Difference between revisions of "Directory:Jon Awbrey/Papers/Riffs and Rotes"

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{{DISPLAYTITLE:Riffs and Rotes}}
 
{{DISPLAYTITLE:Riffs and Rotes}}
__TOC__
+
<div class="nonumtoc">__TOC__</div>
  
 
==Idea==
 
==Idea==
  
Let <math>\text{p}_i</math> be the <math>i^\text{th}</math> prime, where the positive integer <math>i</math> is called the ''index'' of the prime  <math>\text{p}_i</math> and the indices are taken in such a way that <math>\text{p}_1 = 2.</math>  Thus the sequence of primes begins as follows:
+
Let <math>\text{p}_i\!</math> be the <math>i^\text{th}\!</math> prime, where the positive integer <math>i\!</math> is called the ''index'' of the prime  <math>\text{p}_i\!</math> and the indices are taken in such a way that <math>\text{p}_1 = 2.\!</math>  Thus the sequence of primes begins as follows:
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
Line 21: Line 21:
 
|}
 
|}
  
The prime factorization of a positive integer <math>n</math> can be written in the following form:
+
The prime factorization of a positive integer <math>n\!</math> can be written in the following form:
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
| <math>n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},</math>
+
| <math>n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},\!</math>
 
|}
 
|}
  
where <math>\text{p}_{i(k)}^{j(k)}</math> is the <math>k^\text{th}</math> prime power in the factorization and <math>\ell</math> is the number of distinct prime factors dividing <math>n.</math>  The factorization of <math>1</math> is defined as <math>1</math> in accord with the convention that an empty product is equal to <math>1.</math>
+
where <math>\text{p}_{i(k)}^{j(k)}\!</math> is the <math>k^\text{th}\!</math> prime power in the factorization and <math>\ell\!</math> is the number of distinct prime factors dividing <math>n.\!</math>  The factorization of <math>1\!</math> is defined as <math>1\!</math> in accord with the convention that an empty product is equal to <math>1.\!</math>
  
Let <math>I(n)</math> be the set of indices of primes that divide  <math>n</math> and let <math>j(i, n)</math> be the number of times that <math>\text{p}_i</math> divides <math>n.</math>  Then the prime factorization of <math>n</math> can be written in the following alternative form:
+
Let <math>I(n)\!</math> be the set of indices of primes that divide  <math>n\!</math> and let <math>j(i, n)\!</math> be the number of times that <math>\text{p}_i\!</math> divides <math>n.\!</math>  Then the prime factorization of <math>n\!</math> can be written in the following alternative form:
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
| <math>n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.</math>
+
| <math>n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.\!</math>
 
|}
 
|}
  
Line 40: Line 40:
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
9876543210
+
123456789
& = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721
+
& = & 3^2 \cdot 3607 \cdot 3803
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1.
+
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1.
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
  
Each index <math>i</math> and exponent <math>j</math> appearing in the prime factorization of a positive integer <math>n</math> is itself a positive integer, and thus has a prime factorization of its own.
+
Each index <math>i\!</math> and exponent <math>j\!</math> appearing in the prime factorization of a positive integer <math>n\!</math> is itself a positive integer, and thus has a prime factorization of its own.
  
Continuing with the same example, the index <math>32277</math> has the factorization <math>3 \cdot 7 \cdot 29 \cdot 53 = \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1.</math>  Taking this information together with previously known factorizations allows the following replacements to be made:
+
Continuing with the same example, the index <math>504\!</math> has the factorization <math>2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\!</math> and the index <math>529\!</math> has the factorization <math>{23}^2 = \text{p}_9^2.\!</math>  Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
Line 55: Line 55:
 
2 & \mapsto & \text{p}_1^1
 
2 & \mapsto & \text{p}_1^1
 
\\[6pt]
 
\\[6pt]
3 & \mapsto & \text{p}_2^1
+
504 & \mapsto & \text{p}_1^3 \text{p}_2^2 \text{p}_4^1
 
\\[6pt]
 
\\[6pt]
7 & \mapsto & \text{p}_4^1
+
529 & \mapsto & \text{p}_9^2
\\[6pt]
+
\end{array}</math>
32277 & \mapsto & \text{p}_2^1 \text{p}_4^1 \text{p}_{10}^1 \text{p}_{16}^1
+
|}
 +
 
 +
This leads to the following development:
 +
 
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{lll}
 +
123456789
 +
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
  
==Riffs in Numerical Order==
+
Continuing to replace every index and exponent with its factorization produces the following development:
  
{| align="center" border="1" cellpadding="10"
+
{| align="center" cellpadding="6" width="90%"
|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
+
|
| valign="bottom" |
+
<math>\begin{array}{lll}
<p>&nbsp;</p><br>
+
123456789
<p><math>1\!</math></p><br>
+
& = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1
<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
+
\\[18pt]
| valign="bottom" |
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^3 \text{p}_2^2 \text{p}_4^1}^1 \text{p}_{\text{p}_9^2}^1
<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
+
\\[18pt]
<p><math>\text{p}\!</math></p><br>
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_2^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^2}^1}^1 \text{p}_{\text{p}_{\text{p}_2^2}^{\text{p}_1^1}}^1
<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
+
\\[18pt]
| valign="bottom" |
+
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1} \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^{\text{p}_1^1}}^1
<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
+
\end{array}</math>
<p><math>\text{p}_\text{p}\!</math></p><br>
+
|}
<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
+
 
 +
The <math>1\!</math>'s that appear as indices and exponents are formally redundant, conveying no information apart from the places they occupy in the resulting syntactic structure.  Leaving them tacit produces the following expression:
 +
 
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{lll}
 +
123456789
 +
& = & \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}^{\text{p}_{\text{p}}} \text{p}_{\text{p}}^{\text{p}} \text{p}_{\text{p}^{\text{p}}}} \text{p}_{\text{p}_{\text{p}_{\text{p}}^{\text{p}}}^{\text{p}}}
 +
\end{array}</math>
 +
|}
 +
 
 +
The pattern of indices and exponents illustrated here is called a ''doubly recursive factorization'', or ''DRF''.  Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the DRF of <math>n.\!</math> &nbsp; If <math>\mathbb{M}</math> is the set of positive integers, <math>\mathcal{L}</math> is the set of DRF expressions, and the mapping defined by the factorization process is denoted <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L},</math> then the doubly recursive factorization of <math>n\!</math> is denoted <math>\operatorname{drf}(n).\!</math>
 +
 
 +
The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.
 +
 
 +
{| align=center cellpadding="6" width="90%"
 +
|-
 +
| <math>\operatorname{riff}(123456789)</math> is the following digraph:
 +
|-
 +
| align=center | [[Image:Riff 123456789 Big.jpg|220px]]
 +
|-
 +
| <math>\operatorname{rote}(123456789)</math> is the following graph:
 +
|-
 +
| align=center | [[Image:Rote 123456789 Big.jpg|345px]]
 +
|}
 +
 
 +
==Riffs in Numerical Order==
 +
 
 +
{| align="center" border="1" cellpadding="12"
 +
|+ style="height:25px" | <math>\text{Riffs in Numerical Order}\!</math>
 
| valign="bottom" |
 
| valign="bottom" |
<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
+
<p>&nbsp;</p><br>
<p><math>\text{p}^\text{p}\!</math></p><br>
+
<p><math>1\!</math></p><br>
<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
+
<p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p>
 
| valign="bottom" |
 
| valign="bottom" |
<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
+
<p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
+
<p><math>\text{p}\!</math></p><br>
 +
<p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
 +
<p><math>\text{p}_\text{p}\!</math></p><br>
 +
<p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
 +
<p><math>\text{p}^\text{p}\!</math></p><br>
 +
<p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
 +
| valign="bottom" |
 +
<p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
 +
<p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
 
<p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p>
 
|-
 
|-
Line 323: Line 374:
  
 
{| align="center" border="1" cellpadding="6"
 
{| align="center" border="1" cellpadding="6"
 +
|+ style="height:25px" | <math>\text{Rotes in Numerical Order}\!</math>
 
| valign="bottom" |
 
| valign="bottom" |
 
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
 
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
Line 574: Line 626:
 
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
 
<p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p>
 +
|}
 +
 +
==Prime Animations==
 +
 +
===Riffs 1 to 60===
 +
 +
{| align="center"
 +
| [[Image:Animation Riff 60 x 0.16.gif]]
 +
|}
 +
 +
===Rotes 1 to 60===
 +
 +
{| align="center"
 +
| [[Image:Animation Rote 60 x 0.16.gif]]
 
|}
 
|}
  
Line 584: Line 650:
 
* '''Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.'''
 
* '''Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.'''
  
* [http://oeis.org/wiki/A061396 OEIS Wiki Entry for A061396].
+
* [http://oeis.org/A061396 OEIS Entry for A061396].
  
 
{| align="center" border="1" width="96%"
 
{| align="center" border="1" width="96%"
Line 724: Line 790:
 
* '''Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.'''
 
* '''Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.'''
  
* [http://oeis.org/wiki/A062504 OEIS Wiki Entry for A062504].
+
* [http://oeis.org/A062504 OEIS Entry for A062504].
  
 
{| align="center"
 
{| align="center"
Line 1,129: Line 1,195:
 
* '''Nodes in riff (rooted index-functional forest) for n.'''
 
* '''Nodes in riff (rooted index-functional forest) for n.'''
  
* [http://oeis.org/wiki/A062537 OEIS Wiki Entry for A062537].
+
* [http://oeis.org/A062537 OEIS Entry for A062537].
  
 
{| align="center" border="1" cellpadding="10"
 
{| align="center" border="1" cellpadding="10"
Line 1,390: Line 1,456:
 
* '''Smallest j with n nodes in its riff (rooted index-functional forest).'''
 
* '''Smallest j with n nodes in its riff (rooted index-functional forest).'''
  
* [http://oeis.org/wiki/A062860 OEIS Wiki Entry for A062860].
+
* [http://oeis.org/A062860 OEIS Entry for A062860].
  
 
{| align="center" border="1" cellpadding="10"
 
{| align="center" border="1" cellpadding="10"
Line 1,441: Line 1,507:
 
* '''a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.'''
 
* '''a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.'''
  
* [http://oeis.org/wiki/A109301 OEIS Wiki Entry for A109301].
+
* [http://oeis.org/A109301 OEIS Entry for A109301].
  
 
; Example
 
; Example
Line 1,742: Line 1,808:
 
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br>
 
<p><math>a(60) ~=~ 3</math></p>
 
<p><math>a(60) ~=~ 3</math></p>
 +
|}
 +
 +
==Miscellaneous Examples==
 +
 +
{| align="center" border="1" width="96%"
 +
|+ style="height:24px" | <math>\text{Integers, Riffs, Rotes}\!</math>
 +
|- style="height:50px; background:#f0f0ff"
 +
|
 +
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
 +
| width="10%" | <math>\text{Integer}\!</math>
 +
| width="45%" | <math>\text{Riff}\!</math>
 +
| width="45%" | <math>\text{Rote}\!</math>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>1\!</math>
 +
| width="45%" | &nbsp;
 +
| width="45%" | [[Image:Rote 1 Big.jpg|15px]]
 +
|-
 +
| <math>2\!</math>
 +
| [[Image:Riff 2 Big.jpg|15px]]
 +
| [[Image:Rote 2 Big.jpg|30px]]
 +
|-
 +
| <math>3\!</math>
 +
| [[Image:Riff 3 Big.jpg|30px]]
 +
| [[Image:Rote 3 Big.jpg|30px]]
 +
|-
 +
| <math>4\!</math>
 +
| [[Image:Riff 4 Big.jpg|30px]]
 +
| [[Image:Rote 4 Big.jpg|48px]]
 +
|-
 +
| <math>360\!</math>
 +
| [[Image:Riff 360 Big.jpg|120px]]
 +
| [[Image:Rote 360 Big.jpg|135px]]
 +
|-
 +
| <math>2010\!</math>
 +
| [[Image:Riff 2010 Big.jpg|138px]]
 +
| [[Image:Rote 2010 Big.jpg|144px]]
 +
|-
 +
| <math>2011\!</math>
 +
| [[Image:Riff 2011 Big.jpg|84px]]
 +
| [[Image:Rote 2011 Big.jpg|120px]]
 +
|-
 +
| <math>2012\!</math>
 +
| [[Image:Riff 2012 Big.jpg|100px]]
 +
| [[Image:Rote 2012 Big.jpg|125px]]
 +
|-
 +
| <math>2500\!</math>
 +
| [[Image:Riff 2500 Big.jpg|66px]]
 +
| [[Image:Rote 2500 Big.jpg|125px]]
 +
|-
 +
| <math>802701\!</math>
 +
| [[Image:Riff 802701 Big.jpg|156px]]
 +
| [[Image:Rote 802701 Big.jpg|245px]]
 +
|-
 +
| <math>123456789\!</math>
 +
| [[Image:Riff 123456789 Big.jpg|162px]]
 +
| [[Image:Rote 123456789 Big.jpg|256px]]
 +
|}
 
|}
 
|}

Latest revision as of 22:00, 30 January 2016

Idea

Let \(\text{p}_i\!\) be the \(i^\text{th}\!\) prime, where the positive integer \(i\!\) is called the index of the prime \(\text{p}_i\!\) and the indices are taken in such a way that \(\text{p}_1 = 2.\!\) Thus the sequence of primes begins as follows:

\(\begin{matrix} \text{p}_1 = 2, & \text{p}_2 = 3, & \text{p}_3 = 5, & \text{p}_4 = 7, & \text{p}_5 = 11, & \text{p}_6 = 13, & \text{p}_7 = 17, & \text{p}_8 = 19, & \ldots \end{matrix}\)

The prime factorization of a positive integer \(n\!\) can be written in the following form:

\(n ~=~ \prod_{k = 1}^{\ell} \text{p}_{i(k)}^{j(k)},\!\)

where \(\text{p}_{i(k)}^{j(k)}\!\) is the \(k^\text{th}\!\) prime power in the factorization and \(\ell\!\) is the number of distinct prime factors dividing \(n.\!\) The factorization of \(1\!\) is defined as \(1\!\) in accord with the convention that an empty product is equal to \(1.\!\)

Let \(I(n)\!\) be the set of indices of primes that divide \(n\!\) and let \(j(i, n)\!\) be the number of times that \(\text{p}_i\!\) divides \(n.\!\) Then the prime factorization of \(n\!\) can be written in the following alternative form:

\(n ~=~ \prod_{i \in I(n)} \text{p}_{i}^{j(i, n)}.\!\)

For example:

\(\begin{matrix} 123456789 & = & 3^2 \cdot 3607 \cdot 3803 & = & \text{p}_2^2 \text{p}_{504}^1 \text{p}_{529}^1. \end{matrix}\)

Each index \(i\!\) and exponent \(j\!\) appearing in the prime factorization of a positive integer \(n\!\) is itself a positive integer, and thus has a prime factorization of its own.

Continuing with the same example, the index \(504\!\) has the factorization \(2^3 \cdot 3^2 \cdot 7 = \text{p}_1^3 \text{p}_2^2 \text{p}_4^1\!\) and the index \(529\!\) has the factorization \({23}^2 = \text{p}_9^2.\!\) Taking this information together with previously known factorizations allows the following replacements to be made in the expression above:

\(\begin{array}{rcl} 2 & \mapsto & \text{p}_1^1 \'"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"' '"`UNIQ-MathJax4-QINU`"' :{| border="1" cellpadding="20" | [[Image:Rote 802701 Big.jpg|330px]] |} '"`UNIQ-MathJax5-QINU`"' <br> {| align="center" border="1" cellpadding="6" |+ style="height:25px" | \(a(n) = \text{Rote Height of}~ n\)

Rote 1 Big.jpg


\(1\!\)


\(a(1) ~=~ 0\)

Rote 2 Big.jpg


\(\text{p}\!\)


\(a(2) ~=~ 1\)

Rote 3 Big.jpg


\(\text{p}_\text{p}\!\)


\(a(3) ~=~ 2\)

Rote 4 Big.jpg


\(\text{p}^\text{p}\!\)


\(a(4) ~=~ 2\)

Rote 5 Big.jpg


\(\text{p}_{\text{p}_\text{p}}\!\)


\(a(5) ~=~ 3\)

Rote 6 Big.jpg


\(\text{p} \text{p}_\text{p}\!\)


\(a(6) ~=~ 2\)

Rote 7 Big.jpg


\(\text{p}_{\text{p}^\text{p}}\!\)


\(a(7) ~=~ 3\)

Rote 8 Big.jpg


\(\text{p}^{\text{p}_\text{p}}\!\)


\(a(8) ~=~ 3\)

Rote 9 Big.jpg


\(\text{p}_\text{p}^\text{p}\!\)


\(a(9) ~=~ 2\)

Rote 10 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(10) ~=~ 3\)

Rote 11 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(11) ~=~ 4\)

Rote 12 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}\!\)


\(a(12) ~=~ 2\)

Rote 13 Big.jpg


\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(13) ~=~ 3\)

Rote 14 Big.jpg


\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(14) ~=~ 3\)

Rote 15 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(15) ~=~ 3\)

Rote 16 Big.jpg


\(\text{p}^{\text{p}^\text{p}}\!\)


\(a(16) ~=~ 3\)

Rote 17 Big.jpg


\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(a(17) ~=~ 4\)

Rote 18 Big.jpg


\(\text{p} \text{p}_\text{p}^\text{p}\!\)


\(a(18) ~=~ 2\)

Rote 19 Big.jpg


\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(a(19) ~=~ 4\)

Rote 20 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(20) ~=~ 3\)

Rote 21 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(21) ~=~ 3\)

Rote 22 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(22) ~=~ 4\)

Rote 23 Big.jpg


\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(a(23) ~=~ 3\)

Rote 24 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)


\(a(24) ~=~ 3\)

Rote 25 Big.jpg


\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(a(25) ~=~ 3\)

Rote 26 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(26) ~=~ 3\)

Rote 27 Big.jpg


\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(a(27) ~=~ 3\)

Rote 28 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(28) ~=~ 3\)

Rote 29 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(a(29) ~=~ 4\)

Rote 30 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(30) ~=~ 3\)

Rote 31 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)


\(a(31) ~=~ 5\)

Rote 32 Big.jpg


\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(32) ~=~ 4\)

Rote 33 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(33) ~=~ 4\)

Rote 34 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(a(34) ~=~ 4\)

Rote 35 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(a(35) ~=~ 3\)

Rote 36 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)


\(a(36) ~=~ 2\)

Rote 37 Big.jpg


\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)


\(a(37) ~=~ 3\)

Rote 38 Big.jpg


\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(a(38) ~=~ 4\)

Rote 39 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(39) ~=~ 3\)

Rote 40 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)


\(a(40) ~=~ 3\)

Rote 41 Big.jpg


\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)


\(a(41) ~=~ 4\)

Rote 42 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(a(42) ~=~ 3\)

Rote 43 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)


\(a(43) ~=~ 4\)

Rote 44 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(44) ~=~ 4\)

Rote 45 Big.jpg


\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(45) ~=~ 3\)

Rote 46 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(a(46) ~=~ 3\)

Rote 47 Big.jpg


\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(a(47) ~=~ 4\)

Rote 48 Big.jpg


\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)


\(a(48) ~=~ 3\)

Rote 49 Big.jpg


\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)


\(a(49) ~=~ 3\)

Rote 50 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(a(50) ~=~ 3\)

Rote 51 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(a(51) ~=~ 4\)

Rote 52 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(a(52) ~=~ 3\)

Rote 53 Big.jpg


\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)


\(a(53) ~=~ 4\)

Rote 54 Big.jpg


\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(a(54) ~=~ 3\)

Rote 55 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(a(55) ~=~ 4\)

Rote 56 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(a(56) ~=~ 3\)

Rote 57 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(a(57) ~=~ 4\)

Rote 58 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(a(58) ~=~ 4\)

Rote 59 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)


\(a(59) ~=~ 5\)

Rote 60 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(a(60) ~=~ 3\)

Miscellaneous Examples

\(\text{Integers, Riffs, Rotes}\!\)
\(\text{Integer}\!\) \(\text{Riff}\!\) \(\text{Rote}\!\)
\(1\!\)   Rote 1 Big.jpg
\(2\!\) Riff 2 Big.jpg Rote 2 Big.jpg
\(3\!\) Riff 3 Big.jpg Rote 3 Big.jpg
\(4\!\) Riff 4 Big.jpg Rote 4 Big.jpg
\(360\!\) 120px 135px
\(2010\!\) Riff 2010 Big.jpg Rote 2010 Big.jpg
\(2011\!\) Riff 2011 Big.jpg Rote 2011 Big.jpg
\(2012\!\) Riff 2012 Big.jpg Rote 2012 Big.jpg
\(2500\!\) Riff 2500 Big.jpg Rote 2500 Big.jpg
\(802701\!\) Riff 802701 Big.jpg Rote 802701 Big.jpg
\(123456789\!\) Riff 123456789 Big.jpg Rote 123456789 Big.jpg