7L 3s: Difference between revisions
Wikispaces>JosephRuhf **Imported revision 565465405 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 598695606 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt> | : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-07 11:59:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>598695606</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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The generator range reflects two extremes: one where L=s (3\10), and another where s=0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 -- five degrees of [[17edo]]: | The generator range reflects two extremes: one where L=s (3\10), and another where s=0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 -- five degrees of [[17edo]]: | ||
||||||||||||~ Generator (fraction of octave) ||~ Generator (cents) ||~ Comments || | ||||||||||||~ Generator (fraction of octave) ||~ Generator (cents) ||~ ~11:10:9 (cents) ||~ ~18:14:11 (cents) ||~ Comments || | ||
|| 3\10 || || || || || || 360 ||= || | || 3\10 || || || || || || 360 || 188.7+171.3 || 411.21+428.79 ||= || | ||
|| || || || || || 17\57 || 357.895 || || | || || || || || || 17\57 || 357.895 || 187.65+170.25 || 412.265+429.84 || || | ||
|| || || || || 14\47 || || 357.45 || || | || || || || || 14\47 || || 357.45 || 187.42+170.02 || 412.49+430.06 || || | ||
|| || || || 11\37 || || || 356.76 || || | || || || || 11\37 || || || 356.76 || 187.08+169.68 || 412.83+430.41 || || | ||
|| || || 8\27 || || || || 355.56 ||= Optimum rank range (L/s=3/2) || | || || || 8\27 || || || || 355.56 || 186.48+169.08 || 413.43+431.01 ||= Optimum rank range (L/s=3/2) || | ||
|| || || || || || || 355.11 || || | || || || || || || || 355.11 || 186.25+168.85 || 413.66+431.235 || || | ||
|| || || || || 21\71 || || 354.93 ||= || | || || || || || 21\71 || || 354.93 || 186.165+168.765 || 413.75+431.32 ||= || | ||
|| || || || || || 34\115 || 354.78 ||= Golden neutral thirds || | || || || || || || 34\115 || 354.78 || 186.09+168.69 || 413,82+431.4 ||= Golden neutral thirds || | ||
|| || || || 13\44 || || || 354. | || || || || 13\44 || || || 354.545 || 185.97+168.57 || 413.94+431.515 ||= || | ||
|| || || || || || || 354.19 || || | || || || || || || || 354.19 || 185.8+168.4 || 414.12+431.69 || || | ||
|| || 5\17 || || || || || 352.94 ||= Boundary of propriety: generators larger than this are proper || | || || 5\17 || || || || || 352.94 || 185.17+167.77 || 414.74+432.32 ||= Boundary of propriety: generators larger than this are proper || | ||
|| || || || 12\41 || || || 351.22 ||= || | || || || || 12\41 || || || 351.22 || 184.31+166.91 || 415.6+433.18 ||= || | ||
|| || || || || || 31\106 || 350.94 ||= No-5's neutral thirds (observing 81/80) is around here || | || || || || || || 31\106 || 350.94 || 184.17+166.77 || 415.74+433.32 ||= No-5's neutral thirds (observing 81/80) is around here || | ||
|| || || || || 19\65 || || 350.77 ||= || | || || || || || 19\65 || || 350.77 || 184.08+166.685 || 415.83+433.4 ||= || | ||
|| || || || || || || 350.64 ||= <span style="display: block; text-align: center;">L/s = e</span> || | || || || || || || || 350.64 || 184.02+166.62 || 415.89+433.47 ||= <span style="display: block; text-align: center;">L/s = e</span> || | ||
|| || || 7\24 || || || || 350 ||= || | || || || 7\24 || || || || 350 || 183.7+166.3 || 416.21+433.79 ||= || | ||
|| || || || || || || 349.72 ||= <span style="display: block; text-align: center;">L/s = pi</span> || | || || || || || || || 349.72 || 183.55+166.16 || 416.35+433.93 ||= <span style="display: block; text-align: center;">L/s = pi</span> || | ||
|| || || || || 16\55 || || 349.09 || || | || || || || || 16\55 || || 349.09 || 183.24+165.85 || 416.67+434.24 || || | ||
|| || || || 9\31 || || || 348.39 ||= Mohajira (tempering out 81/80) is around here || | || || || || 9\31 || || || 348.39 || 182.89+165.493 || 417.02+434.595 ||= Mohajira (tempering out 81/80) is around here || | ||
|| || || || || 11\38 || || 347.37 || || | || || || || || 11\38 || || 347.37 || 182.38+164.98 || 417.52+435.1 || || | ||
|| || || || || || 13\45 || 346.67 || || | || || || || || || 13\45 || 346.67 || 182.03+164.63 || 417.87+435.455 || || | ||
|| 2\7 || || || || || || 342.86 ||= || | || 2\7 || || || || || || 342.86 || 180.13+162.73 || 419.78+437.36 ||= || | ||
The scale produced by stacks of 5\17 is the [[17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic pairs|chromatic pair]]. | The scale produced by stacks of 5\17 is the [[17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic pairs|chromatic pair]]. | ||
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</th> | </th> | ||
<th>Generator (cents)<br /> | <th>Generator (cents)<br /> | ||
</th> | |||
<th>~11:10:9 (cents)<br /> | |||
</th> | |||
<th>~18:14:11 (cents)<br /> | |||
</th> | </th> | ||
<th>Comments<br /> | <th>Comments<br /> | ||
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</td> | </td> | ||
<td>360<br /> | <td>360<br /> | ||
</td> | |||
<td>188.7+171.3<br /> | |||
</td> | |||
<td>411.21+428.79<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 153: | Line 161: | ||
</td> | </td> | ||
<td>357.895<br /> | <td>357.895<br /> | ||
</td> | |||
<td>187.65+170.25<br /> | |||
</td> | |||
<td>412.265+429.84<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>357.45<br /> | <td>357.45<br /> | ||
</td> | |||
<td>187.42+170.02<br /> | |||
</td> | |||
<td>412.49+430.06<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>356.76<br /> | <td>356.76<br /> | ||
</td> | |||
<td>187.08+169.68<br /> | |||
</td> | |||
<td>412.83+430.41<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>355.56<br /> | <td>355.56<br /> | ||
</td> | |||
<td>186.48+169.08<br /> | |||
</td> | |||
<td>413.43+431.01<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Optimum rank range (L/s=3/2)<br /> | <td style="text-align: center;">Optimum rank range (L/s=3/2)<br /> | ||
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</td> | </td> | ||
<td>355.11<br /> | <td>355.11<br /> | ||
</td> | |||
<td>186.25+168.85<br /> | |||
</td> | |||
<td>413.66+431.235<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>354.93<br /> | <td>354.93<br /> | ||
</td> | |||
<td>186.165+168.765<br /> | |||
</td> | |||
<td>413.75+431.32<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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</td> | </td> | ||
<td>354.78<br /> | <td>354.78<br /> | ||
</td> | |||
<td>186.09+168.69<br /> | |||
</td> | |||
<td>413,82+431.4<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Golden neutral thirds<br /> | <td style="text-align: center;">Golden neutral thirds<br /> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td>354. | <td>354.545<br /> | ||
</td> | |||
<td>185.97+168.57<br /> | |||
</td> | |||
<td>413.94+431.515<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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</td> | </td> | ||
<td>354.19<br /> | <td>354.19<br /> | ||
</td> | |||
<td>185.8+168.4<br /> | |||
</td> | |||
<td>414.12+431.69<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>352.94<br /> | <td>352.94<br /> | ||
</td> | |||
<td>185.17+167.77<br /> | |||
</td> | |||
<td>414.74+432.32<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Boundary of propriety: generators larger than this are proper<br /> | <td style="text-align: center;">Boundary of propriety: generators larger than this are proper<br /> | ||
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</td> | </td> | ||
<td>351.22<br /> | <td>351.22<br /> | ||
</td> | |||
<td>184.31+166.91<br /> | |||
</td> | |||
<td>415.6+433.18<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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</td> | </td> | ||
<td>350.94<br /> | <td>350.94<br /> | ||
</td> | |||
<td>184.17+166.77<br /> | |||
</td> | |||
<td>415.74+433.32<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">No-5's neutral thirds (observing 81/80) is around here<br /> | <td style="text-align: center;">No-5's neutral thirds (observing 81/80) is around here<br /> | ||
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</td> | </td> | ||
<td>350.77<br /> | <td>350.77<br /> | ||
</td> | |||
<td>184.08+166.685<br /> | |||
</td> | |||
<td>415.83+433.4<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 387: | Line 447: | ||
</td> | </td> | ||
<td>350.64<br /> | <td>350.64<br /> | ||
</td> | |||
<td>184.02+166.62<br /> | |||
</td> | |||
<td>415.89+433.47<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = e</span><br /> | <td style="text-align: center;"><span style="display: block; text-align: center;">L/s = e</span><br /> | ||
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</td> | </td> | ||
<td>350<br /> | <td>350<br /> | ||
</td> | |||
<td>183.7+166.3<br /> | |||
</td> | |||
<td>416.21+433.79<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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</td> | </td> | ||
<td>349.72<br /> | <td>349.72<br /> | ||
</td> | |||
<td>183.55+166.16<br /> | |||
</td> | |||
<td>416.35+433.93<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br /> | <td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br /> | ||
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</td> | </td> | ||
<td>349.09<br /> | <td>349.09<br /> | ||
</td> | |||
<td>183.24+165.85<br /> | |||
</td> | |||
<td>416.67+434.24<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>348.39<br /> | <td>348.39<br /> | ||
</td> | |||
<td>182.89+165.493<br /> | |||
</td> | |||
<td>417.02+434.595<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Mohajira (tempering out 81/80) is around here<br /> | <td style="text-align: center;">Mohajira (tempering out 81/80) is around here<br /> | ||
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</td> | </td> | ||
<td>347.37<br /> | <td>347.37<br /> | ||
</td> | |||
<td>182.38+164.98<br /> | |||
</td> | |||
<td>417.52+435.1<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>346.67<br /> | <td>346.67<br /> | ||
</td> | |||
<td>182.03+164.63<br /> | |||
</td> | |||
<td>417.87+435.455<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>342.86<br /> | <td>342.86<br /> | ||
</td> | |||
<td>180.13+162.73<br /> | |||
</td> | |||
<td>419.78+437.36<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||