MOS substitution: Difference between revisions
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|+ style="font-size: 105%;" | The three subst 2'''L'''(1'''m'''2'''s''') scales | |+ style="font-size: 105%;" | The three subst 2'''L'''(1'''m'''2'''s''') scales | ||
|- | |- | ||
! rowspan="2" | [[UDP]] for filling MOS | ! rowspan="2" | [[UDP]] for filling MOS | ||
! rowspan="2" | Filling MOS | ! rowspan="2" | Filling MOS | ||
! colspan="2" | Step pattern | ! colspan="2" | Step pattern | ||
! rowspan="2" | Denoted as | ! rowspan="2" | Denoted as | ||
|- | |- | ||
! Template MOS: | ! Template MOS: | ||
| <code>LXLXX</code> | | <code>LXLXX</code> | ||
|- | |- | ||
| 2{{pipe}}0 | | 2{{pipe}}0 | ||
| style="text-align: right;" | <code>mss</code> | | style="text-align: right;" | <code>mss</code> | ||
| colspan="2" style="text-align: right;" | <code>LmLss</code> | | colspan="2" style="text-align: right;" | <code>LmLss</code> | ||
| subst 2L(1m2s 2{{pipe}}0) | | subst 2L(1m2s 2{{pipe}}0) | ||
|- | |- | ||
| 1{{pipe}}1 | | 1{{pipe}}1 | ||
| style="text-align: right;" | <code>sms</code> | | style="text-align: right;" | <code>sms</code> | ||
| colspan="2" style="text-align: right;" | <code>LsLms</code> | | colspan="2" style="text-align: right;" | <code>LsLms</code> | ||
| subst 2L(1m1s 1{{pipe}}1) | | subst 2L(1m1s 1{{pipe}}1) | ||
|- | |- | ||
| 0{{pipe}}2 | | 0{{pipe}}2 | ||
| style="text-align: right;" | <code>ssm</code> | | style="text-align: right;" | <code>ssm</code> | ||
| colspan="2" style="text-align: right;" | <code>LsLsm</code> | | colspan="2" style="text-align: right;" | <code>LsLsm</code> | ||
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|- | |- | ||
| 2 || <code>mssmss</code> || 4{{pipe}}0(2) | | 2 || <code>mssmss</code> || 4{{pipe}}0(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LmLsLsLmLss</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''m''' | | colspan="2" | GS({{nowrap| '''L''' + '''m''' | '''L''' + '''s''' | '''L''' + '''s''' }}) || yes | ||
|- | |- | ||
| 1 || <code>smssms</code> || 2{{pipe}}2(2) | | 1 || <code>smssms</code> || 2{{pipe}}2(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LsLmLsLsLms</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''s''' | | colspan="2" | GS({{nowrap| '''L''' + '''s''' | '''L''' + '''m''' | '''L''' + '''s''' }}) || yes | ||
|- | |- | ||
| 0 || <code>ssmssm</code> || 0{{pipe}}4(2) | | 0 || <code>ssmssm</code> || 0{{pipe}}4(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LsLsLmLsLsm</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''s''' | | colspan="2" | GS({{nowrap| '''L''' + '''s''' | '''L''' + '''s''' | '''L''' + '''m''' }}) || yes | ||
|} | |} | ||
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|- | |- | ||
| 3 || <code>msss</code> || 6{{pipe}}0(2) | | 3 || <code>msss</code> || 6{{pipe}}0(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LmLssLsLmsLss</code> | ||
| colspan="2" | GS | | colspan="2" | GS({{nowrap| (2'''L''' + '''m''' + 2'''s''')<sup>3</sup> | 2'''L''' + 3'''s''' }}) || yes | ||
|- | |- | ||
| 2 || <code>smss</code> || 4{{pipe}}2(2) | | 2 || <code>smss</code> || 4{{pipe}}2(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LsLmsLsLsmLss</code> | ||
| colspan="2" | GS | | colspan="2" | GS({{nowrap| (2'''L''' + '''m''' + 2'''s''')<sup>2</sup> | 2'''L''' + 3'''s''' | 2'''L''' + '''m''' + 2'''s''' }}) || yes | ||
|- | |- | ||
| 1 || <code>ssms</code> || 2{{pipe}}4(2) | | 1 || <code>ssms</code> || 2{{pipe}}4(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LsLsmLsLssLms</code> | ||
| colspan="2" | GS({{nowrap|2'''L''' + '''m''' + 2'''s''' | | colspan="2" | GS({{nowrap| 2'''L''' + '''m''' + 2'''s''' | 2'''L''' + 3'''s''' | (2'''L''' + '''m''' + 2'''s''')<sup>2</sup> }}) || yes | ||
|- | |- | ||
| 0 || <code>sssm</code> || 0{{pipe}}6(2) | | 0 || <code>sssm</code> || 0{{pipe}}6(2) | ||
| colspan="2" | | colspan="2" style="text-align: right;" | <code>LsLssLmLssLsm</code> | ||
| colspan="2" | GS({{nowrap|2'''L''' + 3'''s''' | | colspan="2" | GS({{nowrap| 2'''L''' + 3'''s''' | (2'''L''' + '''m''' + 2'''s''')<sup>3</sup> }}) || yes | ||
|} | |} | ||
Here the notation ''G''<sup>''k''</sup> denotes repeating the generator ''G'' ''k'' times in the generator sequence. | Here the notation ''G''<sup>''k''</sup> denotes repeating the generator ''G'' ''k'' times in the generator sequence. | ||
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| 4 || <code>LsLss</code> || 12{{pipe}}0(3) | | 4 || <code>LsLss</code> || 12{{pipe}}0(3) | ||
| colspan="2" style="text-align: right;" | <code>mLsmLsmsLmsLmssmLsmLss</code> | | colspan="2" style="text-align: right;" | <code>mLsmLsmsLmsLmssmLsmLss</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''m''' + '''s''' | | colspan="2" | GS({{nowrap| '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''m''' + 2'''s''' }}) || yes | ||
|- | |- | ||
| 3 || <code>LssLs</code> || 9{{pipe}}3(3) | | 3 || <code>LssLs</code> || 9{{pipe}}3(3) | ||
| colspan="2" style="text-align: right;" | <code>mLsmsLmsLmssmLsmLsmsLs</code> | | colspan="2" style="text-align: right;" | <code>mLsmsLmsLmssmLsmLsmsLs</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''m''' + '''s''' | | colspan="2" | GS({{nowrap| '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''m''' + 2'''s''' | '''L''' + '''m''' + '''s''' }}) || yes | ||
|- | |- | ||
| 2 || <code>sLsLs</code> || 6{{pipe}}6(3) | | 2 || <code>sLsLs</code> || 6{{pipe}}6(3) | ||
| colspan="2" style="text-align: right;" | <code>msLmsLmssmLsmLsmsLmsLs</code> | | colspan="2" style="text-align: right;" | <code>msLmsLmssmLsmLsmsLmsLs</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''m''' + '''s''' | | colspan="2" | GS({{nowrap| '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''m''' + 2'''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' }}) || yes | ||
|- | |- | ||
| 1 || <code>sLssL</code> || 3{{pipe}}9(3) | | 1 || <code>sLssL</code> || 3{{pipe}}9(3) | ||
| colspan="2" style="text-align: right;" | <code>msLmssmLsmLsmsLmsLmssL</code> | | colspan="2" style="text-align: right;" | <code>msLmssmLsmLsmsLmsLmssL</code> | ||
| colspan="2" | GS({{nowrap|'''L''' + '''m''' + '''s''' | | colspan="2" | GS({{nowrap| '''L''' + '''m''' + '''s''' | '''m''' + 2'''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' }}) || yes | ||
|- | |- | ||
| 0 || <code>ssLsL</code> || 0{{pipe}}12(3) | | 0 || <code>ssLsL</code> || 0{{pipe}}12(3) | ||
| colspan="2" style="text-align: right;" | <code>mssmLsmLsmsLmsLmssmLsL</code> | | colspan="2" style="text-align: right;" | <code>mssmLsmLsmsLmsLmssmLsL</code> | ||
| colspan="2" | GS({{nowrap|'''m''' + 2'''s''' | | colspan="2" | GS({{nowrap| '''m''' + 2'''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' | '''L''' + '''m''' + '''s''' }}) || no | ||
|} | |} | ||
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then it is a MOS substitution scale, namely subst(({{nowrap|''a'' + ''b''}})'''X'''''c'''''s'''(''i''), '''X''', ''a'''''L'''''b'''''m'''(''j'')) for some brightnesses ''i'' and ''j''. | then it is a MOS substitution scale, namely subst(({{nowrap|''a'' + ''b''}})'''X'''''c'''''s'''(''i''), '''X''', ''a'''''L'''''b'''''m'''(''j'')) for some brightnesses ''i'' and ''j''. | ||
In particular, all [[monotone-MOS scale]]s (i.e. such that the results of {{nowrap|'''L''' {{=}} '''m''' | In particular, all [[monotone-MOS scale]]s (i.e. such that the results of {{nowrap|'''L''' {{=}} '''m''' | '''m''' {{=}} '''s'''}}, and {{nowrap|'''s''' {{=}} '''0'''}} temperings are MOSes) arise from MOS substitution in this way. | ||
=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences === | === If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences === | ||