Comparison of mode notation systems: Difference between revisions
Wikispaces>TallKite **Imported revision 580857115 - Original comment: ** |
Wikispaces>TallKite **Imported revision 580871995 - 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:TallKite|TallKite]] and made on <tt>2016-04- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-22 00:10:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>580871995</tt>.<br> | ||
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|| Locrian || 7th Meantone[7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | || Locrian || 7th Meantone[7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | ||
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. **Unlike modal UDP notation, the generator isn't always chroma-positive.** This is necessary to keep the same generator for different MOS's of the same [[Regular Temperaments|temperament]], which guarantees that the smaller MOS will always be a subset of the larger MOS. | The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. **Unlike modal UDP notation, the generator isn't always chroma-positive.** This is necessary to keep the same generator for different MOS's of the same [[Regular Temperaments|temperament]], which guarantees that the smaller MOS will always be a subset of the larger MOS. | ||
For example, Meantone[5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes' progression is from flatter to sharper. Because Meantone[5] and Meantone[7]have the same generator, C 2nd Meantone[5] = CDFGAC is a subset of C 2nd Meantone[7] = CDEFGABC. | For example, Meantone[5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes' progression is from flatter to sharper. Because Meantone[5] and Meantone[7]have the same generator, C 2nd Meantone[5] = CDFGAC is a subset of C 2nd Meantone[7] = CDEFGABC. | ||
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|| Meantone [11] || ??? || C G D A E B F# C# G# D# A# || | || Meantone [11] || ??? || C G D A E B F# C# G# D# A# || | ||
|| Meantone [12] if generator < 700¢ || E# A# D# G# C# F# B E A D G C || C G D A E B F# C# G# D# A# E# || | || Meantone [12] if generator < 700¢ || E# A# D# G# C# F# B E A D G C || C G D A E B F# C# G# D# A# E# || | ||
||< Meantone [12] if generator > 700¢ || C G D A E B F# C# G# D# A# E# ||= | ||< Meantone [12] if generator > 700¢ || C G D A E B F# C# G# D# A# E# ||= C G D A E B F# C# G# D# A# E# || | ||
An even larger problem is that Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. In other words the notation is overly tuning-dependent. One might address this problem by constraining meantone's fifth to a reasonable range of < 700¢. Likewise one could constrain superpyth's fifth to be > 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either < 700¢ or > 700¢. | An even larger problem is that Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. In other words the notation is overly tuning-dependent. One might address this problem by constraining meantone's fifth to a reasonable range of < 700¢. Likewise one could constrain superpyth's fifth to be > 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either < 700¢ or > 700¢. Every single UDP mode of Dominant[12] is ambiguous. For example "Dominant 8|3" could mean either "4th Dominant[12]" or "9th Dominant[12]". | ||
A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas GMN is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, when comparing different MOS's of a temperament, with GMN but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and GMN uses the more intuitive one-based counting. | A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas GMN is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, when comparing different MOS's of a temperament, with GMN but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and GMN uses the more intuitive one-based counting. | ||
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<br /> | <br /> | ||
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. <strong>Unlike modal UDP notation, the generator isn't always chroma-positive.</strong> This is necessary to keep the same generator for different MOS's of the same <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>, which guarantees that the smaller MOS will always be a subset of the larger MOS. <br /> | The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. <strong>Unlike modal UDP notation, the generator isn't always chroma-positive.</strong> This is necessary to keep the same generator for different MOS's of the same <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>, which guarantees that the smaller MOS will always be a subset of the larger MOS.<br /> | ||
<br /> | <br /> | ||
For example, Meantone[5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes' progression is from flatter to sharper. Because Meantone[5] and Meantone[7]have the same generator, C 2nd Meantone[5] = CDFGAC is a subset of C 2nd Meantone[7] = CDEFGABC.<br /> | For example, Meantone[5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes' progression is from flatter to sharper. Because Meantone[5] and Meantone[7]have the same generator, C 2nd Meantone[5] = CDFGAC is a subset of C 2nd Meantone[7] = CDEFGABC.<br /> | ||
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<td>C G D A E B F# C# G# D# A# E#<br /> | <td>C G D A E B F# C# G# D# A# E#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">C G D A E B F# C# G# D# A# E#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
An even larger problem is that Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. In other words the notation is overly tuning-dependent. One might address this problem by constraining meantone's fifth to a reasonable range of &lt; 700¢. Likewise one could constrain superpyth's fifth to be &gt; 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either &lt; 700¢ or &gt; 700¢. | An even larger problem is that Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. In other words the notation is overly tuning-dependent. One might address this problem by constraining meantone's fifth to a reasonable range of &lt; 700¢. Likewise one could constrain superpyth's fifth to be &gt; 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either &lt; 700¢ or &gt; 700¢. Every single UDP mode of Dominant[12] is ambiguous. For example &quot;Dominant 8|3&quot; could mean either &quot;4th Dominant[12]&quot; or &quot;9th Dominant[12]&quot;.<br /> | ||
<br /> | <br /> | ||
A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas GMN is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, when comparing different MOS's of a temperament, with GMN but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and GMN uses the more intuitive one-based counting.<br /> | A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas GMN is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, when comparing different MOS's of a temperament, with GMN but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and GMN uses the more intuitive one-based counting.<br /> | ||