「スーパーパーティキュラー」の版間の差分
Dummy index (トーク | 投稿記録) 編集の要約なし |
Dummy index (トーク | 投稿記録) |
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== 一般化 == | == 一般化 == | ||
Taylorは一般化した用語について記述している。 | Taylorは一般化した用語について記述している。 | ||
* （実のところ ''n'' | * （実のところ ''n'' の値ひとつづつに対応した用語があるのだが省略） | ||
* ''superbipartient'' (or ''odd-particulars''（''隣接奇数比''）) 分子割る分母が1余り2である、つまりデルタ2比のうち 5/3 以降が該当する。 | * ''superbipartient'' (or ''odd-particulars''（''隣接奇数比''）) 分子割る分母が1余り2である、つまりデルタ2比のうち 5/3 以降が該当する。 | ||
* ''supertripartient'' (or ''throdd-particulars'') 分子割る分母が1余り3である、つまりデルタ3比のうち 7/4 以降が該当する。 | * ''supertripartient'' (or ''throdd-particulars'') 分子割る分母が1余り3である、つまりデルタ3比のうち 7/4 以降が該当する。 | ||
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Generalisation in the "meta" direction gives rise to [[square superparticular]]s and then [[ultraparticular]]s, under the idea that if a superparticular is the difference between two adjacent harmonics then a square superparticular is the difference between two adjacent superparticulars and an ultraparticular is the difference between two adjacent square superparticulars. This gives rise to descriptions of infinite comma families of which many known commas are examples. A notable property is that just as "all [[superpartient ratio]]s can be constructed as products of [consecutive] superparticular numbers", all ratios between two superparticular intervals (e.g ([[8/7]])/([[11/10]]) = 80/77) can be constructed as a product of consecutive [[square superparticular]] numbers (e.g [[64/63]] * [[81/80]] * [[100/99]] = S8 * S9 * S10), for the same algebraic reason as in the corresponding case of [[superpartient ratio]]s. (There is a corresponding analogy with ultraparticulars too, for the same reason.) | Generalisation in the "meta" direction gives rise to [[square superparticular]]s and then [[ultraparticular]]s, under the idea that if a superparticular is the difference between two adjacent harmonics then a square superparticular is the difference between two adjacent superparticulars and an ultraparticular is the difference between two adjacent square superparticulars. This gives rise to descriptions of infinite comma families of which many known commas are examples. A notable property is that just as "all [[superpartient ratio]]s can be constructed as products of [consecutive] superparticular numbers", all ratios between two superparticular intervals (e.g ([[8/7]])/([[11/10]]) = 80/77) can be constructed as a product of consecutive [[square superparticular]] numbers (e.g [[64/63]] * [[81/80]] * [[100/99]] = S8 * S9 * S10), for the same algebraic reason as in the corresponding case of [[superpartient ratio]]s. (There is a corresponding analogy with ultraparticulars too, for the same reason.) | ||
<pre> | |||
unit fraction (= one part) 1/1 1/2 1/3 1/4 1/5 1/6 1/7 | |||
superparticular 2/1 3/2 4/3 5/4 6/5 7/6 | |||
square superparticular 4/3 9/8 16/15 25/24 36/35 | |||
ultraparticular 32/27 135/128 128/125 875/864 | |||
</pre> | |||
== See also == | == See also == | ||