Comparison of mode notation systems: Difference between revisions
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 23:45:27 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593194590</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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C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C | C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C | ||
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc. | C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc. | ||
==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**== | ==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**== | ||
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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, 7th mode not 3rd mode) | ||
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 (harmonic minor) | ||
6th Meantone [7] #3: C Db E F G Ab Bb C | 6th Meantone [7] #3: C Db E F G Ab Bb C (phrygian dominant) | ||
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") | The 3rd scale breaks the pattern to avoid an altered tonic ("3rd Meantone [7] #1"). | ||
Ascending melodic minor modes: | |||
1st Meantone [7] #6 #7: C D E F# G# A B C | 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 | 2nd Meantone [7] #6 #7: C Db Eb Fb Gb Ab Bb C | ||
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7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C | 7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C | ||
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wA ------ yC# | wA ------ yC# | ||
Using ups and downs to mean "raised/lowered by ~ | Using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4": | ||
|| scale name || Ls pattern || example in C || genchains || | || scale name || Ls pattern || example in C || genchains || | ||
|| 1st Blackwood [10] || Ls Ls Ls Ls Ls || 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] || Ls Ls Ls Ls Ls || 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]]**__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 can be named Meantone [6], Meantone [8], etc. | 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. Curly brackets can be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}. | ||
However chromatic alterations create genchains with gaps that are very difficult to name, such as F C * D A E B F# * G#. Is it F Meantone {8} #2 (G is a 2nd above F) or F Meantone {8} #3? These scales must be named as MOS scales with notes added or removed, using "add" and "no", analogous to altered jazz chords. As with MODMOS scales, there is often more than one name for a scale. | |||
|| scale || genchain || | || scale || genchain || name || | ||
|| octotonic: | || octotonic: || || || | ||
|| C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone | || C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone {8} || | ||
||= " ||= " || C 1st Meantone [7] add b4 | || " || " || C 2nd 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 | ||= " ||= " || C 1st Meantone [7] add b4 || | ||
|| A B C D D# E F G# A || F C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #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 G# A || C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 no6 | || A B C D D# E F G# A || F C * D __**A**__ E B * * G# D# || A 5th Meantone [7] #7 add #4 || | ||
|| nonotonic: | || 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 F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone [7] add #4, #7 | || nonotonic: || || || | ||
||= " ||= " || A 2nd Meantone [7] add #4, b7 | || A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone {9} || | ||
||= " ||= " || A 1st Meantone [7] add b4, b7 | || " || " || A 3rd 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 | ||= " ||= " || A 2nd Meantone [7] add #4, b7 || | ||
|| hexatonic: | ||= " ||= " || A 1st Meantone [7] add b4, b7 || | ||
|| F G A C D E F || __**F**__ C G D A E || F | || 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 || | ||
||= " ||= " || F 1st Meantone [7] no4 | || hexatonic: || || || | ||
|| G A C D E F# G || C __**G**__ D A E * F# || G 2nd Meantone [7] no3 | || F G A C D E F || __**F**__ C G D A E || F 1st Meantone {6} || | ||
|| pentatonic: | || " || " || F 2nd Meantone [7] no4 || | ||
|| F G A C E F || __**F**__ C G * A E || F 2nd Meantone [7] no4 no6 | ||= " ||= " || F 1st Meantone [7] no4 || | ||
||= " ||= " || F 1st Meantone [7] no4 no6 | || G A C D E F# G || C __**G**__ D A E * F# || G 2nd Meantone [7] no3 || | ||
|| A B C E F A || F C * * __**A**__ E B || A 5th Meantone [7] no4 no7 | || pentatonic: || || || | ||
|| F G A C E F || __**F**__ C G * A E || F 2nd 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 || | |||
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. | ||
The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be "F 1st Meantone [5] no5 add b6" and "A 3rd Meantone [5] no4 no7 add #5, #2". | The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be "F 1st Meantone [5] no5 add b6" and "A 3rd Meantone [5] no4 no7 add #5, #2". | ||
A | Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4. | ||
A | Another category is scales that would be MOS, but the generator is too sharp or flat. For example, a chain of 750¢ fifths: F C G D A E B is tuned A C B D F E G A. This scale is best named as Meantone [5] with added notes: Which brings us to... | ||
==[[#How to name rank-2 scales-Non-MOS scales]]__Numbering considerations__== | |||
As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what #4 means in a pentatonic or hexatonic context. | |||
If the scale is written using heptatonically using 7 /note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, e.g. H J K L #M, it would be 1st Meantone [5] #5. | |||
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===__**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?**__=== | ||
There are centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine): | |||
"Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio __**<span class="mw-redirect">3:2</span>**__ (i.e., the untempered perfect __**fifth**__)." -- [[https://en.wikipedia.org/wiki/Pythagorean_tuning|en.wikipedia.org/wiki/Pythagorean_tuning]] | "Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio __**<span class="mw-redirect">3:2</span>**__ (i.e., the untempered perfect __**fifth**__)." -- [[https://en.wikipedia.org/wiki/Pythagorean_tuning|en.wikipedia.org/wiki/Pythagorean_tuning]] | ||
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===__**Then why not always choose the larger of the two generators?**__=== | ===__**Then why not always choose the larger of the two generators?**__=== | ||
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.) | 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.) | ||
===__Why not always choose the chroma-positive generator?__=== | ===__Why not always choose the chroma-positive generator?__=== | ||
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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. | 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. | 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. | ||
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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:32:&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:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Kite Giedraitis method">Kite Giedraitis method</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><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:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-MODMOS scales">MODMOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Fractional-octave periods">Fractional-octave periods</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Rank-2 scales that are neither MOS nor MODMOS">Rank-2 scales that are neither MOS nor MODMOS</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Numbering considerations">Numbering considerations</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Explanation / Rationale">Explanation / Rationale</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><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: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><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: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><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: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><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: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><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:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 1em;"><a href="#Jake Freivald method">Jake Freivald method</a></div> | |||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><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:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Request for admins">Request for admins</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --></div> | ||
<!-- ws:end:WikiTextTocRule:49 --><!-- 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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C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.<br /> | C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Kite Giedraitis method-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule: | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Kite Giedraitis method-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:50:&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:50 --><strong><u>MODMOS scales</u></strong></h2> | |||
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<a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales are named as chromatic alterations of a MOS scale, similar to UDP notation. &quot;#&quot; means raised by L-s. For <em>some-temperament-name</em>[N], &quot;#&quot; means moved N steps on the genchain, forwards if the generator is chroma-positive, otherwise backwards.<br /> | <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales are named as chromatic alterations of a MOS scale, similar to UDP notation. &quot;#&quot; means raised by L-s. For <em>some-temperament-name</em>[N], &quot;#&quot; means moved N steps on the genchain, forwards if the generator is chroma-positive, otherwise backwards.<br /> | ||
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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)<br /> | 7th Meantone [7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern, 7th mode not 3rd mode)<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 (harmonic minor)<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 (phrygian dominant)<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 /> | The 3rd scale breaks the pattern to avoid an altered tonic (&quot;3rd Meantone [7] #1&quot;).<br /> | ||
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Ascending melodic minor modes:<br /> | |||
1st Meantone [7] #6 #7: C D E F# G# A B C<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 /> | 2nd Meantone [7] #6 #7: C Db Eb Fb Gb Ab Bb C<br /> | ||
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7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C<br /> | 7th Meantone [7] #6 #7: C D Eb F Gb Ab Bb C<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Kite Giedraitis method-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:51:&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:51 --><strong><u>Fractional-octave periods</u></strong></h2> | ||
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Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.<br /> | Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.<br /> | ||
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wA ------ yC#<br /> | wA ------ yC#<br /> | ||
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Using ups and downs to mean &quot;raised/lowered by ~ | Using ups and downs to mean &quot;raised/lowered by 2/5 of an octave minus ~5/4&quot;:<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: | <!-- 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:52:&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:52 --><strong><u>Rank-2 scales that are neither MOS nor MODMOS</u></strong></h2> | ||
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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. | 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. Curly brackets can be used to distinguish them from MOS scales: Meantone {6} and Meantone {8}.<br /> | ||
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However chromatic alterations create genchains with gaps that are very difficult to name, such as F C * D A E B F# * G#. Is it F Meantone {8} #2 (G is a 2nd above F) or F Meantone {8} #3? These scales must be named as MOS scales with notes added or removed, using &quot;add&quot; and &quot;no&quot;, analogous to altered jazz chords. As with MODMOS scales, there is often more than one name for a scale.<br /> | |||
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</td> | </td> | ||
<td>name<br /> | <td>name<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>octotonic:<br /> | <td>octotonic:<br /> | ||
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<td>F <u><strong>C</strong></u> G D A E B F#<br /> | <td>F <u><strong>C</strong></u> G D A E B F#<br /> | ||
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<td>C 2nd Meantone {8}<br /> | |||
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<td>&quot;<br /> | |||
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<td>&quot;<br /> | |||
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<td>C 2nd Meantone [7] add #4<br /> | <td>C 2nd Meantone [7] add #4<br /> | ||
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<td>C 1st Meantone [7] add b4<br /> | <td>C 1st Meantone [7] add b4<br /> | ||
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<td>C 3rd Meantone [7] add #4<br /> | <td>C 3rd Meantone [7] add #4<br /> | ||
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<td>A 5th Meantone [7] #7 add #4<br /> | <td>A 5th Meantone [7] #7 add #4<br /> | ||
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<td>A 5th Meantone [7] #7 add #4 no6<br /> | <td>A 5th Meantone [7] #7 add #4 no6<br /> | ||
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</tr> | </tr> | ||
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<td>nonotonic:<br /> | <td>nonotonic:<br /> | ||
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<td>G D <u><strong>A</strong></u> E B F# C# G# D#<br /> | <td>G D <u><strong>A</strong></u> E B F# C# G# D#<br /> | ||
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<td>A 3rd Meantone | <td>A 3rd Meantone {9}<br /> | ||
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<td>A 3rd Meantone [ | </tr> | ||
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<td>&quot;<br /> | |||
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<td>&quot;<br /> | |||
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<td>A 3rd Meantone [7] add #4, #7<br /> | |||
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<td>A 2nd Meantone [7] add #4, b7<br /> | <td>A 2nd Meantone [7] add #4, b7<br /> | ||
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<td>A 1st Meantone [7] add b4, b7<br /> | <td>A 1st Meantone [7] add b4, b7<br /> | ||
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<td>A 5th Meantone [7] add #4, #7<br /> | <td>A 5th Meantone [7] add #4, #7<br /> | ||
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</tr> | </tr> | ||
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<td>hexatonic:<br /> | <td>hexatonic:<br /> | ||
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<td><br /> | <td><br /> | ||
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<td><u><strong>F</strong></u> C G D A E<br /> | <td><u><strong>F</strong></u> C G D A E<br /> | ||
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<td>F 1st Meantone {6}<br /> | |||
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<td>&quot;<br /> | |||
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<td>&quot;<br /> | |||
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<td>F 2nd Meantone [7] no4<br /> | <td>F 2nd Meantone [7] no4<br /> | ||
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<td>F 1st Meantone [7] no4<br /> | <td>F 1st Meantone [7] no4<br /> | ||
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<td>G 2nd Meantone [7] no3<br /> | <td>G 2nd Meantone [7] no3<br /> | ||
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</tr> | </tr> | ||
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<td>pentatonic:<br /> | <td>pentatonic:<br /> | ||
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<td>F 2nd Meantone [7] no4 no6<br /> | <td>F 2nd Meantone [7] no4 no6<br /> | ||
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</td> | </td> | ||
<td>F 1st Meantone [7] no4 no6<br /> | <td>F 1st Meantone [7] no4 no6<br /> | ||
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<td>A 5th Meantone [7] no4 no7<br /> | <td>A 5th Meantone [7] no4 no7<br /> | ||
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The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be &quot;F 1st Meantone [5] no5 add b6&quot; and &quot;A 3rd Meantone [5] no4 no7 add #5, #2&quot;.<br /> | The pentatonic scales could be notated as Meantone [5], but this would be more awkward. The last two examples would be &quot;F 1st Meantone [5] no5 add b6&quot; and &quot;A 3rd Meantone [5] no4 no7 add #5, #2&quot;.<br /> | ||
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A | Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone [7] #7 no2 add #4.<br /> | ||
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A | Another category is scales that would be MOS, but the generator is too sharp or flat. For example, a chain of 750¢ fifths: F C G D A E B is tuned A C B D F E G A. This scale is best named as Meantone [5] with added notes: Which brings us to...<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Kite Giedraitis method-Numbering considerations"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- 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-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:53 --><u>Numbering considerations</u></h2> | |||
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As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what #4 means in a pentatonic or hexatonic context.<br /> | |||
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If the scale is written using heptatonically using 7 /note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, e.g. H J K L #M, it would be 1st Meantone [5] #5.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Kite Giedraitis method-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:14 --><!-- 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-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:54 --><u>Explanation / Rationale</u></h2> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Kite Giedraitis method-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3> | ||
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Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br /> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Kite Giedraitis method-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:18 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3> | ||
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There are 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;Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <u><strong><span class="mw-redirect">3:2</span></strong></u> (i.e., the untempered perfect <u><strong>fifth</strong></u>).&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</a><br /> | &quot;Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <u><strong><span class="mw-redirect">3:2</span></strong></u> (i.e., the untempered perfect <u><strong>fifth</strong></u>).&quot; -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</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 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Kite Giedraitis method-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:20 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Kite Giedraitis method-Explanation / Rationale-Why not always choose the chroma-positive generator?"></a><!-- ws:end:WikiTextHeadingRule:22 --><u>Why not always choose the chroma-positive generator?</u></h3> | ||
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See below.<br /> | See below.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Kite Giedraitis method-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:24 --><u><strong>Why not just use UDP notation?</strong></u></h3> | ||
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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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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 /> | 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 /> | 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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Jake Freivald method"></a><!-- ws:end:WikiTextHeadingRule:26 -->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:28:&lt;h2&gt; --><h2 id="toc14"><a name="Jake Freivald method-Extending to non-MOS"></a><!-- ws:end:WikiTextHeadingRule:28 -->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:30:&lt;h1&gt; --><h1 id="toc15"><a name="Request for admins"></a><!-- ws:end:WikiTextHeadingRule:30 -->Request for admins</h1> | ||
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