S-expression: Difference between revisions

Line 8:

Square-particulars are important structurally because they are the intervals between consecutive superparticular intervals while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''n'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward.

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:

Line 15:

<nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.

Furthermore, defining another sequence of commas with [[#Sk/S(k + 2) (semiparticulars)|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.

== S''k''*S(''k'' + 1) (triangle-particulars) ==

If we examine (k+1)/k then we can notice that if we equate (k+2)/(k+1) with k/(k-1), we have:

(k+2)/(k+1) * (k+1)/k = (k+1)/k * k/(k-1)

Which is to say that if we temper S''k''*S(''k'' + 1) = (k/(k-1))/((k+1)/k) * ((k+1)/k)/((k+2)/(k+1)) = (k/(k-1))/((k+2)/(k+1)) then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and S(''k''+1) individually unless other considerations seem to force your hand. Another reason commas of this form are of note is they are always [[superparticular]]. It is also an interesting consequence that if we temper S''k''*S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of k/(k-1), (k+1)/k and (k+2)/(k+1) ''must'' be mapped inconsistently, because if (k+1)/k is mapped above (k+2)/(k+1) ~ k/(k-1) we have (k+1)/k > k/(k-1) and if it is mapped below we have (k+1)/k < (k+2)/(k+1). A short proof of the superparticularity of S''k''*S(''k'' + 1) is as follows:

S''k''*S(''k'' + 1) = (k/(k-1))/((k+2)/(k+1)) = (k(k+1))/((k-1)(k+2)) = (k2 + k)/(k2 + k - 2)

Then notice that k2 + k is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "semi-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these commas (or intervals) are [[triangular number]]s!

For completeness, all the commas of this form are included, because these "commas" (intervals rather) have structural importance for JI, and for the possibility of consistency of mappings for the above reason.

{| class="wikitable center-all

|-

! S-expression

! Interval relation

! Comma

|-

| S2*S3

| ([[3/1]])/([[2/1]])

| [[3/2]]

|-

| S3*S4

| ([[3/2]])/([[5/4]])

| [[6/5]]

|-

| S4*S5

| ([[4/3]])/([[6/5]])

| [[10/9]]

|-

| S5*S6

| ([[5/4]])/([[7/6]])

| [[15/14]]

|-

| S6*S7

| ([[6/5]])/([[8/7]])

| [[21/20]]

|-

| S7*S8 = S4/S6

| ([[7/6]])([[9/8]])

| [[28/27]]

|-

| S8*S9 = S6

| ([[8/7]])/([[10/9]])

| [[36/35]]

|-

| S9*S10

| ([[9/8]])/([[11/10]])

| [[45/44]]

|-

| S10*S11

| ([[10/9]])/([[12/11]])

| [[55/54]]

|-

| S11*S12

| ([[11/10]])/([[13/12]])

| [[66/65]]

|-

| S12*S13

| ([[12/11]])/([[14/13]])

| [[78/77]]

|-

| S13*S14

| ([[13/12]])/([[15/14]])

| [[91/90]]

|-

| S14*S15

| ([[14/13]])/([[16/15]])

| [[105/104]]

|-

| S15*S16

| ([[15/14]])/([[17/16]])

| [[120/119]]

|-

| S16*S17

| (16/15)/(18/17)

| [[136/135]]

|-

| S17*S18

| ([[17/16]])/([[19/18]])

| [[153/152]]

|-

| S18*S19

| ([[18/17]])/([[20/19]])

| [[171/170]]

|-

| S19*S20

| ([[19/18]])/([[21/20]])

| [[190/189]]

|-

| S20*S21

| ([[20/19]])/([[22/21]])

| [[210/209]]

|-

| S21*S22

| ([[21/20]])/([[23/22]])

| [[231/230]]

|-

| S22*S23

| ([[22/21]])/([[24/23]])

| [[253/252]]

|-

| S23*S24

| ([[23/22]])/([[25/24]])

| [[276/275]]

|-

| S24*S25

| ([[24/23]])/([[26/25]])

| [[300/299]]

|-

| S25*S26 = S10/S12

| ([[25/24]])/([[27/26]])

| [[325/324]]

|-

| S26*S27

| ([[26/25]])/([[28/27]])

| [[351/350]]

|}

Also included are some higher-up [[23-limit]] triangle-particulars (as many of the prior intervals were quite large):

{| class="wikitable center-all

|-

! S-expression

! Interval relation

! Comma

|-

| S33*S34

| ([[33/32]])/([[35/34]])

| [[561/560]]

|-

| S34*S35

| ([[34/33]])/([[36/35]])

| [[595/594]]

|-

| S49*S50 = S35

| ([[49/48]])/([[51/50]])

| [[1225/1224]]

|-

| S50*S51

| (50/49)/(52/51)

| [[1275/1274]]

|-

| S55*S56

| ([[55/54]])/([[57/56]])

| [[1540/1539]]

|-

| S64*S65

| ([[64/63]])/([[66/65]])

| [[2080/2079]]

|-

| S76*S77

| ([[76/75]])/([[78/77]])

| [[2926/2925]]

|-

| S169*S170

| (169/168)/(171/170)

| [[14365/14364]]

|-

| S208*S209

| ([[208/207]])/([[210/209]])

| [[21736/21735]]

|}

(Note: after 75, 76, 77, 78, consecutive harmonics in the 23-limit become very sparse. The last few streaks are related to [[311edo]], thus explaining its consistency, as it tempers all commas from [[561/560]] onwards.)

== S''k''/S(''k'' + 1) (ultraparticulars) ==

Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. An arithmetic of ~~rather interesting commas whose elements are~~ square-~~particulars~~ 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.)

Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that 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.

{| class="wikitable center-all

@@ Line 8: / Line 8: @@
 Square-particulars are important structurally because they are the intervals between consecutive superparticular intervals while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''n'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward.
-It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:
+It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:
 <math>{\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3}</math>
@@ Line 15: / Line 15: @@
 <nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.
+Furthermore, defining another sequence of commas with [[#Sk/S(k + 2) (semiparticulars)|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.
+== S''k''*S(''k'' + 1) (triangle-particulars) ==
+If we examine (k+1)/k then we can notice that if we equate (k+2)/(k+1) with k/(k-1), we have:
+(k+2)/(k+1) * (k+1)/k = (k+1)/k * k/(k-1)
+Which is to say that if we temper S''k''*S(''k'' + 1) = (k/(k-1))/((k+1)/k) * ((k+1)/k)/((k+2)/(k+1)) = (k/(k-1))/((k+2)/(k+1)) then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and S(''k''+1) individually unless other considerations seem to force your hand. Another reason commas of this form are of note is they are always [[superparticular]]. It is also an interesting consequence that if we temper S''k''*S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of k/(k-1), (k+1)/k and (k+2)/(k+1) ''must'' be mapped inconsistently, because if (k+1)/k is mapped above (k+2)/(k+1) ~ k/(k-1) we have (k+1)/k > k/(k-1) and if it is mapped below we have (k+1)/k < (k+2)/(k+1). A short proof of the superparticularity of S''k''*S(''k'' + 1) is as follows:
+S''k''*S(''k'' + 1) = (k/(k-1))/((k+2)/(k+1)) = (k(k+1))/((k-1)(k+2)) = (k<sup>2</sup> + k)/(k<sup>2</sup> + k - 2)
+Then notice that k<sup>2</sup> + k is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "semi-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these commas (or intervals) are [[triangular number]]s!
+For completeness, all the commas of this form are included, because these "commas" (intervals rather) have structural importance for JI, and for the possibility of consistency of mappings for the above reason.
+{| class="wikitable center-all
+|-
+! S-expression
+! Interval relation
+! Comma
+|-
+| S2*S3
+| ([[3/1]])/([[2/1]])
+| [[3/2]]
+|-
+| S3*S4
+| ([[3/2]])/([[5/4]])
+| [[6/5]]
+|-
+| S4*S5
+| ([[4/3]])/([[6/5]])
+| [[10/9]]
+|-
+| S5*S6
+| ([[5/4]])/([[7/6]])
+| [[15/14]]
+|-
+| S6*S7
+| ([[6/5]])/([[8/7]])
+| [[21/20]]
+|-
+| S7*S8 = S4/S6
+| ([[7/6]])([[9/8]])
+| [[28/27]]
+|-
+| S8*S9 = S6
+| ([[8/7]])/([[10/9]])
+| [[36/35]]
+|-
+| S9*S10
+| ([[9/8]])/([[11/10]])
+| [[45/44]]
+|-
+| S10*S11
+| ([[10/9]])/([[12/11]])
+| [[55/54]]
+|-
+| S11*S12
+| ([[11/10]])/([[13/12]])
+| [[66/65]]
+|-
+| S12*S13
+| ([[12/11]])/([[14/13]])
+| [[78/77]]
+|-
+| S13*S14
+| ([[13/12]])/([[15/14]])
+| [[91/90]]
+|-
+| S14*S15
+| ([[14/13]])/([[16/15]])
+| [[105/104]]
+|-
+| S15*S16
+| ([[15/14]])/([[17/16]])
+| [[120/119]]
+|-
+| S16*S17
+| (16/15)/(18/17)
+| [[136/135]]
+|-
+| S17*S18
+| ([[17/16]])/([[19/18]])
+| [[153/152]]
+|-
+| S18*S19
+| ([[18/17]])/([[20/19]])
+| [[171/170]]
+|-
+| S19*S20
+| ([[19/18]])/([[21/20]])
+| [[190/189]]
+|-
+| S20*S21
+| ([[20/19]])/([[22/21]])
+| [[210/209]]
+|-
+| S21*S22
+| ([[21/20]])/([[23/22]])
+| [[231/230]]
+|-
+| S22*S23
+| ([[22/21]])/([[24/23]])
+| [[253/252]]
+|-
+| S23*S24
+| ([[23/22]])/([[25/24]])
+| [[276/275]]
+|-
+| S24*S25
+| ([[24/23]])/([[26/25]])
+| [[300/299]]
+|-
+| S25*S26 = S10/S12
+| ([[25/24]])/([[27/26]])
+| [[325/324]]
+|-
+| S26*S27
+| ([[26/25]])/([[28/27]])
+| [[351/350]]
+|}
+Also included are some higher-up [[23-limit]] triangle-particulars (as many of the prior intervals were quite large):
+{| class="wikitable center-all
+|-
+! S-expression
+! Interval relation
+! Comma
+|-
+| S33*S34
+| ([[33/32]])/([[35/34]])
+| [[561/560]]
+|-
+| S34*S35
+| ([[34/33]])/([[36/35]])
+| [[595/594]]
+|-
+| S49*S50 = S35
+| ([[49/48]])/([[51/50]])
+| [[1225/1224]]
+|-
+| S50*S51
+| (50/49)/(52/51)
+| [[1275/1274]]
+|-
+| S55*S56
+| ([[55/54]])/([[57/56]])
+| [[1540/1539]]
+|-
+| S64*S65
+| ([[64/63]])/([[66/65]])
+| [[2080/2079]]
+|-
+| S76*S77
+| ([[76/75]])/([[78/77]])
+| [[2926/2925]]
+|-
+| S169*S170
+| (169/168)/(171/170)
+| [[14365/14364]]
+|-
+| S208*S209
+| ([[208/207]])/([[210/209]])
+| [[21736/21735]]
+|}
+(Note: after 75, 76, 77, 78, consecutive harmonics in the 23-limit become very sparse. The last few streaks are related to [[311edo]], thus explaining its consistency, as it tempers all commas from [[561/560]] onwards.)
 == S''k''/S(''k'' + 1) (ultraparticulars) ==
-Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. An arithmetic of rather interesting commas whose elements are square-particulars 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.)
+Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that 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.
 {| class="wikitable center-all