Comparison of mode notation systems: Difference between revisions
Wikispaces>TallKite **Imported revision 593126030 - Original comment: ** |
Wikispaces>TallKite **Imported revision 593128374 - 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>2016-09-23 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-23 04:54:02 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593128374</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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|| 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__.** | 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 (see below). **__Unlike modal UDP notation, the generator isn't always chroma-positive__.** There are many disadvantages of chroma-positive generators, see [[http://xenharmonic.wikispaces.com/Kite%20Giedraitis%20method-Explanation%20/Kite%20Giedraitis%20method-Explanation%20/%20Rationale-Why%20not%20just%20use%20UDP%20notation?|Explanation / Rationale-Why not just use UDP notation?]] | ||
Pentatonic meantone scales: | Pentatonic meantone scales: | ||
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==[[#How to name rank-2 scales-Non-MOS scales]]**__Rank-2 scales that are neither MOS nor MODMOS__**== | ==[[#How to name rank-2 scales-Non-MOS scales]]**__Rank-2 scales that are neither MOS nor MODMOS__**== | ||
One category are scales with too many or too few notes to be MOS. If they have an unbroken genchain, they | One category are scales with too many or too few notes to be MOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. However chromatic modifications create genchains with gaps that are very difficult to name. These scales must be named as altered MOS scales, using "add" and "no" to add or subtract notes, analogous to altered jazz chords. As with MODMOS scales, there is often more than one name for a scale. | ||
Reserving the //name//[//number//] format for only MOS and MODMOS scales | There's an advantage to naming even unbroken-genchain scales as altered MOS scales. Reserving the //name//[//number//] format for only MOS and MODMOS scales identifies what scale sizes can be MOS in unfamiliar temperaments. For example, Porcupine [8] is MOS but Porcupine [9] isn't. Writing Porcupine [9] as an altered Porcupine [8] indicates this. However there's also an advantage to the brevity and clarity of Porcupine [9]. | ||
|| scale || genchain || name || | || scale || genchain || name || alternate name || | ||
|| octotonic: || || || | || octotonic: || || || || | ||
|| C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone [7] add #4 || | || C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone [7] add #4 || C 2nd Meantone [8] || | ||
||= " ||= " || C 1st Meantone [7] add b4 || | ||= " ||= " || C 1st Meantone [7] add b4 || || | ||
|| C D E F F# G A Bb C || Bb F __**C**__ G D A E * F# || C 3rd Meantone [7] add #4 || | || C D E F F# G A Bb C || Bb F __**C**__ G D A E * F# || C 3rd Meantone [7] add #4 || || | ||
|| A B C D D# E F G# A || F C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 || | || A B C D D# E F G# A || F C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 || || | ||
|| A B C D D# E G# A || C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 no6 || | || A B C D D# E G# A || C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 no6 || || | ||
|| nonotonic: || || || | || nonotonic: || || || || | ||
|| A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone [7] add #4, #7 || | || A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone [7] add #4, #7 || A 3rd Meantone [9] || | ||
||= " ||= " || A 2nd Meantone [7] add #4, b7 || | ||= " ||= " || A 2nd Meantone [7] add #4, b7 || || | ||
||= " ||= " || A 1st Meantone [7] add b4, b7 || | ||= " ||= " || A 1st Meantone [7] add b4, b7 || || | ||
|| A B C D D# E F G G# A || F C G D __**A**__ E B * * G# D# || A 5th Meantone [7] add #4, #7 || | || A B C D D# E F G G# A || F C G D __**A**__ E B * * G# D# || A 5th Meantone [7] add #4, #7 || || | ||
|| hexatonic: || || || | || hexatonic: || || || || | ||
|| F G A C D E F || __**F**__ C G D A E || F 2nd Meantone [7] no4 || | || F G A C D E F || __**F**__ C G D A E || F 2nd Meantone [7] no4 || F 1st Meantone [6] || | ||
||= " ||= " || F 1st Meantone [7] no4 || | ||= " ||= " || F 1st Meantone [7] no4 || || | ||
|| G A C D E F# G || C __**G**__ D A E * F# || G 2nd Meantone [7] no3 || | || G A C D E F# G || C __**G**__ D A E * F# || G 2nd Meantone [7] no3 || || | ||
|| pentatonic: || || || | || pentatonic: || || || || | ||
|| F G A C E F || __**F**__ C G * A E || F 2nd Meantone [7] no4 no6 || | || F G A C E F || __**F**__ C G * A E || F 2nd Meantone [7] no4 no6 || || | ||
||= " ||= " || F 1st Meantone [7] no4 no6 || | ||= " ||= " || F 1st Meantone [7] no4 no6 || || | ||
|| A B C E F A || F C * * __**A**__ E B || A 5th Meantone [7] no4 no7 || | || A B C E F A || F C * * __**A**__ E B || A 5th Meantone [7] no4 no7 || || | ||
In the 2nd example, "add b4" means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th. | In the 2nd example, "add b4" means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th. | ||
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|| Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E || | || Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E || | ||
|| Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# || | || Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# || | ||
|| Meantone[12] in 31edo ||= 4/3 || E# A# D# G# C# F# | || Meantone[12] in 31edo ||= 4/3 || E# A# D# G# C# F# | ||
B E A D G C ||= 3/2 || C G D A E B F# C# G# | B E A D G C ||= 3/2 || C G D A E B F# C# G# | ||
D# A# E# || | D# A# E# || | ||
|| Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C# | || Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C# | ||
G# D# A# E# B# | G# D# A# E# B# | ||
FxCx Gx Dx Ax Ex ||= 3/2 || C G D A E B F# C# G# | FxCx Gx Dx Ax Ex ||= 3/2 || C G D A E B F# C# G# | ||
D# A# E# B# Fx Cx Gx | D# A# E# B# Fx Cx Gx | ||
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An even larger problem is that the notation is overly tuning-dependent. 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. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 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 more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant [12]" or "9th Dominant [12]". Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it. | An even larger problem is that the notation is overly tuning-dependent. 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. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 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 more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant [12]" or "9th Dominant [12]". Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it. | ||
Three other problems with UDP are more issues of taste. 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). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Shrutal [10] 6|2(2), use four numbers where only two are needed. | |||
Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C. | |||
Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. | |||
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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><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> | 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 (see below). <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> There are many disadvantages of chroma-positive generators, see <a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/Kite%20Giedraitis%20method-Explanation%20/Kite%20Giedraitis%20method-Explanation%20/%20Rationale-Why%20not%20just%20use%20UDP%20notation?" rel="nofollow">Explanation / Rationale-Why not just use UDP notation?</a><br /> | ||
<br /> | <br /> | ||
Pentatonic meantone scales:<br /> | Pentatonic meantone scales:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:49:&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:49 --><strong><u>Rank-2 scales that are neither MOS nor MODMOS</u></strong></h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Rank-2 scales that are neither MOS nor MODMOS"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:49:&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:49 --><strong><u>Rank-2 scales that are neither MOS nor MODMOS</u></strong></h2> | ||
<br /> | <br /> | ||
One category are scales with too many or too few notes to be MOS. If they have an unbroken genchain, they | One category are scales with too many or too few notes to be MOS. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. However chromatic modifications create genchains with gaps that are very difficult to name. These scales must be named as altered MOS scales, using &quot;add&quot; and &quot;no&quot; to add or subtract notes, analogous to altered jazz chords. As with MODMOS scales, there is often more than one name for a scale.<br /> | ||
<br /> | <br /> | ||
Reserving the <em>name</em>[<em>number</em>] format for only MOS and MODMOS scales | There's an advantage to naming even unbroken-genchain scales as altered MOS scales. Reserving the <em>name</em>[<em>number</em>] format for only MOS and MODMOS scales identifies what scale sizes can be MOS in unfamiliar temperaments. For example, Porcupine [8] is MOS but Porcupine [9] isn't. Writing Porcupine [9] as an altered Porcupine [8] indicates this. However there's also an advantage to the brevity and clarity of Porcupine [9].<br /> | ||
<br /> | <br /> | ||
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</td> | </td> | ||
<td>name<br /> | <td>name<br /> | ||
</td> | |||
<td>alternate name<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>octotonic:<br /> | <td>octotonic:<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>C 2nd Meantone [7] add #4<br /> | <td>C 2nd Meantone [7] add #4<br /> | ||
</td> | |||
<td>C 2nd Meantone [8]<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>C 1st Meantone [7] add b4<br /> | <td>C 1st Meantone [7] add b4<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>C 3rd Meantone [7] add #4<br /> | <td>C 3rd Meantone [7] add #4<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 5th Meantone [7] #7 add #4<br /> | <td>A 5th Meantone [7] #7 add #4<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 5th Meantone [7] #7 add #4 no6<br /> | <td>A 5th Meantone [7] #7 add #4 no6<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>nonotonic:<br /> | <td>nonotonic:<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>A 3rd Meantone [7] add #4, #7<br /> | <td>A 3rd Meantone [7] add #4, #7<br /> | ||
</td> | |||
<td>A 3rd Meantone [9]<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 2nd Meantone [7] add #4, b7<br /> | <td>A 2nd Meantone [7] add #4, b7<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 1st Meantone [7] add b4, b7<br /> | <td>A 1st Meantone [7] add b4, b7<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 5th Meantone [7] add #4, #7<br /> | <td>A 5th Meantone [7] add #4, #7<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>hexatonic:<br /> | <td>hexatonic:<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>F 2nd Meantone [7] no4<br /> | <td>F 2nd Meantone [7] no4<br /> | ||
</td> | |||
<td>F 1st Meantone [6]<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>F 1st Meantone [7] no4<br /> | <td>F 1st Meantone [7] no4<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>G 2nd Meantone [7] no3<br /> | <td>G 2nd Meantone [7] no3<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>pentatonic:<br /> | <td>pentatonic:<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</td> | </td> | ||
<td>F 2nd Meantone [7] no4 no6<br /> | <td>F 2nd Meantone [7] no4 no6<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>F 1st Meantone [7] no4 no6<br /> | <td>F 1st Meantone [7] no4 no6<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 5th Meantone [7] no4 no7<br /> | <td>A 5th Meantone [7] no4 no7<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">4/3<br /> | <td style="text-align: center;">4/3<br /> | ||
</td> | </td> | ||
<td>E# A# D# G# C# F# <br /> | <td>E# A# D# G# C# F#<br /> | ||
B E A D G C<br /> | B E A D G C<br /> | ||
</td> | </td> | ||
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<td style="text-align: center;">3/2<br /> | <td style="text-align: center;">3/2<br /> | ||
</td> | </td> | ||
<td>C G D A E B F# C# <br /> | <td>C G D A E B F# C#<br /> | ||
G# D# A# E# B# <br /> | G# D# A# E# B#<br /> | ||
FxCx Gx Dx Ax Ex<br /> | FxCx Gx Dx Ax Ex<br /> | ||
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An even larger problem is that the notation is overly tuning-dependent. 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. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 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 more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example &quot;Dominant 8|3&quot; could mean either &quot;4th Dominant [12]&quot; or &quot;9th Dominant [12]&quot;. Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.<br /> | An even larger problem is that the notation is overly tuning-dependent. 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. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 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 more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example &quot;Dominant 8|3&quot; could mean either &quot;4th Dominant [12]&quot; or &quot;9th Dominant [12]&quot;. Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.<br /> | ||
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Three other problems with UDP are more issues of taste. 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). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Shrutal [10] 6|2(2), use four numbers where only two are needed. <br /> | |||
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Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.<br /> | |||
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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 Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.<br /> | |||
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