Comparison of mode notation systems: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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=__**Kite** Giedraitis method__= | =__**Kite** Giedraitis method__= | ||
==__<span style="font-size: 1.3em; line-height: 1.5;">Proposed method of naming all possible rank-2 scales</span>__== | ==__<span style="font-size: 1.3em; line-height: 1.5;">Proposed method of naming all possible rank-2 scales</span>__== | ||
**Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator: | **Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator: | ||
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|| Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | || Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || | ||
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. ** | 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 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. | |||
For | For more on the disadvantages of chroma-positive generators, see [[http://xenharmonic.wikispaces.com/Kite%20Giedraitis%20method-Explanation%20/Kite%20Giedraitis%20method-Explanation%20/%20Rationale-Why%20not%20just%20use%20UDP%20notation?|Explanation / Rationale-Why not just use UDP notation?]] | ||
Pentatonic meantone scales: | Pentatonic meantone scales: | ||
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|| 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: | 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 | C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C | ||
The modes would follow a more regular pattern if using octotonic fourth-based notation: | The modes would follow a more regular pattern if using octotonic fourth-based 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, | 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, forwards if the generator is chroma-positive, otherwise backwards. | ||
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). | 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). | ||
|| old scale name || example in A || new scale name || | || old scale name || example in A || genchain || new scale name || | ||
|| Harmonic minor || A B C D E F G# A || 5th Meantone [7] #7 || | || Harmonic minor || A B C D E F G# A || F C * D __**A**__ E B * * G# || 5th Meantone [7] #7 || | ||
|| Ascending melodic minor || A B C D E F# G# A || 5th Meantone [7] #6 #7 || | || Ascending melodic minor || A B C D E F# G# A || C * D __**A**__ E B F# * G# || 5th Meantone [7] #6 #7 || | ||
||= " ||= " || 2nd Meantone [7] b3 || | ||= " ||= " ||= " || 2nd Meantone [7] b3 || | ||
||= " ||= " || 4th Meantone [7] #7 || | ||= " ||= " ||= " || 4th Meantone [7] #7 || | ||
|| Double harmonic minor || A B C D# E F G# A || 5th Meantone [7] #4 #7 || | || Double harmonic minor || A B C D# E F G# A || F C * * __**A**__ E B * * G# D# || 5th Meantone [7] #4 #7 || | ||
||= " ||= " || 1st Meantone [7] b3 b6 || | ||= " ||= " || || 1st Meantone [7] b3 b6 || | ||
|| Double harmonic major || A Bb C# D E F G# A || 2nd Meantone [7] b2 b6 || | || Double harmonic major || A Bb C# D E F G# A || Bb F * * D __**A**__ E * * C# G# || 2nd Meantone [7] b2 b6 || | ||
||= " ||= " || 6th Meantone [7] #3 #7 || | ||= " ||= " || || 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 || | || <span class="mw-redirect">Hungarian gypsy </span>minor || A B C D# E F G A || F C G * __**A**__ E B * * * D# || 5th Meantone [7] #4 || | ||
|| Phrygian dominant || A Bb C# D E F G A || 6th Meantone [7] #3 || | || Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || 6th Meantone [7] #3 || | ||
As can be seen from the genchains, the harmonic minor and the phrygian dominant are modes of each other, as are the double harmonic minor and the double harmonic major. Unfortunately the scale names do not indicate this. | |||
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 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. | ||
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: | ||
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==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**== | ==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**== | ||
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. | Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. | ||
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5 | F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5 | ||
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4 | F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4 | ||
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3 | F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3 | ||
When the period is an octave, the genweb octave-reduces to a single horizontal genchain: | When the period is an octave, the genweb octave-reduces to a single horizontal genchain: | ||
F --- C --- G --- D --- A --- E --- B | F --- C --- G --- D --- A --- E --- B | ||
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Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8. | Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8. | ||
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete. | In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete. A MODMOS scale would have gaps, but no incomplete columns. | ||
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names: | If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names: | ||
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wC ---- wG ---- wD ---- wA ---- wE | wC ---- wG ---- wD ---- wA ---- wE | ||
As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. | As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 not Ty4 as the period: | ||
wC ---- wG ---- wD ----- wA ---- wE | wC ---- wG ---- wD ----- wA ---- wE | ||
gGb --- gDb --- gAb --- gEb --- gBb | gGb --- gDb --- gAb --- gEb --- gBb | ||
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==[[#How to name rank-2 scales-Non-MOS scales]]**__Non-MOS | ==[[#How to name rank-2 scales-Non-MOS scales]]**__Non-MOS scales__**== | ||
Non-MOS scales with an unbroken genchain could be named Meantone [6], Meantone [8], etc. However this would imply a hexatonic or octotonic notation. Furthermore, chromatic modifications create genchains with gaps that are very difficult to name. Instead, they are named as altered MOS scales, using "add" and "no" to add or remove notes, analogous to altered jazz chords. | |||
As before, there is more than one name for a scale. | |||
|| scale || genchain || name || alternative name || | |||
|| scale || genchain || name || name || | |||
|| C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone [7] add #4 || C 2nd Meantone [8] || | || C D E F F# G A B C || F __**C**__ G D A E B F# || C 2nd Meantone [7] add #4 || C 2nd Meantone [8] || | ||
||= " ||= " || C 1st Meantone [7] add b4 || || | ||= " ||= " || C 1st Meantone [7] add b4 || || | ||
|| C D E F F# G A Bb C || Bb F __**C**__ G D A E * F# || C 3rd Meantone [7] add #4 || | || C D E F F# G A Bb C || Bb F __**C**__ G D A E * F# || C 3rd Meantone [7] add #4 || || | ||
|| A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone [7] add #4, #7 || A 3rd Meantone [9] || | || A B C# D D# E F# G G# A || G D __**A**__ E B F# C# G# D# || A 3rd Meantone [7] add #4, #7 || A 3rd Meantone [9] || | ||
||= " ||= " || A 2nd Meantone [7] add #4, b7 || || | ||= " ||= " || A 2nd Meantone [7] add #4, b7 || || | ||
|| A B C D D# E F G G# A || F C G D __**A**__ E B * * G# D# || A 5th Meantone [7] add #4, #7 || | || A B C D E F# G G# A || C G D A E B F# * G# || || || | ||
|| F G A C D E F || __**F**__ C G D A E || | || 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 G A C E F || __**F**__ C G * A E | || F G A C D E F || __**F**__ C G D A E || F 2nd Meantone [7] no4 || F 1st Meantone [6] || | ||
|| G A B D E F# G || __**G**__ D A E B F# || | || F G A C E F || __**F**__ C G * A E || F 2nd Meantone [7] no4 no6 || || | ||
|| G A C D E F# G || C __**G**__ D A E * F# | || G A B D E F# G || __**G**__ D A E B F# || G 2nd Meantone [7] no4 || G 1st Meantone [7] no4 || | ||
|| 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 || || | || A B C E F A || F C * * __**A**__ E B || A 5th Meantone [7] no4 no7 || || | ||
In the 2nd example, "b4" means a 4th flattened relative to the 4th | In the 2nd example, "add b4" means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th. | ||
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|| Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E || | || Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E || | ||
|| Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# || | || Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# || | ||
|| Meantone[12] in 31edo ||= 4/3 || E# A# D# G# C# F# B E | || Meantone[12] in 31edo ||= 4/3 || E# A# D# G# C# F# | ||
A D G C ||= 3/2 || C G D A E B F# C# G# | B E A D G C ||= 3/2 || C G D A E B F# C# G# | ||
D# A# E# || | D# A# E# || | ||
|| Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C# G# | || Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C# | ||
D# A# E# B# Fx Cx Gx | G# D# A# E# B# Fx | ||
Dx Ax Ex ||= 3/2 || C G D A E B F# C# G# | Cx Gx Dx Ax Ex ||= 3/2 || C G D A E B F# C# G# | ||
D# A# E# B# Fx Cx Gx | D# A# E# B# Fx Cx Gx | ||
Dx Ax Ex || | Dx Ax Ex || | ||
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<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-MODMOS scales">MODMOS scales</a></div> | <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-MODMOS scales">MODMOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Fractional-octave periods">Fractional-octave periods</a></div> | <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Fractional-octave periods">Fractional-octave periods</a></div> | ||
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Non-MOS | <!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Non-MOS scales">Non-MOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Explanation / Rationale">Explanation / Rationale</a></div> | <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Kite Giedraitis method-Explanation / Rationale">Explanation / Rationale</a></div> | ||
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><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:37 --><!-- ws:start:WikiTextTocRule:38: --><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> | ||
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<!-- 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> | ||
<br /> | <br /> | ||
<strong>Mode numbers</strong> provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br /> | <strong>Mode numbers</strong> provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br /> | ||
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<br /> | <br /> | ||
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. <strong>Unlike modal UDP notation, the generator isn't always chroma-positive.</strong> This is necessary to keep the same generator for different MOS's of the same <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>, which guarantees that the smaller MOS will always be a subset of the larger MOS.<br /> | The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. <strong><u>Unlike modal UDP notation, the generator isn't always chroma-positive</u>.</strong> 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 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 /> | 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 /> | |||
For more on the disadvantages of chroma-positive generators, see <a class="wiki_link_ext" href="http://xenharmonic.wikispaces.com/Kite%20Giedraitis%20method-Explanation%20/Kite%20Giedraitis%20method-Explanation%20/%20Rationale-Why%20not%20just%20use%20UDP%20notation?" rel="nofollow">Explanation / Rationale-Why not just use UDP notation?</a><br /> | |||
<br /> | <br /> | ||
Pentatonic meantone scales:<br /> | Pentatonic meantone scales:<br /> | ||
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</table> | </table> | ||
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation: <br /> | The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:<br /> | ||
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C<br /> | C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C<br /> | ||
The modes would follow a more regular pattern if using octotonic fourth-based notation:<br /> | The modes would follow a more regular pattern if using octotonic fourth-based notation:<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:47:&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:47 --><strong><u>MODMOS scales</u></strong></h2> | <!-- 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:47:&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:47 --><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, | 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, forwards if the generator is chroma-positive, otherwise backwards.<br /> | ||
<br /> | <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 /> | 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 /> | ||
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</td> | </td> | ||
<td>example in A<br /> | <td>example in A<br /> | ||
</td> | |||
<td>genchain<br /> | |||
</td> | </td> | ||
<td>new scale name<br /> | <td>new scale name<br /> | ||
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</td> | </td> | ||
<td>A B C D E F G# A<br /> | <td>A B C D E F G# A<br /> | ||
</td> | |||
<td>F C * D <u><strong>A</strong></u> E B * * G#<br /> | |||
</td> | </td> | ||
<td>5th Meantone [7] #7<br /> | <td>5th Meantone [7] #7<br /> | ||
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</td> | </td> | ||
<td>A B C D E F# G# A<br /> | <td>A B C D E F# G# A<br /> | ||
</td> | |||
<td>C * D <u><strong>A</strong></u> E B F# * G#<br /> | |||
</td> | </td> | ||
<td>5th Meantone [7] #6 #7<br /> | <td>5th Meantone [7] #6 #7<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">&quot;<br /> | |||
</td> | |||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">&quot;<br /> | |||
</td> | |||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | </td> | ||
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</td> | </td> | ||
<td>A B C D# E F G# A<br /> | <td>A B C D# E F G# A<br /> | ||
</td> | |||
<td>F C * * <u><strong>A</strong></u> E B * * G# D#<br /> | |||
</td> | </td> | ||
<td>5th Meantone [7] #4 #7<br /> | <td>5th Meantone [7] #4 #7<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td>1st Meantone [7] b3 b6<br /> | <td>1st Meantone [7] b3 b6<br /> | ||
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</td> | </td> | ||
<td>A Bb C# D E F G# A<br /> | <td>A Bb C# D E F G# A<br /> | ||
</td> | |||
<td>Bb F * * D <u><strong>A</strong></u> E * * C# G#<br /> | |||
</td> | </td> | ||
<td>2nd Meantone [7] b2 b6<br /> | <td>2nd Meantone [7] b2 b6<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">&quot;<br /> | <td style="text-align: center;">&quot;<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td>6th Meantone [7] #3 #7<br /> | <td>6th Meantone [7] #3 #7<br /> | ||
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</td> | </td> | ||
<td>A B C D# E F G A<br /> | <td>A B C D# E F G A<br /> | ||
</td> | |||
<td>F C G * <u><strong>A</strong></u> E B * * * D#<br /> | |||
</td> | </td> | ||
<td>5th Meantone [7] #4<br /> | <td>5th Meantone [7] #4<br /> | ||
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</td> | </td> | ||
<td>A Bb C# D E F G A<br /> | <td>A Bb C# D E F G A<br /> | ||
</td> | |||
<td>Bb F * G D <u><strong>A</strong></u> E * * C#<br /> | |||
</td> | </td> | ||
<td>6th Meantone [7] #3<br /> | <td>6th Meantone [7] #3<br /> | ||
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</table> | </table> | ||
As can be seen from the genchains, the harmonic minor and the phrygian dominant are modes of each other, as are the double harmonic minor and the double harmonic major. Unfortunately the scale names do not indicate this.<br /> | |||
<br /> | <br /> | ||
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 /> | 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 /> | ||
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 /> | ||
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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:48:&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:48 --><strong><u>Fractional-octave periods</u></strong></h2> | <!-- 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:48:&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:48 --><strong><u>Fractional-octave periods</u></strong></h2> | ||
<br /> | <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 /> | 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 /> | ||
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5<br /> | F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5<br /> | ||
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4<br /> | F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4<br /> | ||
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3<br /> | F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3<br /> | ||
<br /> | <br /> | ||
When the period is an octave, the genweb octave-reduces to a single horizontal genchain: <br /> | When the period is an octave, the genweb octave-reduces to a single horizontal genchain:<br /> | ||
F --- C --- G --- D --- A --- E --- B<br /> | F --- C --- G --- D --- A --- E --- B<br /> | ||
<br /> | <br /> | ||
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Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.<br /> | Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.<br /> | ||
<br /> | <br /> | ||
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.<br /> | In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete. A MODMOS scale would have gaps, but no incomplete columns.<br /> | ||
<br /> | <br /> | ||
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:<br /> | If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:<br /> | ||
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wC ---- wG ---- wD ---- wA ---- wE<br /> | wC ---- wG ---- wD ---- wA ---- wE<br /> | ||
<br /> | <br /> | ||
As always, y means &quot;81/80 below w&quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. | As always, y means &quot;81/80 below w&quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 not Ty4 as the period:<br /> | ||
wC ---- wG ---- wD ----- wA ---- wE<br /> | wC ---- wG ---- wD ----- wA ---- wE<br /> | ||
gGb --- gDb --- gAb --- gEb --- gBb<br /> | gGb --- gDb --- gAb --- gEb --- gBb<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Non-MOS | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Kite Giedraitis method-Non-MOS scales"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:49:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales&quot; title=&quot;Anchor: How to name rank-2 scales-Non-MOS scales&quot;/&gt; --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:49 --><strong><u>Non-MOS scales</u></strong></h2> | ||
<br /> | <br /> | ||
Non-MOS scales with an unbroken genchain could be named Meantone [6], Meantone [8], etc. However this would imply a hexatonic or octotonic notation. Furthermore, chromatic modifications create genchains with gaps that are very difficult to name. Instead, they are named as altered MOS scales, using &quot;add&quot; and &quot;no&quot; to add or remove notes, analogous to altered jazz chords.<br /> | |||
<br /> | <br /> | ||
As before, there is more than one name for a scale.<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
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<td>name<br /> | <td>name<br /> | ||
</td> | </td> | ||
<td>name<br /> | <td>alternative name<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>C 3rd Meantone [7] add #4<br /> | <td>C 3rd Meantone [7] add #4<br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>A 2nd Meantone [7] add #4, b7<br /> | <td>A 2nd Meantone [7] add #4, b7<br /> | ||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>A B C D E F# G G# A<br /> | |||
</td> | |||
<td>C G D A E B F# * G#<br /> | |||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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<td>A 5th Meantone [7] add #4, #7<br /> | <td>A 5th Meantone [7] add #4, #7<br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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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 /> | ||
</td> | </td> | ||
<td><br /> | <td>F 2nd Meantone [7] no4<br /> | ||
</td> | </td> | ||
<td>F 1st Meantone [6]<br /> | <td>F 1st Meantone [6]<br /> | ||
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</td> | </td> | ||
<td><u><strong>F</strong></u> C G * A E<br /> | <td><u><strong>F</strong></u> C G * A E<br /> | ||
</td> | |||
<td>F 2nd Meantone [7] no4 no6<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><u><strong>G</strong></u> D A E B F#<br /> | <td><u><strong>G</strong></u> D A E B F#<br /> | ||
</td> | </td> | ||
<td><br /> | <td>G 2nd Meantone [7] no4<br /> | ||
</td> | </td> | ||
<td>G 1st Meantone [ | <td>G 1st Meantone [7] no4<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>C <u><strong>G</strong></u> D A E * F#<br /> | <td>C <u><strong>G</strong></u> D A E * F#<br /> | ||
</td> | |||
<td>G 2nd Meantone [7] no3<br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</table> | </table> | ||
In the 2nd example, &quot;b4&quot; means a 4th flattened relative to the 4th | In the 2nd example, &quot;add b4&quot; means add a 4th flattened relative to the Lydian mode's 4th, not the perfect 4th.<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
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<td style="text-align: center;">4/3<br /> | <td style="text-align: center;">4/3<br /> | ||
</td> | </td> | ||
<td>E# A# D# G# C# F# | <td>E# A# D# G# C# F# <br /> | ||
A D G C<br /> | B E A D G C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">3/2<br /> | <td style="text-align: center;">3/2<br /> | ||
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<td style="text-align: center;">3/2<br /> | <td style="text-align: center;">3/2<br /> | ||
</td> | </td> | ||
<td>C G D A E B F# C | <td>C G D A E B F# C# <br /> | ||
D# A# E# B# Fx | G# D# A# E# B# Fx <br /> | ||
Dx Ax Ex<br /> | Cx Gx Dx Ax Ex<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">3/2<br /> | <td style="text-align: center;">3/2<br /> | ||
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