Comparison of mode notation systems: Difference between revisions
Wikispaces>TallKite **Imported revision 581087571 - 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>2016-04-25 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-04-25 03:56:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>581092615</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> | ||
<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">**This page is a work in progress...** | <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">**This page is a work in progress...** | ||
Question: number MODMOS modes by position in compacted or uncompacted genchain? | |||
Question: resolve the ambiguity for MODMOS mode numbers? | |||
**Mode numbers** provide 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: | **Mode numbers** provide 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: | ||
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|| old scale name || new scale name || Ls pattern || example in C || --------- genchain ------- || | || old scale name || new scale name || Ls pattern || example in C || --------- genchain ------- || | ||
|| major pentatonic || 1st Meantone[5] || ssL sL || C D E G A C ||> __**C**__ G D A E || | || major pentatonic || 1st Meantone[5] || ssL sL || C D E G A C ||> __**C**__ G D A E || | ||
||= | ||= || 2nd Meantone[5] || sLs sL || C D F G A C ||> F __**C**__ G D A -- || | ||
||= | ||= || 3rd Meantone[5] || sLs Ls || C D F G Bb C || -------- Bb F __**C**__ G D || | ||
|| minor pentatonic || 4th Meantone[5] || Lss Ls || C Eb F G Bb C || ---- Eb Bb F __**C**__ G || | || minor pentatonic || 4th Meantone[5] || Lss Ls || C Eb F G Bb C || ---- Eb Bb F __**C**__ G || | ||
||= | ||= || 5th Meantone[5] || LsL ss || C Eb F Ab Bb C || Ab Eb Bb F __**C**__ || | ||
Chromatic meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged. | Chromatic meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged. | ||
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|| Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || Bb F C G D __**A**__ E || 6th Meantone[7,+3] || | || Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || Bb F C G D __**A**__ E || 6th Meantone[7,+3] || | ||
|| Japanese pentatonic || A B C E F A || F C * * __**A**__ E B || __**A**__ E B F# C# || 1st Meantone[5,-4,-5] || | || Japanese pentatonic || A B C E F A || F C * * __**A**__ E B || __**A**__ E B F# C# || 1st Meantone[5,-4,-5] || | ||
|| (a mode of the above) || F A B C E F || __**F**__ C * * A E B || | || (a mode of the above) || F A B C E F || __**F**__ C * * A E B || A E B __**F#**__ C# || 4th Meantone[5,-4,-5] || | ||
Use the compacted genchain to find the mode number so that harmonic minor and melodic minor are both the 5th mode, just as standard minor is. Also so that two notes a (possibly aug or dim) fifth apart will still have adjacent mode numbers. | Use the compacted genchain to find the mode number so that harmonic minor and melodic minor are both the 5th mode, just as standard minor is. Also so that two notes a (possibly aug or dim) fifth apart will still have adjacent mode numbers. | ||
In the Japanese pentatonic scale, "[5,-4,-5]" moves the notes 7 steps in the genchain, not 5, because heptatonic scale degrees are used, even though the scale is pentatonic. The rationale for this is that the notation uses 7 letters, so the notation is still essentially heptatonic. In other words, F is the 5th note of the scale, but F is the 6th letter counting from the tonic A. If the notation used only 5 letters, perhaps H J K L M, the alteration would be written "b5". | |||
//A point to consider:// | //A point to consider:// | ||
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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 a just 5/4 = 386¢. L = 146¢ and s = 94¢ | The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains. | ||
|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || | |||
|| 1st Diminished[8] || sLsL sLsL || C Db Eb E Gb G A Bb C ||= __**C**__ G || Eb Bb || Gb Db || A E || | |||
|| 2nd Diminished[8] || LsLs LsLs || C D Eb F F# Ab A B C ||= F __**C**__ || Ab Eb || B F# || D A || | |||
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order 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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==[[#How to name rank-2 scales-Non-MOS scales]]**__Non-MOS non-MODMOS scales__**== | ==[[#How to name rank-2 scales-Non-MOS scales]]**__Non-MOS non-MODMOS scales__**== | ||
Compact the genchain to remove any gaps via | Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples: | ||
C D E F F# G A B C | C D E F F# G A B C has a genchain F __**C**__ G D A E B F#, and is named C 2nd Meantone[8]. | ||
C D E F F# G A Bb C, with genchain Bb F __**C**__ G D A E * F#. Alter Bb to get an unbroken genchain: F __**C**__ G D A E B F#. The scale is C 2nd Meantone[8] | C D E F F# G A Bb C, with genchain Bb F __**C**__ G D A E * F#. Alter Bb to get an unbroken genchain: F __**C**__ G D A E B F#. The scale is C 2nd Meantone[8,-7]. | ||
A B C D D# E F G G# A, with genchain F C G D __**A**__ E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D __**A**__ E B F# C# G# D#, giving the name A 3rd Meantone[9] | A B C D D# E F G G# A, with genchain F C G D __**A**__ E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D __**A**__ E B F# C# G# D#, giving the name A 3rd Meantone[9,-6,-7]. | ||
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] | 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,no4]. | ||
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===**__Why not number the modes in the order they occur in the scale?__**=== | ===**__Why not number the modes in the order they occur in the scale?__**=== | ||
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. | ||
Genchain-based: genchain = 1 2 3 4 5 6 7, Lydian scale = 1 3 5 7 2 4 6 1, major scale = 2 4 6 1 3 5 7 2 | |||
Scale-based: | |||
The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The disadvantage of genchain-based numbering is that the mode numbers are harder to find in the scale. However this is arguably an advantage, because in the course of | |||
===__**Why make an exception for 3/2 vs 4/3 as the generator?**__=== | ===__**Why make an exception for 3/2 vs 4/3 as the generator?**__=== | ||
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"The term "well temperament" or "good temperament" usually means some sort of <span class="new">irregular temperament</span> in which the tempered __**fifths**__ are of different sizes." -- [[https://en.wikipedia.org/wiki/Well_temperament|en.wikipedia.org/wiki/Well_temperament]] | "The term "well temperament" or "good temperament" usually means some sort of <span class="new">irregular temperament</span> in which the tempered __**fifths**__ are of different sizes." -- [[https://en.wikipedia.org/wiki/Well_temperament|en.wikipedia.org/wiki/Well_temperament]] | ||
"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike __wise__ consistencies, it doesn't reduce memorization, because everyone already knows that the generator is historically 3/2. | |||
===__**Then why not always choose the larger of the two generators?**__=== | ===__**Then why not always choose the larger of the two generators?**__=== | ||
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===__**Why not just use UDP notation?**__=== | ===__**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. | One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. | ||
|| scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain || | || scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain || | ||
|| Meantone [2] || 3/2 || C G || 3/2 || C G || | || Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E || | ||
|| Meantone [ | || Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# || | ||
|| Meantone [ | || 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# | |||
D# A# E# || | |||
|| Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C# G# | |||
D# A# E# B# Fx Cx Gx | |||
Dx Ax Ex ||= 3/2 || C G D A E B F# C# G# | |||
D# A# E# B# Fx Cx Gx | |||
Dx Ax Ex || | |||
|| scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain || | |||
|| Porcupine[7] || 9/5 || Bb Ab^ Gv F Eb^ Dv C || 10/9 || C Dv Eb^ F Gv Ab^ Bb || | |||
|| Porcupine[8] || 10/9 || C Dv Eb^ F Gv Ab^ Bb Cv || 10/9 || C Dv Eb^ F Gv Ab^ Bb Cv || | |||
|| Meantone [7] || 3/2 || C G D A E B F# || 3/2 || C G D A E B F# || | || Meantone [7] || 3/2 || C G D A E B F# || 3/2 || C G D A E B F# || | ||
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<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><strong>This page is a work in progress...</strong><br /> | <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><strong>This page is a work in progress...</strong><br /> | ||
Question: number MODMOS modes by position in compacted or uncompacted genchain?<br /> | |||
Question: resolve the ambiguity for MODMOS mode numbers?<br /> | |||
<br /> | <br /> | ||
<strong>Mode numbers</strong> provide 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 /> | <strong>Mode numbers</strong> provide 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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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>2nd Meantone[5]<br /> | <td>2nd Meantone[5]<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>3rd Meantone[5]<br /> | <td>3rd Meantone[5]<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>5th Meantone[5]<br /> | <td>5th Meantone[5]<br /> | ||
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<td><u><strong>F</strong></u> C * * A E B<br /> | <td><u><strong>F</strong></u> C * * A E B<br /> | ||
</td> | </td> | ||
<td> | <td>A E B <u><strong>F#</strong></u> C#<br /> | ||
</td> | </td> | ||
<td>4th Meantone[5,-4,-5]<br /> | <td>4th Meantone[5,-4,-5]<br /> | ||
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Use the compacted genchain to find the mode number so that harmonic minor and melodic minor are both the 5th mode, just as standard minor is. Also so that two notes a (possibly aug or dim) fifth apart will still have adjacent mode numbers.<br /> | Use the compacted genchain to find the mode number so that harmonic minor and melodic minor are both the 5th mode, just as standard minor is. Also so that two notes a (possibly aug or dim) fifth apart will still have adjacent mode numbers.<br /> | ||
<br /> | <br /> | ||
In the Japanese pentatonic scale, &quot;[5,-4,-5]&quot; moves the notes 7 steps in the genchain, not 5, because heptatonic scale degrees are used, even though the scale is pentatonic. The rationale for this is that the notation uses 7 letters, so the notation is still essentially heptatonic. In other words, F is the 5th note of the scale, but F is the 6th letter counting from the tonic A. If the notation used only 5 letters, perhaps H J K L M, the alteration would be written &quot;b5&quot;.<br /> | |||
<br /> | <br /> | ||
<em>A point to consider:</em><br /> | <em>A point to consider:</em><br /> | ||
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<br /> | <br /> | ||
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢ | The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains.<br /> | ||
<table class="wiki_table"> | |||
<tr> | |||
<td>scale name<br /> | |||
</td> | |||
<td>Ls pattern<br /> | |||
</td> | |||
<td>example in C<br /> | |||
</td> | |||
<td>1st chain<br /> | |||
</td> | |||
<td>2nd chain<br /> | |||
</td> | |||
<td>3rd chain<br /> | |||
</td> | |||
<td>4th chain<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1st Diminished[8]<br /> | |||
</td> | |||
<td>sLsL sLsL<br /> | |||
</td> | |||
<td>C Db Eb E Gb G A Bb C<br /> | |||
</td> | |||
<td style="text-align: center;"><u><strong>C</strong></u> G<br /> | |||
</td> | |||
<td>Eb Bb<br /> | |||
</td> | |||
<td>Gb Db<br /> | |||
</td> | |||
<td>A E<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2nd Diminished[8]<br /> | |||
</td> | |||
<td>LsLs LsLs<br /> | |||
</td> | |||
<td>C D Eb F F# Ab A B C<br /> | |||
</td> | |||
<td style="text-align: center;">F <u><strong>C</strong></u><br /> | |||
</td> | |||
<td>Ab Eb<br /> | |||
</td> | |||
<td>B F#<br /> | |||
</td> | |||
<td>D A<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.<br /> | |||
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<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:20:&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:20 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2> | <!-- 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:20:&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:20 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2> | ||
Compact the genchain to remove any gaps via | Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples:<br /> | ||
<br /> | <br /> | ||
C D E F F# G A B C | C D E F F# G A B C has a genchain F <u><strong>C</strong></u> G D A E B F#, and is named C 2nd Meantone[8].<br /> | ||
<br /> | <br /> | ||
C D E F F# G A Bb C, with genchain Bb F <u><strong>C</strong></u> G D A E * F#. Alter Bb to get an unbroken genchain: F <u><strong>C</strong></u> G D A E B F#. The scale is C 2nd Meantone[8] | C D E F F# G A Bb C, with genchain Bb F <u><strong>C</strong></u> G D A E * F#. Alter Bb to get an unbroken genchain: F <u><strong>C</strong></u> G D A E B F#. The scale is C 2nd Meantone[8,-7].<br /> | ||
<br /> | <br /> | ||
A B C D D# E F G G# A, with genchain F C G D <u><strong>A</strong></u> E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D <u><strong>A</strong></u> E B F# C# G# D#, giving the name A 3rd Meantone[9] | A B C D D# E F G G# A, with genchain F C G D <u><strong>A</strong></u> E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D <u><strong>A</strong></u> E B F# C# G# D#, giving the name A 3rd Meantone[9,-6,-7].<br /> | ||
<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] | 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,no4].<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | ||
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. <br /> | ||
Genchain-based: genchain = 1 2 3 4 5 6 7, Lydian scale = 1 3 5 7 2 4 6 1, major scale = 2 4 6 1 3 5 7 2<br /> | |||
Scale-based:<br /> | |||
<br /> | |||
The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The disadvantage of genchain-based numbering is that the mode numbers are harder to find in the scale. However this is arguably an advantage, because in the course of<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:10 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:10 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3> | ||
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&quot;The term &quot;well temperament&quot; or &quot;good temperament&quot; usually means some sort of <span class="new">irregular temperament</span> in which the tempered <u><strong>fifths</strong></u> are of different sizes.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Well_temperament" rel="nofollow">en.wikipedia.org/wiki/Well_temperament</a><br /> | &quot;The term &quot;well temperament&quot; or &quot;good temperament&quot; usually means some sort of <span class="new">irregular temperament</span> in which the tempered <u><strong>fifths</strong></u> are of different sizes.&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Well_temperament" rel="nofollow">en.wikipedia.org/wiki/Well_temperament</a><br /> | ||
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&quot;A foolish consistency is the hobgoblin of little minds&quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike <u>wise</u> consistencies, it doesn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:12 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:12 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Why not just use UDP notation?</strong></u></h3> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Why not just use UDP notation?</strong></u></h3> | ||
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 /> | One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction.<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Meantone [2]<br /> | <td>Meantone[5] in 31edo<br /> | ||
</td> | |||
<td style="text-align: center;">4/3<br /> | |||
</td> | |||
<td>E A D G C<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
<td>C G D A E<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone[7] in 31edo<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
<td>C G D A E B F#<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
<td>C G D A E B F#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone[12] in 31edo<br /> | |||
</td> | |||
<td style="text-align: center;">4/3<br /> | |||
</td> | |||
<td>E# A# D# G# C# F# B E<br /> | |||
A D G C<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
<td>C G D A E B F# C# G# <br /> | |||
D# A# E#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>Meantone[19] in 31edo<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
<td>C G D A E B F# C# G#<br /> | |||
D# A# E# B# Fx Cx Gx<br /> | |||
Dx Ax Ex<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
<td>C G D A E B F# C# G# <br /> | |||
D# A# E# B# Fx Cx Gx <br /> | |||
Dx Ax Ex<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
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<table class="wiki_table"> | |||
<tr> | |||
<td>scale<br /> | |||
</td> | </td> | ||
<td> | <td>UDP generator<br /> | ||
</td> | </td> | ||
<td> | <td>UDP genchain<br /> | ||
</td> | </td> | ||
<td> | <td>Mode Numbers generator<br /> | ||
</td> | </td> | ||
<td> | <td>Mode Numbers genchain<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>Porcupine[7]<br /> | ||
</td> | </td> | ||
<td> | <td>9/5<br /> | ||
</td> | </td> | ||
<td> | <td>Bb Ab^ Gv F Eb^ Dv C<br /> | ||
</td> | </td> | ||
<td> | <td>10/9<br /> | ||
</td> | </td> | ||
<td>C | <td>C Dv Eb^ F Gv Ab^ Bb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>Porcupine[8]<br /> | ||
</td> | </td> | ||
<td> | <td>10/9<br /> | ||
</td> | </td> | ||
<td> | <td>C Dv Eb^ F Gv Ab^ Bb Cv<br /> | ||
</td> | </td> | ||
<td> | <td>10/9<br /> | ||
</td> | </td> | ||
<td>C | <td>C Dv Eb^ F Gv Ab^ Bb Cv<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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