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Directory:Jon Awbrey/Papers/Riffs and Rotes (view source)

Revision as of 15:36, 25 January 2010

10,069 bytes added ,  15:36, 25 January 2010
→‎A062504
Line 1,062: Line 1,062:  
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
 
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
 
\end{array}</math>
 
\end{array}</math>
 +
|}
 +
 +
{| align="center" border="1" width="90%"
 +
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
 +
|- style="height:50px; background:#f0f0ff"
 +
|
 +
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
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| width="10%" | <math>\text{Integer}\!</math>
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| width="25%" | <math>\text{Factorization}\!</math>
 +
| width="15%" | <math>\text{Notation}\!</math>
 +
| width="25%" | <math>\text{Riff Digraph}\!</math>
 +
| width="25%" | <math>\text{Rote Graph}\!</math>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>1\!</math>
 +
| width="25%" | <math>1\!</math>
 +
| width="15%" | &nbsp;
 +
| width="25%" | &nbsp;
 +
| width="25%" | [[Image:Rote 1 Big.jpg|20px]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>2\!</math>
 +
| width="25%" | <math>\text{p}_1^1\!</math>
 +
| width="15%" | <math>\text{p}\!</math>
 +
| width="25%" | [[Image:Riff 2 Big.jpg|20px]]
 +
| width="25%" | [[Image:Rote 2 Big.jpg|40px]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>3\!</math>
 +
| width="25%" |
 +
<math>\begin{array}{lll}
 +
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
 +
\end{array}</math>
 +
| width="15%" | <math>\text{p}_\text{p}\!</math>
 +
| width="25%" | [[Image:Riff 3 Big.jpg|40px]]
 +
| width="25%" | [[Image:Rote 3 Big.jpg|40px]]
 +
|-
 +
| <math>4\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}^\text{p}\!</math>
 +
| [[Image:Riff 4 Big.jpg|40px]]
 +
| [[Image:Rote 4 Big.jpg|65px]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>5\!</math>
 +
| width="25%" |
 +
<math>\begin{array}{lll}
 +
\text{p}_3^1
 +
& = & \text{p}_{\text{p}_2^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 +
\end{array}</math>
 +
| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
 +
| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
 +
| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
 +
|-
 +
| <math>6\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_2^1
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
 +
\end{array}</math>
 +
| <math>\text{p} \text{p}_{\text{p}}\!</math>
 +
| [[Image:Riff 6 Big.jpg|65px]]
 +
| [[Image:Rote 6 Big.jpg|80px]]
 +
|-
 +
| <math>7\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_4^1
 +
& = & \text{p}_{\text{p}_1^2}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 7 Big.jpg|65px]]
 +
| [[Image:Rote 7 Big.jpg|65px]]
 +
|-
 +
| <math>8\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^3
 +
& = & \text{p}_1^{\text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 8 Big.jpg|65px]]
 +
| [[Image:Rote 8 Big.jpg|65px]]
 +
|-
 +
| <math>9\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^2
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_\text{p}^\text{p}\!</math>
 +
| [[Image:Riff 9 Big.jpg|40px]]
 +
| [[Image:Rote 9 Big.jpg|80px]]
 +
|-
 +
| <math>16\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^4
 +
& = & \text{p}_1^{\text{p}_1^2}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 16 Big.jpg|65px]]
 +
| [[Image:Rote 16 Big.jpg|90px]]
 +
|}
 +
|-
 +
|
 +
{| cellpadding="12" style="text-align:center; width:100%"
 +
| width="10%" | <math>10\!</math>
 +
| width="25%" |
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_3^1
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1
 +
\\[12pt]
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
 +
\end{array}</math>
 +
| width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math>
 +
| width="25%" | [[Image:Riff 10 Big.jpg|90px]]
 +
| width="25%" | [[Image:Rote 10 Big.jpg|80px]]
 +
|-
 +
| <math>11\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_5^1
 +
& = & \text{p}_{\text{p}_3^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math>
 +
| [[Image:Riff 11 Big.jpg|90px]]
 +
| [[Image:Rote 11 Big.jpg|40px]]
 +
|-
 +
| <math>12\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^2 \text{p}_2^1
 +
& = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math>
 +
| [[Image:Riff 12 Big.jpg|65px]]
 +
| [[Image:Rote 12 Big.jpg|105px]]
 +
|-
 +
| <math>13\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_6^1
 +
& = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 13 Big.jpg|65px]]
 +
| [[Image:Rote 13 Big.jpg|80px]]
 +
|-
 +
| <math>14\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_4^1
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1
 +
\\[12pt]
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 14 Big.jpg|90px]]
 +
| [[Image:Rote 14 Big.jpg|105px]]
 +
|-
 +
| <math>17\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_7^1
 +
& = & \text{p}_{\text{p}_4^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 17 Big.jpg|90px]]
 +
| [[Image:Rote 17 Big.jpg|65px]]
 +
|-
 +
| <math>18\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^1 \text{p}_2^2
 +
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math>
 +
| [[Image:Riff 18 Big.jpg|65px]]
 +
| [[Image:Rote 18 Big.jpg|120px]]
 +
|-
 +
| <math>19\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_8^1
 +
& = & \text{p}_{\text{p}_1^3}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 19 Big.jpg|90px]]
 +
| [[Image:Rote 19 Big.jpg|65px]]
 +
|-
 +
| <math>23\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_9^1
 +
& = & \text{p}_{\text{p}_2^2}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 +
| [[Image:Riff 23 Big.jpg|65px]]
 +
| [[Image:Rote 23 Big.jpg|80px]]
 +
|-
 +
| <math>25\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_3^2
 +
& = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math>
 +
| [[Image:Riff 25 Big.jpg|65px]]
 +
| [[Image:Rote 25 Big.jpg|80px]]
 +
|-
 +
| <math>27\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^3
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 27 Big.jpg|65px]]
 +
| [[Image:Rote 27 Big.jpg|80px]]
 +
|-
 +
| <math>32\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^5
 +
& = & \text{p}_1^{\text{p}_3^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 32 Big.jpg|90px]]
 +
| [[Image:Rote 32 Big.jpg|65px]]
 +
|-
 +
| <math>49\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_4^2
 +
& = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math>
 +
| [[Image:Riff 49 Big.jpg|65px]]
 +
| [[Image:Rote 49 Big.jpg|80px]]
 +
|-
 +
| <math>53\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_{16}^1
 +
& = & \text{p}_{\text{p}_1^4}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 53 Big.jpg|90px]]
 +
| [[Image:Rote 53 Big.jpg|90px]]
 +
|-
 +
| <math>64\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^6
 +
& = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math>
 +
| [[Image:Riff 64 Big.jpg|65px]]
 +
| [[Image:Rote 64 Big.jpg|105px]]
 +
|-
 +
| <math>81\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_2^4
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2}
 +
\\[12pt]
 +
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math>
 +
| [[Image:Riff 81 Big.jpg|65px]]
 +
| [[Image:Rote 81 Big.jpg|105px]]
 +
|-
 +
| <math>128\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^7
 +
& = & \text{p}_1^{\text{p}_4^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 128 Big.jpg|90px]]
 +
| [[Image:Rote 128 Big.jpg|90px]]
 +
|-
 +
| <math>256\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^8
 +
& = & \text{p}_1^{\text{p}_1^3}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math>
 +
| [[Image:Riff 256 Big.jpg|90px]]
 +
| [[Image:Rote 256 Big.jpg|90px]]
 +
|-
 +
| <math>512\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^9
 +
& = & \text{p}_1^{\text{p}_2^2}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math>
 +
| [[Image:Riff 512 Big.jpg|65px]]
 +
| [[Image:Rote 512 Big.jpg|105px]]
 +
|-
 +
| <math>65536\!</math>
 +
|
 +
<math>\begin{array}{lll}
 +
\text{p}_1^{16}
 +
& = & \text{p}_1^{\text{p}_1^4}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}}
 +
\\[12pt]
 +
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}
 +
\end{array}</math>
 +
| <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math>
 +
| [[Image:Riff 65536 Big.jpg|90px]]
 +
| [[Image:Rote 65536 Big.jpg|115px]]
 +
|}
 
|}
 
|}
  
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/106408"