Don Page comma: Difference between revisions

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~~By a~~ '''Don Page comma''' is ~~meant~~ a [[comma]] computed from two other intervals by the method suggested by the Don Page paper, [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small'']. If ''a'' and ''b'' are two rational numbers greater than 1, define ~~{{nowrap| ''~~r'' {~~{=}} (~~(''a'' - 1)(''b'' + 1)) / (~~(''~~b'' - 1~~)(''a'' + 1)~~) }}. ~~Suppose~~ ''r'' reduced to lowest terms is ''p''/''q'', ~~and ''a'' and ''b'' are written in [[monzo]] form as '''u''' and '''v'''. Then~~ the Don Page comma is defined as DPC(''a'', ''b'') = ''q'''''u''' - ''p'''''v''', or else minus that if the size in [[cent]]s is less than zero. The reason for introducing monzos is purely numerical; if monzos are not used the numerators and denominators of the Don Page comma quickly become so large they cannot easily be handled. In ratio form, the Don Page comma can be written {{nowrap| ''a''''q''/''b''''p'' }}, or the reciprocal of that if ~~that~~ is less than 1.

A '''Don Page comma''' or '''bimodular comma''' is a [[comma]] computed from two other intervals by the method suggested by the Don Page paper, [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small'']. If ''a'' and ''b'' are two rational numbers greater than 1, define <math>r=\frac{\left(a-1\right)\left(b+1\right)}{\left(a+1\right)\left(b-1\right)}</math>. If we write ''r'' reduced to lowest terms as ''p''/''q'', the Don Page comma is defined as DPC(''a'', ''b'') = {{nowrap| ''a''''q''/''b''''p'' }}, or else the reciprocal of that if it is less than 1. We may also express it in monzo form as ''q'''''u''' - ''p'''''v''' for ''a'' and ''b'' written in [[monzo]] form as '''u''' and '''v'''

== Bimodular approximants ==

If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The bimodular approximant function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') = ''x'' + ''x''3/''x'' + x5/5 + …, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''2 + O(''x''3) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''2 + O(''x''3) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)2/4 + O(x3) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.

If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The bimodular approximant function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') {{=}} ''x'' + ''x''3/''x'' + ''x''5/5 + … }}, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''2 + O(''x''3) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''2 + O(''x''3) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)2/4 + O(''x''3) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.

If ''r'' is as above we have that {{nowrap| ''r'' {{=}} bim(''a'')/bim(''b'') }}, and depending on common factors the corresponding Don Page comma is equal to an ''n''-th power of {{nowrap| ''a''bim(''b'') / ''b''bim(''a'') {{=}} mib('''u''')'''v'''/mib('''v''')'''u''' }} for some ''n''. If we set ''a'' = 1 + ''x'', ''b'' = 1 + ''y'', then ''r'' = ''r''(''x'', ''y'') is an analytic function of two complex variables with a power series expansion around {{nowrap| ''x'' {{=}} 0 }}, {{nowrap| ''y'' {{=}} 0 }}. This expansion begins as ''r''(''x'', ''y'') = 1 - (''xy''3 - ''x''3''y'')/24 + (3''xy''4 + ''x''2''y''3 - ''x''3''y''2 - 3''x''4''y'')/48 + …, with its first nonconstant term of total degree four, and so when ''x'' and ''y'' are small, ''r''(''x'', ''y'') will be close to 1. The ''n''-th power will still be close to 1, and if there are common factors in the denominators of the exponents, the resulting comma will be less complex (lower in height), and this is the really interesting case. For example, if ''a'' = 7/6 and ''b'' = 27/25, we obtain (7/6)1/26/(27/25)1/13, the lcm of 13 and 26 is 26, and taking the 26th power gives us 4375/4374. The common factor of 13 leads to the resulting comma (4375/4374) being relatively simple.

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What is going on here becomes clearer if we shift to BMC rather than DPC. If bim(''x'') was an exact logarithmic function rather than an approximation, then the Don Page commas would all be 1. They measure the deviation between an approximate relationship between intervals and an exact one. For example, {{nowrap| bim(11/9) {{=}} 1/10 }} and {{nowrap| bim(3/2) = 1/5 }}, and two 11/9 intervals fall short of 3/2 by {{nowrap|(3/2)/(11/9)2 {{=}} BMC(1/10, 1/5) {{=}} 243/242 }}. Not all relationships between intervals of this sort arise from bimodular approximation. The syntonic comma, 81/80, is how much two 9/8 intervals exceed 5/4, and how much two 10/9 intervals fall short of it. But {{nowrap| bim(10/9) {{=}} 1/19 }} and {{nowrap| bim(9/8) {{=}} 1/17 }}, neither of which will add up to {{nowrap| bim(5/4) {{=}} 1/9 }}. Instead {{nowrap| mib(1/18) {{=}} 19/17 }} will give {{nowrap| BMC(1/18, 1/9) {{=}} 1445/1444 }}, a whole other deal. To get 81/80, note that {{nowrap| bim(4/3) {{=}} 1/7 }} and {{nowrap| bim(9/5) {{=}} 2/7 }}, and {{nowrap| BMC(1/7, 2/7) {{=}} 81/80 }}.

For {{nowrap| n > 1 }} {{nowrap| BMC(1/''n'', 1/(2''n'')) }} goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939…, with {{nowrap| BMC(1/13, 1/26) }} being our example 4375/4374. Similarly, {{nowrap| BMC(1/''n'', 1/(3''n'')) }} goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565, …, and {{nowrap| BMC(2/''n'', 3/''n'') }} goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125, ….

For {{nowrap| ''n'' > 1 }} {{nowrap| BMC(1/''n'', 1/(2''n'')) }} goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939…, with {{nowrap| BMC(1/13, 1/26) }} being our example 4375/4374. Similarly, {{nowrap| BMC(1/''n'', 1/(3''n'')) }} goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565, …, and {{nowrap| BMC(2/''n'', 3/''n'') }} goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125, ….

We might also note that successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a some tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, not surprising when we note that in this case the first non-constant term of the one-variable expansion is of order five.

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-By a '''Don Page comma''' is meant a [[comma]] computed from two other intervals by the method suggested by the Don Page paper, [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small'']. If ''a'' and ''b'' are two rational numbers greater than 1, define {{nowrap| ''r'' {{=}} ((''a'' - 1)(''b'' + 1)) / ((''b'' - 1)(''a'' + 1)) }}. Suppose ''r'' reduced to lowest terms is ''p''/''q'', and ''a'' and ''b'' are written in [[monzo]] form as '''u''' and '''v'''. Then the Don Page comma is defined as DPC(''a'', ''b'') = ''q'''''u''' - ''p'''''v''', or else minus that if the size in [[cent]]s is less than zero. The reason for introducing monzos is purely numerical; if monzos are not used the numerators and denominators of the Don Page comma quickly become so large they cannot easily be handled. In ratio form, the Don Page comma can be written {{nowrap| ''a''<sup>''q''</sup>/''b''<sup>''p''</sup> }}, or the reciprocal of that if that is less than 1.
+A '''Don Page comma''' or '''bimodular comma''' is a [[comma]] computed from two other intervals by the method suggested by the Don Page paper, [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small'']. If ''a'' and ''b'' are two rational numbers greater than 1, define <math>r=\frac{\left(a-1\right)\left(b+1\right)}{\left(a+1\right)\left(b-1\right)}</math>. If we write ''r'' reduced to lowest terms as ''p''/''q'', the Don Page comma is defined as DPC(''a'', ''b'') = {{nowrap| ''a''<sup>''q''</sup>/''b''<sup>''p''</sup> }}, or else the reciprocal of that if it is less than 1. We may also express it in monzo form as ''q'''''u''' - ''p'''''v''' for ''a'' and ''b'' written in [[monzo]] form as '''u''' and '''v'''
 == Bimodular approximants ==
-If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The bimodular approximant function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') = ''x'' + ''x''<sup>3</sup>/''x'' + x<sup>5</sup>/5 + …, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)<sup>2</sup>/4 + O(x<sup>3</sup>) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.
+{{Todo|clarify|inline=1|text=rewrite GWS gibberish}}
+If ''x'' is near to 1, then ln(''x'')/2 is approximated by {{nowrap| bim(''x'') {{=}} (''x'' - 1)/(''x'' + 1) }}, the bimodular approximant function, which is the {{w|Padé approximant}} of order (1, 1) to ln(''x'')/2 near 1. The bimodular approximant function is a {{w|Möbius transformation}} and hence has an inverse, which we denote {{nowrap| mib(''x'') {{=}} (1 + ''x'')/(1 - ''x'') }}, which is the (1, 1) Padé approximant around 0 for exp(2''x''). Then {{nowrap| bim(exp(2''x'')) {{=}} tanh(''x'') }}, and therefore {{nowrap| ln(mib(''x''))/2 {{=}} artanh(''x'') {{=}} ''x'' + ''x''<sup>3</sup>/''x'' + ''x''<sup>5</sup>/5 + … }}, from which it is apparent that bim(''x'') approximates ln(''x'')/2, and mib(''x'') approximates exp(2''x''), to the second order; we may draw the same conclusion by directly comparing the series for exp(2''x'') = 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) with {{nowrap| mib(''x'') {{=}} 1 + 2''x'' + 2''x''<sup>2</sup> + O(''x''<sup>3</sup>) }} and {{nowrap| ln(''x'')/2 {{=}} (''x'' - 1)/2 - (''x'' - 1)<sup>2</sup>/4 + O(''x''<sup>3</sup>) }}, which is the same to the second order as bim(''x''). Using mib, we may also define {{nowrap| BMC(''a'', ''b'') = DPC(mib(''a''), mib(''b'')) }}, where BMC is an acronym for ''bimodular comma''.
 If ''r'' is as above we have that {{nowrap| ''r'' {{=}} bim(''a'')/bim(''b'') }}, and depending on common factors the corresponding Don Page comma is equal to an ''n''-th power of {{nowrap| ''a''<sup>bim(''b'')</sup> / ''b''<sup>bim(''a'')</sup> {{=}} mib('''u''')<sup>'''v'''</sup>/mib('''v''')<sup>'''u'''</sup> }} for some ''n''. If we set ''a'' = 1 + ''x'', ''b'' = 1 + ''y'', then ''r'' = ''r''(''x'', ''y'') is an analytic function of two complex variables with a power series expansion around {{nowrap| ''x'' {{=}} 0 }}, {{nowrap| ''y'' {{=}} 0 }}. This expansion begins as ''r''(''x'', ''y'') = 1 - (''xy''<sup>3</sup> - ''x''<sup>3</sup>''y'')/24 + (3''xy''<sup>4</sup> + ''x''<sup>2</sup>''y''<sup>3</sup> - ''x''<sup>3</sup>''y''<sup>2</sup> - 3''x''<sup>4</sup>''y'')/48 + …, with its first nonconstant term of total degree four, and so when ''x'' and ''y'' are small, ''r''(''x'', ''y'') will be close to 1. The ''n''-th power will still be close to 1, and if there are common factors in the denominators of the exponents, the resulting comma will be less complex (lower in height), and this is the really interesting case. For example, if ''a'' = 7/6 and ''b'' = 27/25, we obtain (7/6)<sup>1/26</sup>/(27/25)<sup>1/13</sup>, the lcm of 13 and 26 is 26, and taking the 26th power gives us 4375/4374. The common factor of 13 leads to the resulting comma (4375/4374) being relatively simple.
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 What is going on here becomes clearer if we shift to BMC rather than DPC. If bim(''x'') was an exact logarithmic function rather than an approximation, then the Don Page commas would all be 1. They measure the deviation between an approximate relationship between intervals and an exact one. For example, {{nowrap| bim(11/9) {{=}} 1/10 }} and {{nowrap| bim(3/2) = 1/5 }}, and two 11/9 intervals fall short of 3/2 by {{nowrap|(3/2)/(11/9)<sup>2</sup> {{=}} BMC(1/10, 1/5) {{=}} 243/242 }}. Not all relationships between intervals of this sort arise from bimodular approximation. The syntonic comma, 81/80, is how much two 9/8 intervals exceed 5/4, and how much two 10/9 intervals fall short of it. But {{nowrap| bim(10/9) {{=}} 1/19 }} and {{nowrap| bim(9/8) {{=}} 1/17 }}, neither of which will add up to {{nowrap| bim(5/4) {{=}} 1/9 }}. Instead {{nowrap| mib(1/18) {{=}} 19/17 }} will give {{nowrap| BMC(1/18, 1/9) {{=}} 1445/1444 }}, a whole other deal. To get 81/80, note that {{nowrap| bim(4/3) {{=}} 1/7 }} and {{nowrap| bim(9/5) {{=}} 2/7 }}, and {{nowrap| BMC(1/7, 2/7) {{=}} 81/80 }}.
-For {{nowrap| n > 1 }} {{nowrap| BMC(1/''n'', 1/(2''n'')) }} goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939…, with {{nowrap| BMC(1/13, 1/26) }} being our example 4375/4374. Similarly, {{nowrap| BMC(1/''n'', 1/(3''n'')) }} goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565, …, and {{nowrap| BMC(2/''n'', 3/''n'') }} goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125, ….
+For {{nowrap| ''n'' > 1 }} {{nowrap| BMC(1/''n'', 1/(2''n'')) }} goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939…, with {{nowrap| BMC(1/13, 1/26) }} being our example 4375/4374. Similarly, {{nowrap| BMC(1/''n'', 1/(3''n'')) }} goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565, …, and {{nowrap| BMC(2/''n'', 3/''n'') }} goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125, ….
 We might also note that successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a some tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, not surprising when we note that in this case the first non-constant term of the one-variable expansion is of order five.