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]].

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|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]].

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]].

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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 &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]].
-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|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]].
+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]].