140edo: Difference between revisions

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In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap|53 & 87}} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

=== Higher-limit behaviour ===

It is very strong as a high-limit/no-limit system, performing generally very well for its size in extremely high [[odd-limit]]s like 125 but also being a good choice for all odd limits 41 through 51. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than {{nowrap|0.5\140 {{=}} ~4.28{{c}}}} of error, but almost always less than {{nowrap|1\140 {{=}} 8.57{{c}} of error}}*), even though there is far more that it gets right. It is especially notable as a tuning of [[degrees]] (with 1\20 period), [[decoid]] (with 1\10 period) and [[thunderclysmic]] (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of [[5edo]].

<nowiki>*</nowiki> In fact, in the full 125-odd-limit, according to the 113-limit [[patent val]], there is only 10 interval pairs that are mapped with more than 1\140 of error, and they are all intervals of 112 = 121, due to 11 being relatively flat. They are: 121/118, 121/114, 121/109, 121/93, 121/89, 121/83, 121/81, 121/79, 121/73, 121/62 (and their octave-complements). This is remarkable because there is an astounding 1600 interval pairs in the 125-odd-limit. As for inconsistencies, there is 374 inconsistent interval pairs in the 125-odd-limit out of 1600, or around 23%. If we omit intervals of 121, it drops to 329/1543, or around 21%.

If we use the ~~[[val]]~~ {{~~val| 140 223 325 394~~ }} (~~140bbd~~) we ~~obtain a tuning~~ for ~~[[porcupine]] temperament~~; the ~~generator 19\140~~ is 0.~~023 cents flat~~ of ~~the [[POTE generator]]~~.

We can algorithmically determine a tonality diamond it does well in using [[User:Godtone]]'s function <code>orderedapproximator</code> to find odd harmonics in the order in which they are most form-fitting relative to each-other. If we use odds 1 through 15 as a starting point (accepting 140edo's tuning of the 13-limit), then the worst odds are 31, 47, 57, 59, 73, 79, 81, 83, 89, 93, 109, 121. Omitting these from the 125-odd-limit gives us that less than less than 7% of interval pairs are inconsistent (68/987), with only two, 55/54 and 54/53, having more than 6{{cent}} of error (meaning being more than 70% of a step of 140edo out of tune). If we explicitly assume an approximately just tendency on average, by having only odd 1 in the starting set, we get this same set of 12 odds as being the most out of tune.

==== Explanation of the cutoff ====

The reason for picking 12, rather than less or more, is that odd 27 (which we want to include) is included fairly late; according to <code>orderedapproximator</code> (via both starting sets), it is the 14th worst odd w.r.t. the odds in the 125-odd-limit aggregated so far, with the 13th worst being 43, which can be reasoned as favourable to include, as it is only slightly more sharp than 27 so that 43/27 is very accurate (less than 0.1{{cent}} off).

The reason for not picking less than 12 is that immediately after 43 we add an odd we intuitively expect to be bad: 31 * 3 = 93, as 31 is (relatively) very sharp so that we don't want to compound it with the slightly sharp 3, and it's very complex as well so that this damage is more likely to matter. After that is 47, a large prime that is quite sharp, and after that is 121, which we deduced as being very bad, and after that is 83 and 79 (two very large primes which are relatively very off), and after that is 3 * 19 = 57, another case of a sharp prime we don't want to composite with 3. So it seems to, in both directions, be a fairly natural cutoff points.

=== Prime harmonics ===

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140edo can be seen as a [[hemipyth]] analogue of 70edo, which has no exact semifourth or semisixth despite admitting [[interseptimal interval]]s. The slightly sharpened approximation of [[Pythagorean tuning]] given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after [[53edo]] when approximating {{nowrap|log2(3/2)/log2(4/3) {{=}} ~1.409}} as {{nowrap|√2 {{=}} ~1.414…}}, though the theoretical significance is unclear.

=== Misc. ===

If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].

== Approximation to JI ==

@@ Line 5: / Line 5: @@
 In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap|53 &amp; 87}} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
+=== Higher-limit behaviour ===
 It is very strong as a high-limit/no-limit system, performing generally very well for its size in extremely high [[odd-limit]]s like 125 but also being a good choice for all odd limits 41 through 51. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than {{nowrap|0.5\140 {{=}} ~4.28{{c}}}} of error, but almost always less than {{nowrap|1\140 {{=}} 8.57{{c}} of error}}*), even though there is far more that it gets right. It is especially notable as a tuning of [[degrees]] (with 1\20 period), [[decoid]] (with 1\10 period) and [[thunderclysmic]] (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of [[5edo]].
 <nowiki>*</nowiki> In fact, in the full 125-odd-limit, according to the 113-limit [[patent val]], there is only 10 interval pairs that are mapped with more than 1\140 of error, and they are all intervals of 11<sup>2</sup> = 121, due to 11 being relatively flat. They are: 121/118, 121/114, 121/109, 121/93, 121/89, 121/83, 121/81, 121/79, 121/73, 121/62 (and their octave-complements). This is remarkable because there is an astounding 1600 interval pairs in the 125-odd-limit. As for inconsistencies, there is 374 inconsistent interval pairs in the 125-odd-limit out of 1600, or around 23%. If we omit intervals of 121, it drops to 329/1543, or around 21%.
-If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
+We can algorithmically determine a tonality diamond it does well in using [[User:Godtone]]'s function <code>orderedapproximator</code> to find odd harmonics in the order in which they are most form-fitting relative to each-other. If we use odds 1 through 15 as a starting point (accepting 140edo's tuning of the 13-limit), then the worst odds are 31, 47, 57, 59, 73, 79, 81, 83, 89, 93, 109, 121. Omitting these from the 125-odd-limit gives us that less than less than 7% of interval pairs are inconsistent (68/987), with only two, 55/54 and 54/53, having more than 6{{cent}} of error (meaning being more than 70% of a step of 140edo out of tune). If we explicitly assume an approximately just tendency on average, by having only odd 1 in the starting set, we get this same set of 12 odds as being the most out of tune.
+==== Explanation of the cutoff ====
+The reason for picking 12, rather than less or more, is that odd 27 (which we want to include) is included fairly late; according to <code>orderedapproximator</code> (via both starting sets), it is the 14th worst odd w.r.t. the odds in the 125-odd-limit aggregated so far, with the 13th worst being 43, which can be reasoned as favourable to include, as it is only slightly more sharp than 27 so that 43/27 is very accurate (less than 0.1{{cent}} off).
+The reason for not picking less than 12 is that immediately after 43 we add an odd we intuitively expect to be bad: 31 * 3 = 93, as 31 is (relatively) very sharp so that we don't want to compound it with the slightly sharp 3, and it's very complex as well so that this damage is more likely to matter. After that is 47, a large prime that is quite sharp, and after that is 121, which we deduced as being very bad, and after that is 83 and 79 (two very large primes which are relatively very off), and after that is 3 * 19 = 57, another case of a sharp prime we don't want to composite with 3. So it seems to, in both directions, be a fairly natural cutoff points.
 === Prime harmonics ===
@@ Line 21: / Line 27: @@
 edo can be seen as a [[hemipyth]] analogue of 70edo, which has no exact semifourth or semisixth despite admitting [[interseptimal interval]]s. The slightly sharpened approximation of [[Pythagorean tuning]] given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after [[53edo]] when approximating {{nowrap|log<sub>2</sub>(3/2)/log<sub>2</sub>(4/3) {{=}} ~1.409}} as {{nowrap|√2 {{=}} ~1.414…}}, though the theoretical significance is unclear.
+=== Misc. ===
+If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
 == Approximation to JI ==