Comparison of mode notation systems: Difference between revisions
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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-22 19:50:59 UTC</tt>.<br> | ||
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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 proceed 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 5 fifths take one down a semitone, not up, the generator is chroma-negative, and the modes proceed 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 Meantone [5] || LsL ss || C Eb F Ab Bb C || Ab Eb Bb F __**C**__ || | ||= || 5th Meantone [5] || LsL ss || C Eb F Ab Bb C || Ab Eb Bb F __**C**__ || | ||
Chromatic meantone scales | Chromatic meantone scales. | ||
|| scale name || Ls pattern || example in C || genchain || | || scale name || Ls pattern || example in C || genchain || | ||
|| 1st Meantone [12] || | || 1st Meantone [12] || sLsL sLL sLsLL || C C# D D# E E# F# G G# A A# B C || __**C**__ G D A E B F# C# G# D# A# E# || | ||
|| 2nd Meantone [12] || | || 2nd Meantone [12] || sLsL LsL sLsLL || C C# D D# E F F# G G# A A# B C || F __**C**__ G D A E B F# C# G# D# A# || | ||
|| 3rd Meantone [12] || | || 3rd Meantone [12] || sLsL LsL sLLsL || C C# D D# E F F# G G# A Bb B C || Bb F __**C**__ G D A E B F# C# G# D# || | ||
|| 4th Meantone [12] || | || 4th Meantone [12] || sLLs LsL sLLsL || C C# D Eb E F F# G G# A Bb B C || Eb Bb F __**C**__ G D A E B F# C# G# || | ||
|| 5th Meantone [12] || | || 5th Meantone [12] || sLLs LsL LsLsL || C C# D Eb E F F# G Ab A Bb B C || Ab Eb Bb F __**C**__ G D A E B F# C# || | ||
|| 6th Meantone [12] || | || 6th Meantone [12] || LsLs LsL LsLsL || C Db D Eb E F F# G Ab A Bb B C || Db Ab Eb Bb F __**C**__ G D A E B F# || | ||
|| 7th Meantone [12] || | || 7th Meantone [12] || LsLs LLs LsLsL || C Db D Eb E F Gb G Ab A Bb B C || Gb Db Ab Eb Bb F __**C**__ G D A E B || | ||
||= etc. || || || || | ||= etc. || || || || | ||
If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged. | |||
[[Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19) | [[Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19) | ||
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|| 7th Sensi [8] || Lss Ls Lss || C D Eb E# Gb G# A# B C || A# D Gb B Eb G# __**C**__ E# || | || 7th Sensi [8] || Lss Ls Lss || C D Eb E# Gb G# A# B C || A# D Gb B Eb G# __**C**__ E# || | ||
|| 8th Sensi [8] || Ls Lss Lss || C D Eb F Gb G# A# B C || F A# D Gb B Eb G# __**C**__ || | || 8th Sensi [8] || Ls Lss Lss || C D Eb F Gb G# A# B C || F A# D Gb B Eb G# __**C**__ || | ||
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation: | |||
====C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C==== | |||
They would follow a more regular pattern if using octotonic fourth-based notation: | |||
====A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A==== | |||
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc. | |||
Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[xenharmonic/ups and downs notation|ups and downs notation]]. | Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[xenharmonic/ups and downs notation|ups and downs notation]]. | ||
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==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**== | ==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**== | ||
To find a [[MODMOS Scales|MODMOS]] scale's name, apply chromatic alterations to the MOS scale, using scale degrees, similar to UDP notation. "#" means raised by L-s, and for //some-temperament-name//[N], "#" means moved N steps on the genchain, either forwards or backwards. | |||
To find a [[MODMOS Scales|MODMOS]] scale's name, | |||
The | The ascending melodic minor scale is 5th Meantone [7] #6 #7. MODMOS names are ambiguous. This scale could also be written as 2nd Meantone [7] b3 (major scale with a minor 3rd), or as 4th Meantone [7] #7 (dorian with a major 7th). | ||
3rd | |||
4th Meantone [7 | |||
7th | |||
|| old scale name || example in A || new scale name || | |||
|| Harmonic minor || A B C D E F G# A || 5th Meantone [7] #7 || | |||
2nd Meantone [7 | || Ascending melodic minor || A B C D E F# G# A || 5th Meantone [7] #6 #7 || | ||
||= " ||= " || 2nd Meantone [7] b3 || | |||
||= " ||= " || 4th Meantone [7] #7 || | |||
|| Double harmonic minor || A B C D# E F G# A || 5th Meantone [7] #4 #7 || | |||
||= " ||= " || 1st Meantone [7] b3 b6 || | |||
|| Double harmonic major || A Bb C# D E F G# A || 2nd Meantone [7] b2 b6 || | |||
||= " ||= " || 6th Meantone [7] #3 #7 || | |||
|| <span class="mw-redirect">Hungarian gypsy </span>minor || A B C D# E F G A || 5th Meantone [7] #4 || | |||
|| Phrygian dominant || A Bb C# D E F G A || 6th Meantone [7] #3 || | |||
The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from A dorian to A melodic minor. In this context, melodic minor might better be described as an altered dorian scale. | |||
The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is called 1st Meantone [7] b3 b6 and the double harmonic major is 6th Meantone [7] #3 #7, one can't tell that they are modes of each other. | |||
Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes: | Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes: | ||
1st Meantone [7] #2: C D# E F# G A B C | 1st Meantone [7] #2: C D# E F# G A B C | ||
2nd Meantone [7] #:5 C D E F G# A B C | 2nd Meantone [7] #:5 C D E F G# A B C | ||
7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern | 7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern) | ||
4th Meantone [7] #4: C D Eb F# G A Bb C | 4th Meantone [7] #4: C D Eb F# G A Bb C | ||
5th Meantone [7] #7: C D Eb F G Ab B C | 5th Meantone [7] #7: C D Eb F G Ab B C | ||
6th Meantone [7] #3: C Db E F G Ab Bb C | 6th Meantone [7] #3: C Db E F G Ab Bb C | ||
7th Meantone [7] #6: C Db Eb F Gb A Bb C | 7th Meantone [7] #6: C Db Eb F Gb A Bb C | ||
The 3rd scale breaks the pattern to avoid an altered tonic ("3rd Meantone [7] #1") | |||
Melodic minor modes: | |||
1st Meantone [7] #6 #7: C D E F# G# A B C | |||
2nd Meantone [7] #6 #7: C Db Eb Fb Gb Ab Bb C | |||
3rd Meantone [7] #6 #7: C D E F# G A Bb C | |||
4th Meantone [7] #6 #7: C D Eb F G A B C | |||
5th Meantone [7] #6 #7: C D E F G Ab Bb C | |||
6th Meantone [7] #6 #7: C Db Eb F G A Bb C | |||
7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C | |||
A pentatonic scale like C D E G A# is written 1st Meantone [5] #6. But C D Fb G Bb would be 3rd Meantone[5] b4. Scale degrees are heptatonic not pentatonic (#6 not #5) because while the scale is pentatonic, the notation uses 7 letters and is inherently heptatonic. If the scale were written H J K L #M, one would use #5. | |||
==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**== | ==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**== | ||
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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. | ||
Other problems with UDP are more of a taste issue. 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, as noted above, 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. 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. | Other problems with UDP are more of a taste issue. 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, as noted above, 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. 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. | ||
=Jake Freivald method= | =Jake Freivald method= | ||
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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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> </h1> | <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:34:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Kite Giedraitis method">Kite Giedraitis method</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Proposed method of naming all possible rank-2 scales">Proposed method of naming all possible rank-2 scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 4em;"><a href="#Kite Giedraitis method-Proposed method of naming all possible rank-2 scales--C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C">C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C</a></div> | ||
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 4em;"><a href="#Kite Giedraitis method-Proposed method of naming all possible rank-2 scales--A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A">A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A</a></div> | |||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-MODMOS scales">MODMOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Fractional-octave periods">Fractional-octave periods</a></div> | |||
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method- | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Non-MOS non-MODMOS scales">Non-MOS non-MODMOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Explanation / Rationale">Explanation / Rationale</a></div> | ||
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-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?</a></div> | ||
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-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?</a></div> | ||
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Why not | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Then why not always choose the larger of the two generators?">Then why not always choose the larger of the two generators?</a></div> | ||
<!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Why | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Why not always choose the chroma-positive generator?">Why not always choose the chroma-positive generator?</a></div> | ||
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 3em;"><a href="#Kite Giedraitis method-Explanation / Rationale-Why not just use UDP notation?">Why not just use UDP notation?</a></div> | ||
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Jake Freivald method">Jake Freivald method</a></div> | |||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Jake Freivald method-Extending to non-MOS">Extending to non-MOS</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Request for admins">Request for admins</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --></div> | ||
<!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Kite Giedraitis method"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Kite</strong> Giedraitis method</u></h1> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><u><span style="font-size: 1.3em; line-height: 1.5;">Proposed method of naming all possible rank-2 scales</span></u></h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><u><span style="font-size: 1.3em; line-height: 1.5;">Proposed method of naming all possible rank-2 scales</span></u></h2> | ||
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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 proceed 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 5 fifths take one down a semitone, not up, the generator is chroma-negative, and the modes proceed 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 /> | ||
| Line 692: | Line 678: | ||
<br /> | <br /> | ||
Chromatic meantone scales | Chromatic meantone scales.<br /> | ||
| Line 709: | Line 695: | ||
<td>1st Meantone [12]<br /> | <td>1st Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>sLsL sLL sLsLL<br /> | ||
</td> | </td> | ||
<td>C C# D D# E E# F# G G# A A# B C<br /> | <td>C C# D D# E E# F# G G# A A# B C<br /> | ||
| Line 719: | Line 705: | ||
<td>2nd Meantone [12]<br /> | <td>2nd Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>sLsL LsL sLsLL<br /> | ||
</td> | </td> | ||
<td>C C# D D# E F F# G G# A A# B C<br /> | <td>C C# D D# E F F# G G# A A# B C<br /> | ||
| Line 729: | Line 715: | ||
<td>3rd Meantone [12]<br /> | <td>3rd Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>sLsL LsL sLLsL<br /> | ||
</td> | </td> | ||
<td>C C# D D# E F F# G G# A Bb B C<br /> | <td>C C# D D# E F F# G G# A Bb B C<br /> | ||
| Line 739: | Line 725: | ||
<td>4th Meantone [12]<br /> | <td>4th Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>sLLs LsL sLLsL<br /> | ||
</td> | </td> | ||
<td>C C# D Eb E F F# G G# A Bb B C<br /> | <td>C C# D Eb E F F# G G# A Bb B C<br /> | ||
| Line 749: | Line 735: | ||
<td>5th Meantone [12]<br /> | <td>5th Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>sLLs LsL LsLsL<br /> | ||
</td> | </td> | ||
<td>C C# D Eb E F F# G Ab A Bb B C<br /> | <td>C C# D Eb E F F# G Ab A Bb B C<br /> | ||
| Line 759: | Line 745: | ||
<td>6th Meantone [12]<br /> | <td>6th Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>LsLs LsL LsLsL<br /> | ||
</td> | </td> | ||
<td>C Db D Eb E F F# G Ab A Bb B C<br /> | <td>C Db D Eb E F F# G Ab A Bb B C<br /> | ||
| Line 769: | Line 755: | ||
<td>7th Meantone [12]<br /> | <td>7th Meantone [12]<br /> | ||
</td> | </td> | ||
<td> | <td>LsLs LLs LsLsL<br /> | ||
</td> | </td> | ||
<td>C Db D Eb E F Gb G Ab A Bb B C<br /> | <td>C Db D Eb E F Gb G Ab A Bb B C<br /> | ||
| Line 788: | Line 774: | ||
</table> | </table> | ||
If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.<br /> | |||
<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/Sensi">Sensi</a> [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)<br /> | <a class="wiki_link" href="/Sensi">Sensi</a> [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)<br /> | ||
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</table> | </table> | ||
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation: <br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h4&gt; --><h4 id="toc3"><a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales--C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C"></a><!-- ws:end:WikiTextHeadingRule:6 -->C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C</h4> | |||
They would follow a more regular pattern if using octotonic fourth-based notation:<br /> | |||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h4&gt; --><h4 id="toc4"><a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales--A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A"></a><!-- ws:end:WikiTextHeadingRule:8 -->A - A#/Bb - B - B# - Cb - C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A</h4> | |||
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.<br /> | |||
<br /> | |||
<br /> | <br /> | ||
Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using <a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation">ups and downs notation</a>.<br /> | Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using <a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation">ups and downs notation</a>.<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:53:&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:53 --><strong><u>MODMOS scales</u></strong></h2> | ||
<br /> | <br /> | ||
To find a <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scale's name, apply chromatic alterations to the MOS scale, using scale degrees, similar to UDP notation. &quot;#&quot; means raised by L-s, and for <em>some-temperament-name</em>[N], &quot;#&quot; means moved N steps on the genchain, either forwards or backwards.<br /> | |||
To find a <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scale's name, | <br /> | ||
The ascending melodic minor scale is 5th Meantone [7] #6 #7. MODMOS names are ambiguous. This scale could also be written as 2nd Meantone [7] b3 (major scale with a minor 3rd), or as 4th Meantone [7] #7 (dorian with a major 7th).<br /> | |||
<br /> | <br /> | ||
| Line 988: | Line 982: | ||
</td> | </td> | ||
<td>example in A<br /> | <td>example in A<br /> | ||
</td> | </td> | ||
<td>new scale name<br /> | <td>new scale name<br /> | ||
| Line 1,001: | Line 991: | ||
<td>A B C D E F G# A<br /> | <td>A B C D E F G# A<br /> | ||
</td> | </td> | ||
<td>5th Meantone [7] #7<br /> | |||
<td>5th Meantone [7 | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,013: | Line 999: | ||
<td>A B C D E F# G# A<br /> | <td>A B C D E F# G# A<br /> | ||
</td> | </td> | ||
<td>5th Meantone [7] #6 #7<br /> | |||
<td>5th Meantone [7 | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,025: | Line 1,007: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td>2nd Meantone [7] b3<br /> | |||
<td> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,037: | Line 1,015: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td>4th Meantone [7] #7<br /> | |||
<td> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,049: | Line 1,023: | ||
<td>A B C D# E F G# A<br /> | <td>A B C D# E F G# A<br /> | ||
</td> | </td> | ||
<td>5th Meantone [7] #4 #7<br /> | |||
<td>5th Meantone [7 | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,061: | Line 1,031: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
<td>1st Meantone [7] b3 b6<br /> | |||
<td>1st Meantone [7 | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,073: | Line 1,039: | ||
<td>A Bb C# D E F G# A<br /> | <td>A Bb C# D E F G# A<br /> | ||
</td> | </td> | ||
<td> | <td>2nd Meantone [7] b2 b6<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,085: | Line 1,047: | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
< | <td>6th Meantone [7] #3 #7<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,097: | Line 1,055: | ||
<td>A B C D# E F G A<br /> | <td>A B C D# E F G A<br /> | ||
</td> | </td> | ||
<td>5th Meantone [7] #4<br /> | |||
<td>5th Meantone [7 | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,109: | Line 1,063: | ||
<td>A Bb C# D E F G A<br /> | <td>A Bb C# D E F G A<br /> | ||
</td> | </td> | ||
<td>6th Meantone [7] #3<br /> | |||
<td>6th Meantone [7 | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 1,143: | Line 1,069: | ||
<br /> | <br /> | ||
The | The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from A dorian to A melodic minor. In this context, melodic minor might better be described as an altered dorian scale.<br /> | ||
<br /> | <br /> | ||
The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is called 1st Meantone [7 | The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is called 1st Meantone [7] b3 b6 and the double harmonic major is 6th Meantone [7] #3 #7, one can't tell that they are modes of each other. <br /> | ||
<br /> | <br /> | ||
Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:<br /> | Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:<br /> | ||
1st Meantone [7] #2: C D# E F# G A B C<br /> | 1st Meantone [7] #2: C D# E F# G A B C<br /> | ||
2nd Meantone [7] #:5 C D E F G# A B C<br /> | 2nd Meantone [7] #:5 C D E F G# A B C<br /> | ||
7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern | 7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern)<br /> | ||
4th Meantone [7] #4: C D Eb F# G A Bb C<br /> | 4th Meantone [7] #4: C D Eb F# G A Bb C<br /> | ||
5th Meantone [7] #7: C D Eb F G Ab B C<br /> | 5th Meantone [7] #7: C D Eb F G Ab B C<br /> | ||
6th Meantone [7] #3: C Db E F G Ab Bb C<br /> | 6th Meantone [7] #3: C Db E F G Ab Bb C<br /> | ||
7th Meantone [7] #6: C Db Eb F Gb A Bb C<br /> | 7th Meantone [7] #6: C Db Eb F Gb A Bb C<br /> | ||
The 3rd scale breaks the pattern to avoid an altered tonic (&quot;3rd Meantone [7] #1&quot;)<br /> | |||
<br /> | |||
Melodic minor modes:<br /> | |||
1st Meantone [7] #6 #7: C D E F# G# A B C<br /> | |||
2nd Meantone [7] #6 #7: C Db Eb Fb Gb Ab Bb C<br /> | |||
3rd Meantone [7] #6 #7: C D E F# G A Bb C<br /> | |||
4th Meantone [7] #6 #7: C D Eb F G A B C<br /> | |||
5th Meantone [7] #6 #7: C D E F G Ab Bb C<br /> | |||
6th Meantone [7] #6 #7: C Db Eb F G A Bb C<br /> | |||
7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C<br /> | |||
<br /> | <br /> | ||
A pentatonic scale like C D E G A# is written 1st Meantone [5] #6. But C D Fb G Bb would be 3rd Meantone[5] b4. Scale degrees are heptatonic not pentatonic (#6 not #5) because while the scale is pentatonic, the notation uses 7 letters and is inherently heptatonic. If the scale were written H J K L #M, one would use #5.<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- ws:start:WikiTextAnchorRule:54:&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:54 --><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 /> | ||
| Line 1,401: | Line 1,306: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Kite Giedraitis method-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:14 --><!-- ws:start:WikiTextAnchorRule:55:&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:55 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2> | ||
<br /> | <br /> | ||
<strong><u>1st method</u>:</strong><br /> | <strong><u>1st method</u>:</strong><br /> | ||
| Line 1,436: | Line 1,341: | ||
G A C D E F# G, with genchain C <u><strong>G</strong></u> D A E * F# = G 3rd Meantone [6] #7<br /> | G A C D E F# G, with genchain C <u><strong>G</strong></u> D A E * F# = G 3rd Meantone [6] #7<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Kite Giedraitis method-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:16 --><!-- ws:start:WikiTextAnchorRule:56:&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:56 --><u>Explanation / Rationale</u></h2> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Kite Giedraitis method-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:18 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | ||
<br /> | <br /> | ||
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | ||
| Line 1,450: | Line 1,355: | ||
The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.<br /> | The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Kite Giedraitis method-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:20 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3> | ||
<br /> | <br /> | ||
Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):<br /> | Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):<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 a <u>wise</u> consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br /> | &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 a <u>wise</u> consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Kite Giedraitis method-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:22 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3> | ||
<br /> | <br /> | ||
Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)<br /> | Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Kite Giedraitis method-Explanation / Rationale-Why not always choose the chroma-positive generator?"></a><!-- ws:end:WikiTextHeadingRule:24 --><u>Why not always choose the chroma-positive generator?</u></h3> | ||
<br /> | <br /> | ||
See below.<br /> | See below.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Kite Giedraitis method-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:26 --><u><strong>Why not just use UDP notation?</strong></u></h3> | ||
<br /> | <br /> | ||
One problem with <a class="wiki_link" href="/Modal%20UDP%20Notation">UDP</a> 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. In Mode Numbers notation, the direction is unchanging.<br /> | One problem with <a class="wiki_link" href="/Modal%20UDP%20Notation">UDP</a> 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. In Mode Numbers notation, the direction is unchanging.<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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Other problems with UDP are more of a taste issue. 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, as noted above, 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. 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 /> | Other problems with UDP are more of a taste issue. 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, as noted above, 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. 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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Jake Freivald method"></a><!-- ws:end:WikiTextHeadingRule:28 -->Jake Freivald method</h1> | ||
My goals for numbering the modes are to make it as simple as possible for people to identify and use the modes they're talking about. As such, desired characteristics include<br /> | My goals for numbering the modes are to make it as simple as possible for people to identify and use the modes they're talking about. As such, desired characteristics include<br /> | ||
(1) as little knowledge needed as possible, to help the less-sophisticated user,<br /> | (1) as little knowledge needed as possible, to help the less-sophisticated user,<br /> | ||
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I also have built-in checks: I know that if I start and end with the same step size that I'm doing something wrong, and using the technique for meantone[7] gives me the diatonic major scale LLsLLLs, or CDEFGABC.<br /> | I also have built-in checks: I know that if I start and end with the same step size that I'm doing something wrong, and using the technique for meantone[7] gives me the diatonic major scale LLsLLLs, or CDEFGABC.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Jake Freivald method-Extending to non-MOS"></a><!-- ws:end:WikiTextHeadingRule:30 -->Extending to non-MOS</h2> | ||
<span style="line-height: 1.5;">My suggestion is that (a) you still start with the step size that occurs the largest number of times, and (b) you still push the largest cluster of that as far out as possible. </span><br /> | <span style="line-height: 1.5;">My suggestion is that (a) you still start with the step size that occurs the largest number of times, and (b) you still push the largest cluster of that as far out as possible. </span><br /> | ||
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NOTE: NO collapsing genchains. NO generator knowledge needed. No mapping knowledge (or indeed mapping at all) required. Extensible to higher ranks without problems. It doesn't matter whether the scale is a temperament at all.<br /> | NOTE: NO collapsing genchains. NO generator knowledge needed. No mapping knowledge (or indeed mapping at all) required. Extensible to higher ranks without problems. It doesn't matter whether the scale is a temperament at all.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Request for admins"></a><!-- ws:end:WikiTextHeadingRule:32 -->Request for admins</h1> | ||
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Admins: Please delete the blank page at <!-- ws:start:WikiTextUrlRule: | Admins: Please delete the blank page at <!-- ws:start:WikiTextUrlRule:1700:http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position --><a href="http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position">http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position</a><!-- ws:end:WikiTextUrlRule:1700 --><br /> | ||
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