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In mathematics, a '''square [[superparticular]] ratio''', also called a ''square-particular ratio'', is the ratio of two consecutive integer numbers whose higher one is a square number (3:4, 8:9, 15:16...). | |||
== Sk (square-particulars) == | == Sk (square-particulars) == | ||
A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form | A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form | ||
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=== Table of 1/3-square-particulars === | === Table of 1/3-square-particulars === | ||
Below is a table of such commas in the 41-limit: | Below is a table of such commas in the 41-limit: | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
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1/n-square-particulars continue the pattern (1/2-square-particulars, 1/3-square-particulars, ...) to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1). | 1/n-square-particulars continue the pattern (1/2-square-particulars, 1/3-square-particulars, ...) to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1). | ||
In other words, each and every S-expression of a comma as a 1/n-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where EG 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]). | In other words, each and every S-expression of a comma as a 1/n-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where EG 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]). | ||
These commas are important in a few ways: | These commas are important in a few ways: | ||
1. As a generalization of important special cases n=0, n=1 and n=2, (which are almost all superparticular; the only case where they aren't is that n=3 (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas. | 1. As a generalization of important special cases n=0, n=1 and n=2, (which are almost all superparticular; the only case where they aren't is that n=3 (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas. | ||
2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example ([[81/80]])/([[91/90]]) = S81 * S82 * ... * S90 = [[729/728]] = S27. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for S81 * S82 * ... * S90 and thus more generally S''a''..''b'' for S''a'' * S(''a'' + 1) * ... * S''b''. | 2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example ([[81/80]])/([[91/90]]) = S81 * S82 * ... * S90 = [[729/728]] = S27. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for S81 * S82 * ... * S90 and thus more generally S''a''..''b'' for S''a'' * S(''a'' + 1) * ... * S''b''. | ||
3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities. | 3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities. | ||
4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/n-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss. | 4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/n-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss. | ||
5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to... | 5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to... | ||
6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss. | 6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss. | ||
=== Significance/implications for consistency === | === Significance/implications for consistency === | ||
1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1) have implications for the [[consistency]] of the (''k'' + ''n'')-[[odd-limit]] when tempered. Specifically: | 1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1) have implications for the [[consistency]] of the (''k'' + ''n'')-[[odd-limit]] when tempered. Specifically: | ||
If a temperament tempers a 1/''n''-square-particular of the form S''k''*S(''k''+1)*...*S(''k''+''n''-1), it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', S(''k''+1), ..., S(''k''+''n''-1). If it does not, it is ''necessarily'' inconsistent (more formally & weakly, not monotonic) in the (''k'' + ''n'')-odd-limit(*). A proof is as follows: | If a temperament tempers a 1/''n''-square-particular of the form S''k''*S(''k''+1)*...*S(''k''+''n''-1), it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', S(''k''+1), ..., S(''k''+''n''-1). If it does not, it is ''necessarily'' inconsistent (more formally & weakly, not monotonic) in the (''k'' + ''n'')-odd-limit(*). A proof is as follows: | ||
Consider the following sequence of superparticular intervals, all of which in the (''k'' + ''n'')-odd-limit: | Consider the following sequence of superparticular intervals, all of which in the (''k'' + ''n'')-odd-limit: | ||
(''k'' + ''n'')/(''k'' + ''n'' - 1), (''k'' + ''n'' - 1)/(''k'' + ''n'' - 2), ..., (''k'' + 1)/''k'', ''k''/(''k'' - 1) | (''k'' + ''n'')/(''k'' + ''n'' - 1), (''k'' + ''n'' - 1)/(''k'' + ''n'' - 2), ..., (''k'' + 1)/''k'', ''k''/(''k'' - 1) | ||
Because of tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1), we require that (''k'' + ''n'')/(''k'' + ''n'' - 1) = ''k''/(''k'' - 1) consistently. Therefore, if any superparticular ''x''/(''x'' - 1) imbetween (meaning ''k'' + ''n'' > x > ''k'') is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true: | Because of tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1), we require that (''k'' + ''n'')/(''k'' + ''n'' - 1) = ''k''/(''k'' - 1) consistently. Therefore, if any superparticular ''x''/(''x'' - 1) imbetween (meaning ''k'' + ''n'' > x > ''k'') is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true: | ||
mapping((''k'' + ''n'')/(''k'' + ''n'' - 1)) > mapping(''x''/(''x'' - 1)) | mapping((''k'' + ''n'')/(''k'' + ''n'' - 1)) > mapping(''x''/(''x'' - 1)) | ||
mapping(''k''/(''k'' - 1)) < mapping(''x''/(''x'' - 1)) | mapping(''k''/(''k'' - 1)) < mapping(''x''/(''x'' - 1)) | ||
Therefore any superparticular interval ''x''/(''x'' - 1) between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the (''k'' + ''n'')-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering 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 (''k'' - 1)-odd-limit. | Therefore any superparticular interval ''x''/(''x'' - 1) between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the (''k'' + ''n'')-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering 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 (''k'' - 1)-odd-limit. | ||
(* Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.) | (* Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.) | ||
=== Table of 1/4-square-particulars === | |||
Below is a table of 1/4-square-particulars in the 23-limit: | |||
{| class="wikitable center-all | |||
|- | |||
! S-expression | |||
! Interval relation | |||
! Ratio | |||
! Prime limit | |||
|- | |||
| S2*S3*S4*S5 | |||
| ([[2/1]])/([[6/5]]) | |||
| [[5/3]] | |||
| 5 | |||
|- | |||
| S3*S4*S5*S6 | |||
| ([[3/2]])/([[7/6]]) | |||
| [[9/7]] | |||
| 7 | |||
|- | |||
| S4*S5*S6*S7 | |||
| ([[4/3]])/([[8/7]]) | |||
| [[7/6]] | |||
| 7 | |||
|- | |||
| S5*S6*S7*S8 | |||
| ([[5/4]])/([[9/8]]) | |||
| [[10/9]] | |||
| 5 | |||
|- | |||
| S6*S7*S8*S9 | |||
| ([[6/5]])/([[10/9]]) | |||
| [[27/25]] | |||
| 5 | |||
|- | |||
| S7*S8*S9*S10 | |||
| ([[7/6]])/([[11/10]]) | |||
| [[35/33]] | |||
| 11 | |||
|- | |||
| S8*S9*S10*S11 | |||
| ([[8/7]])/([[12/11]]) | |||
| [[22/21]] | |||
| 11 | |||
|- | |||
| S9*S10*S11*S12 | |||
| ([[9/8]])/([[13/12]]) | |||
| [[27/26]] | |||
| 13 | |||
|- | |||
| S10*S11*S12*S13 | |||
| ([[10/9]])/([[14/13]]) | |||
| [[65/63]] | |||
| 13 | |||
|- | |||
| S11*S12*S13*S14 | |||
| ([[11/10]])/([[15/14]]) | |||
| [[77/75]] | |||
| 11 | |||
|- | |||
| S12*S13*S14*S15 | |||
| ([[12/11]])/([[16/15]]) | |||
| [[45/44]] | |||
| 11 | |||
|- | |||
| S13*S14*S15*S16 | |||
| ([[13/12]])/([[17/16]]) | |||
| [[52/51]] | |||
| 17 | |||
|- | |||
| S14*S15*S16*S17 | |||
| ([[14/13]])/([[18/17]]) | |||
| [[119/117]] | |||
| 17 | |||
|- | |||
| S15*S16*S17*S18 | |||
| ([[15/14]])/([[19/18]]) | |||
| [[135/133]] | |||
| 19 | |||
|- | |||
| S16*S17*S18*S19 | |||
| ([[16/15]])/([[20/19]]) | |||
| [[76/75]] | |||
| 19 | |||
|- | |||
| S17*S18*S19*S20 | |||
| ([[17/16]])/([[21/20]]) | |||
| [[85/84]] | |||
| 17 | |||
|- | |||
| S18*S19*S20*S21 | |||
| ([[18/17]])/([[22/21]]) | |||
| [[189/187]] | |||
| 17 | |||
|- | |||
| S19*S20*S21*S22 | |||
| ([[19/18]])/([[23/22]]) | |||
| [[209/207]] | |||
| 23 | |||
|- | |||
| S20*S21*S22*S23 | |||
| ([[20/19]])/([[24/23]]) | |||
| [[115/114]] | |||
| 23 | |||
|- | |||
| S21*S22*S23*S24 | |||
| ([[21/20]])/([[25/24]]) | |||
| [[126/125]] | |||
| 7 | |||
|- | |||
| S22*S23*S24*S25 | |||
| ([[22/21]])/([[26/25]]) | |||
| [[275/273]] | |||
| 13 | |||
|- | |||
| S23*S24*S25*S26 | |||
| ([[23/22]])/([[27/26]]) | |||
| [[299/297]] | |||
| 23 | |||
|- | |||
| S24*S25*S26*S27 | |||
| ([[24/23]])/([[28/27]]) | |||
| [[162/161]] | |||
| 23 | |||
|- | |||
| S35*S36*S37*S38 | |||
| ([[35/34]])/([[39/38]]) | |||
| [[665/663]] | |||
| 19 | |||
|- | |||
| S36*S37*S38*S39 | |||
| ([[36/35]])/([[40/39]]) | |||
| [[351/350]] | |||
| 13 | |||
|- | |||
| S45*S46*S47*S48 | |||
| ([[45/44]])/([[49/48]]) | |||
| [[540/539]] | |||
| 11 | |||
|- | |||
| S46*S47*S48*S49 | |||
| ([[46/45]])/([[50/49]]) | |||
| [[1127/1125]] | |||
| 23 | |||
|- | |||
| S51*S52*S53*S54 | |||
| ([[51/50]])/([[55/54]]) | |||
| [[1377/1375]] | |||
| 17 | |||
|- | |||
| S52*S53*S54*S55 | |||
| ([[52/51]])/([[56/55]]) | |||
| [[715/714]] | |||
| 17 | |||
|- | |||
| S65*S66*S67*S68 | |||
| ([[65/64]])/([[69/68]]) | |||
| [[1105/1104]] | |||
| 23 | |||
|- | |||
| S66*S67*S68*S69 | |||
| ([[66/65]])/([[70/69]]) | |||
| [[2277/2275]] | |||
| 23 | |||
|- | |||
| S77*S78*S79*S80 | |||
| ([[77/76]])/([[81/80]]) | |||
| [[1540/1539]] | |||
| 19 | |||
|- | |||
| S81*S82*S83*S84 | |||
| ([[81/80]])/([[85/84]]) | |||
| [[1701/1700]] | |||
| 17 | |||
|- | |||
| S92*S93*S94*S95 | |||
| ([[92/91]])/([[96/95]]) | |||
| [[2185/2184]] | |||
| 23 | |||
|- | |||
| S96*S97*S98*S99 | |||
| <font style="font-size:0.94em">([[96/95]])/([[100/99]])</font> | |||
| [[2376/2375]] | |||
| 19 | |||
|- | |||
| <font style="font-size:0.79em">S221*S222*S223*S224</font> | |||
| <font style="font-size:0.79em">([[221/220]])/([[225/224]])</font> | |||
| [[12376/12375]] | |||
| 17 | |||
|} | |||
== Sk/S(k + 1) (ultraparticulars) == | == Sk/S(k + 1) (ultraparticulars) == | ||
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|} | |} | ||
The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, | The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27. | ||
This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page, with S12/S13 and S13/S14 expected to have their own pages soon. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go. | This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page, with S12/S13 and S13/S14 expected to have their own pages soon. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go. | ||
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[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||