2|perfect fifths]]. As well as 5-2 telic edos having record-breakingly accurate [[5/4|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.
+-2 telic edos have record-breakingly accurate [[3/2|perfect fifths]]. As well as 5-2 telic edos having record-breakingly accurate [[5/4|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.
+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 [[7edo|7]], [[17edo|17]], [[29edo|29]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[359edo|359]], [[971edo|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|1L 1s (monowood)]], [[2L 3s|2L 3s (pentic)]], [[5L 7s|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)''
+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|1L 1s (monowood)]], [[2L 3s|2L 3s (pentic)]], [[5L 7s|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 ===
-WIP
+TBA
 == See also ==
 * [[Consistent circle]]
 [[Category:EDO theory pages]]
 [[Category:Terms]]