S-expression: Difference between revisions
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=== Table of square-particulars === | === Table of square-particulars === | ||
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|+ style="font-size: 105%;" | 31-limit square-particulars | |||
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! S-expression | ! S-expression | ||
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=== Table of triangle-particulars === | === Table of triangle-particulars === | ||
For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason. | For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason. | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.</ref> | |||
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! S-expression | ! S-expression | ||
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== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1 | == {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} ({{frac|1|3}}-square-particulars) == | ||
This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac| {{sfrac|''k'' − 1|''k'' − 2}} | {{sfrac|''k'' + 2|''k'' + 1}} }}}} which therefore do not (directly) involve the ''k''th harmonic. | This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac| {{sfrac|''k'' − 1|''k'' − 2}} | {{sfrac|''k'' + 2|''k'' + 1}} }}}} which therefore do not (directly) involve the ''k''th harmonic. | ||
=== Significance === | === Significance === | ||
1. Two thirds of all 1 | 1. Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].) | ||
2. They are often implied in a variety of ways by combinations of other commas discussed on this page. | 2. They are often implied in a variety of ways by combinations of other commas discussed on this page. | ||
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3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.) | 3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.) | ||
4. Square-particulars, 1 | 4. Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s. | ||
=== Proof of simplification of 1 | === Proof of simplification of {{frac|1|3}}-square-particulars === | ||
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications: | We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications: | ||
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<math>S(k-1) * Sk * S(k+1) = \frac{9n^2 - 1}{9n^2 - 4}</math> | <math>S(k-1) * Sk * S(k+1) = \frac{9n^2 - 1}{9n^2 - 4}</math> | ||
In other words, what this shows is all 1 | In other words, what this shows is all {{frac|1|3}}-square-particulars of the form {{frac|S(''k'' − 1) * S''k'' * S(''k'' + 1)}} are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd). | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|+ style="font-size: 105%;" | 41-limit {{frac|1|3}}-square-particulars | |||
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! S-expression | ! S-expression | ||
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Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*...*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit. | Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*...*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit. | ||
= | {| class="wikitable center-all\ | ||
|+ style="font-size: 105%;" | 23-limit {{frac|1|4}}-square particulars | |||
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! S-expression | ! S-expression | ||
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| 17 | | 17 | ||
|} | |} | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|+ style="font-size: 105%;" | 23-limit {{frac|1|5}}-square particulars | |||
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! S-expression | ! S-expression | ||
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2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness. | 2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness. | ||
3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} → {{(}}S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>{{)}}}}, and any of the two commas in the latter set imply all the other commas too.) | 3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} → {{(}}S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>{{)}} }}, and any of the two commas in the latter set imply all the other commas too.) | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|+ style="font-size: 105%;" | Ultraparticular ratios | |||
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! S-expression | ! S-expression | ||
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If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so: | If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so: | ||
* [[4/3]] is approximately half of [[9/5]] | * [[4/3]] is approximately half of [[9/5]] | ||
* [[9/7]] is approximately half of [[5/3]](=10/6) | * [[9/7]] is approximately half of [[5/3]] (= 10/6) | ||
* [[5/4]] is approximately half of [[11/7]] | * [[5/4]] is approximately half of [[11/7]] | ||
* [[11/9]] is approximately half of [[3/2]](=12/8) | * [[11/9]] is approximately half of [[3/2]] (= 12/8) | ||
* [[6/5]] is approximately half of [[13/9]] | * [[6/5]] is approximately half of [[13/9]] | ||
* [[13/11]] is approximately half of [[7/5]](=14/10) | * [[13/11]] is approximately half of [[7/5]] (= 14/10) | ||
* [[7/6]] is approximately half of [[15/11]] | * [[7/6]] is approximately half of [[15/11]] | ||
* [[15/13]] is approximately half of [[4/3]](=16/12) | * [[15/13]] is approximately half of [[4/3]] (= 16/12) | ||
* [[8/7]] is approximately half of [[17/13]] | * [[8/7]] is approximately half of [[17/13]] | ||
* [[17/15]] is approximately half of [[9/7]](=18/14) | * [[17/15]] is approximately half of [[9/7]] (= 18/14) | ||
* [[9/8]] is approximately half of [[19/15]] | * [[9/8]] is approximately half of [[19/15]] | ||
* [[19/17]] is approximately half of [[5/4]](=20/16) | * [[19/17]] is approximately half of [[5/4]] (= 20/16) | ||
These properties show a pattern: take some arbitrary [[#Glossary|quodd-particular]] (''k'' + 4)/''k''; observe that we can split it into (''k'' + 4)/(''k'' + 2) * (''k'' + 2)/''k''. | |||
Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''. | Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''. | ||