Comparison of mode notation systems: Difference between revisions
Wikispaces>TallKite **Imported revision 580850565 - 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>2016-04-21 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-21 19:40:27 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>580857115</tt>.<br> | ||
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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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">GMN notation (genchain mode numbering) is a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator: | ||
|| old scale name || new scale name || Ls pattern || example on white keys || genchain || | || old scale name || new scale name || Ls pattern || example on white keys || genchain || | ||
|| Lydian || 1st Meantone[7] || LLLs LLs || F G A B C D E F || __**F**__ C G D A E B || | || Lydian || 1st Meantone[7] || LLLs LLs || F G A B C D E F || __**F**__ C G D A E B || | ||
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These [[MOSScales|MOS scales]] are formed from a segment of the [[periods and generators|generator-chain]], or genchain. The first note in the genchain is the tonic of mode #1, the 2nd note is the tonic of mode #2, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as "fourth meantone heptatonic". If in D, as above, it would be "D fourth meantone heptatonic". | These [[MOSScales|MOS scales]] are formed from a segment of the [[periods and generators|generator-chain]], or genchain. The first note in the genchain is the tonic of mode #1, the 2nd note is the tonic of mode #2, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as "fourth meantone heptatonic". If in D, as above, it would be "D fourth meantone heptatonic". | ||
The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The | The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The modes' progression is from sharper to flatter. | ||
|| old scale name || new scale name || Ls pattern || example in C || ------------------- genchain --------------- || | || old scale name || new scale name || Ls pattern || example in C || ------------------- genchain --------------- || | ||
|| Lydian || 1st Meantone[7] || LLLs LLs || C D E F# G A B C ||> __**C**__ G D A E B F# || | || Lydian || 1st Meantone[7] || LLLs LLs || C D E F# G A B C ||> __**C**__ G D A E B F# || | ||
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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. For example, Meantone[5] is | 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. | |||
Pentatonic meantone scales: | Pentatonic meantone scales: | ||
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|| 5th Shrutal[10] || Lssss-Lssss || C D Eb E F F# Ab A Bb B C || Ab Eb Bb F __**C**__ || D A E B F# || | || 5th Shrutal[10] || Lssss-Lssss || C D Eb E F F# Ab A Bb B C || Ab Eb Bb F __**C**__ || D A E B F# || | ||
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4. L = 146¢ and s = 94¢. There are five genchains. Ups and downs are used to distinguish between 5/4 and 2\5. | There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4. L = 146¢ and s = 94¢. There are five genchains. Ups and downs are used to distinguish between 5/4 and 2\5, which is necessary to avoid duplicate note names. | ||
|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || 5th chain || | || scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || 5th chain || | ||
|| 1st Blackwood[10] || LsLsLs LsLs || C C#v D Ev F F#v G Av A Bv C ||= __**C**__ Ev || D F#v || F Av || G Bv || A C#v || | || 1st Blackwood[10] || LsLsLs LsLs || C C#v D Ev F F#v G Av A Bv C ||= __**C**__ Ev || D F#v || F Av || G Bv || A C#v || | ||
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F G A C E F, with genchain __**F**__ C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6. | F G A C E F, with genchain __**F**__ C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6. | ||
==__Why not just use UDP notation?__== | |||
One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. | |||
|| scale || UDP generator || UDP genchain || GMN generator || GMN genchain || | |||
|| Meantone [2] || 3/2 || C G || 3/2 || C G || | |||
|| Meantone [3] || 4/3 || D G C || 3/2 || C G D || | |||
|| Meantone [5] || 4/3 || E A D G C || 3/2 || C G D A E || | |||
|| Meantone [7] || 3/2 || C G D A E B F# || 3/2 || C G D A E B F# || | |||
A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. GMN notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful. | |||
|| scale || UDP genchain || GMN genchain || | |||
|| Meantone [2] || C G || C G || | |||
|| Meantone [3] || D G C || C G D || | |||
|| Meantone [4] || ??? || C G D A || | |||
|| Meantone [5] || E A D G C || C G D A E || | |||
|| Meantone [6] || ??? || G C D A E B || | |||
|| Meantone [7] || C G D A E B F# || C G D A E B F# || | |||
|| Meantone [8] || ??? || C G D A E B F# C# || | |||
|| Meantone [9] || ??? || C G D A E B F# C# G# || | |||
|| Meantone [10] || ??? || C G D A E B F# C# G# D# || | |||
|| 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¢ || 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¢. In this case the UDP term "Dominant 8|3" is ambiguous. It 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. | |||
Admins: Please delete the blank page at http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Naming Rank-2 Scales</title></head><body> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Naming Rank-2 Scales</title></head><body>GMN notation (genchain mode numbering) is a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br /> | ||
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These <a class="wiki_link" href="/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="/periods%20and%20generators">generator-chain</a>, or genchain. The first note in the genchain is the tonic of mode #1, the 2nd note is the tonic of mode #2, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as &quot;fourth meantone heptatonic&quot;. If in D, as above, it would be &quot;D fourth meantone heptatonic&quot;.<br /> | These <a class="wiki_link" href="/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="/periods%20and%20generators">generator-chain</a>, or genchain. The first note in the genchain is the tonic of mode #1, the 2nd note is the tonic of mode #2, etc., somewhat analogous to harmonica positions. 4th Meantone[7] is spoken as &quot;fourth meantone heptatonic&quot;. If in D, as above, it would be &quot;D fourth meantone heptatonic&quot;.<br /> | ||
<br /> | <br /> | ||
The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The | The same seven modes, all with C as the tonic, to illustrate the difference between modes. Similar modes are grouped together. The modes' progression is from sharper to flatter.<br /> | ||
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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. For example, Meantone[5] is | 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 /> | |||
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 /> | |||
<br /> | <br /> | ||
Pentatonic meantone scales:<br /> | Pentatonic meantone scales:<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule:8:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-MODMOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-MODMOS scales&quot;/&gt; --><a name="How to name rank-2 scales-MODMOS scales"></a><!-- ws:end:WikiTextAnchorRule:8 --><strong><u>MODMOS scales</u></strong></h2> | ||
As in modal UDP notation, these are written as MOS scales with chromatic alterations. To find the scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the compacted genchain, then add the appropriate alterations.<br /> | As in modal UDP notation, these are written as MOS scales with chromatic alterations. To find the scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the compacted genchain, then add the appropriate alterations.<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:9:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-Fractional-octave periods&quot; title=&quot;Anchor: How to name rank-2 scales-Fractional-octave periods&quot;/&gt; --><a name="How to name rank-2 scales-Fractional-octave periods"></a><!-- ws:end:WikiTextAnchorRule:9 --><strong><u>Fractional-octave periods</u></strong></h2> | ||
Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br /> | Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br /> | ||
Eb -- Bb -- F --- C --- G<br /> | Eb -- Bb -- F --- C --- G<br /> | ||
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<br /> | <br /> | ||
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4. L = 146¢ and s = 94¢. There are five genchains. Ups and downs are used to distinguish between 5/4 and 2\5.<br /> | There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is 5/4. L = 146¢ and s = 94¢. There are five genchains. Ups and downs are used to distinguish between 5/4 and 2\5, which is necessary to avoid duplicate note names.<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:10:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-Non-MOS scales&quot;/&gt; --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:10 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2> | ||
Examples:<br /> | Examples:<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
F G A C E F, with genchain <u><strong>F</strong></u> C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6.<br /> | F G A C E F, with genchain <u><strong>F</strong></u> C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6] no 6.<br /> | ||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>Why not just use UDP notation?</u></h2> | |||
One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction.<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>scale<br /> | |||
</td> | |||
<td>UDP generator<br /> | |||
</td> | |||
<td>UDP genchain<br /> | |||
</td> | |||
<td>GMN generator<br /> | |||
</td> | |||
<td>GMN genchain<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [2]<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>C G<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>C G<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [3]<br /> | |||
</td> | |||
<td>4/3<br /> | |||
</td> | |||
<td>D G C<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>C G D<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [5]<br /> | |||
</td> | |||
<td>4/3<br /> | |||
</td> | |||
<td>E A D G C<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>C G D A E<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [7]<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>C G D A E B F#<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>C G D A E B F#<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. GMN notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful.<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>scale<br /> | |||
</td> | |||
<td>UDP genchain<br /> | |||
</td> | |||
<td>GMN genchain<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [2]<br /> | |||
</td> | |||
<td>C G<br /> | |||
</td> | |||
<td>C G<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [3]<br /> | |||
</td> | |||
<td>D G C<br /> | |||
</td> | |||
<td>C G D<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [4]<br /> | |||
</td> | |||
<td>???<br /> | |||
</td> | |||
<td>C G D A<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [5]<br /> | |||
</td> | |||
<td>E A D G C<br /> | |||
</td> | |||
<td>C G D A E<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [6]<br /> | |||
</td> | |||
<td>???<br /> | |||
</td> | |||
<td>G C D A E B<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [7]<br /> | |||
</td> | |||
<td>C G D A E B F#<br /> | |||
</td> | |||
<td>C G D A E B F#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [8]<br /> | |||
</td> | |||
<td>???<br /> | |||
</td> | |||
<td>C G D A E B F# C#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [9]<br /> | |||
</td> | |||
<td>???<br /> | |||
</td> | |||
<td>C G D A E B F# C# G#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [10]<br /> | |||
</td> | |||
<td>???<br /> | |||
</td> | |||
<td>C G D A E B F# C# G# D#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [11]<br /> | |||
</td> | |||
<td>???<br /> | |||
</td> | |||
<td>C G D A E B F# C# G# D# A#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone [12] if generator &lt; 700¢<br /> | |||
</td> | |||
<td>E# A# D# G# C# F# B E A D G C<br /> | |||
</td> | |||
<td>C G D A E B F# C# G# D# A# E#<br /> | |||
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<td style="text-align: left;">Meantone [12] if generator &gt; 700¢<br /> | |||
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<td>C G D A E B F# C# G# D# A# E#<br /> | |||
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<td style="text-align: center;">&quot;<br /> | |||
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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¢. In this case the UDP term &quot;Dominant 8|3&quot; is ambiguous. It could mean either &quot;4th Dominant[12]&quot; or &quot;9th Dominant[12]&quot;<br /> | |||
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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 /> | |||
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