Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 593967506 - Original comment: ** |
Wikispaces>TallKite **Imported revision 594385594 - 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>2016-10- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-05 21:56:50 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>594385594</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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EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | ||
" | "supersharp" EDOs, with fifths wider than 720¢ | ||
"pentatonic" EDOs, with a fifth = 720¢ | "pentatonic" EDOs, with a fifth = 720¢ | ||
"regular" EDOs, with a fifth that hits the "sweet spot" between 720¢ and 686¢ | "regular" EDOs, with a fifth that hits the "sweet spot" between 720¢ and 686¢ | ||
"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | "perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | ||
" | "superflat" EDOs, with a fifth less than 686¢ | ||
This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | ||
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0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven". | 0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven". | ||
Sus chords: as in conventional notation, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the | Sus chords: as in conventional notation, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the superflat EDOs below for an exception. | ||
"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = "aug three chord" (not "aug chord", because that refers to the conventional aug chord M3,A5). Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord. | "Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = "aug three chord" (not "aug chord", because that refers to the conventional aug chord M3,A5). Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord. | ||
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C _ C# Db _ D (31edo) | C _ C# Db _ D (31edo) | ||
The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For | The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For superflat EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For supersharp EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it. | ||
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and | Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and superflat EDOs, substitute M2 for m2.) | ||
||= 5edo ||= pentatonic ||= ||= C/Db ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||= 5edo ||= pentatonic ||= ||= C/Db ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 6edo ||= | ||= 6edo ||= supersharp ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 7edo ||= perfect ||= ||= C/C# ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||= 7edo ||= perfect ||= ||= C/C# ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 8edo ||= | ||= 8edo ||= supersharp ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 9edo ||= | ||= 9edo ||= superflat ||= ||= C/Db ||= C#/D ||= D# ||= ||= ||= ||= ||= ||= ||= || | ||
||= 10edo ||= pentatonic ||= ||= C/Db ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||= || | ||= 10edo ||= pentatonic ||= ||= C/Db ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||= || | ||
||= 11edo ||= | ||= 11edo ||= superflat ||= ||= C ||= D ||= C# ||= D# ||= ||= ||= ||= ||= ||= || | ||
||= 12edo ||= regular ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= || | ||= 12edo ||= regular ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= || | ||
||= 13b-edo ||= | ||= 13b-edo ||= superflat ||= ||= C ||= D ||= * ||= C# ||= D# ||= ||= ||= ||= ||= || | ||
||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= || | ||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= || | ||
||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= || | ||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= || | ||
||= 16edo ||= | ||= 16edo ||= superflat ||= ||= C ||= C#/Db ||= D ||= D# ||= ||= ||= ||= ||= ||= || | ||
||= 17edo ||= regular ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= || | ||= 17edo ||= regular ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= || | ||
||= 18b-edo ||= | ||= 18b-edo ||= superflat ||= ||= C/Db ||= * ||= C#/D ||= * ||= D# ||= ||= ||= ||= ||= || | ||
||= 19edo ||= regular ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= || | ||= 19edo ||= regular ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= || | ||
||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= || | ||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= || | ||
||= 21edo ||= perfect ||= ||= C/C# ||= * ||= * ||= Db/D ||= ||= ||= ||= ||= ||= || | ||= 21edo ||= perfect ||= ||= C/C# ||= * ||= * ||= Db/D ||= ||= ||= ||= ||= ||= || | ||
||= 22edo ||= regular ||= ||= C ||= Db ||= * ||= C# ||= D ||= ||= ||= ||= ||= || | ||= 22edo ||= regular ||= ||= C ||= Db ||= * ||= C# ||= D ||= ||= ||= ||= ||= || | ||
||= 23edo ||= | ||= 23edo ||= superflat ||= ||= C ||= C# ||= Db ||= D ||= D# ||= ||= ||= ||= ||= || | ||
||= 24edo ||= regular ||= ||= C ||= * ||= C#/Db ||= * ||= D ||= ||= ||= ||= ||= || | ||= 24edo ||= regular ||= ||= C ||= * ||= C#/Db ||= * ||= D ||= ||= ||= ||= ||= || | ||
||= 25edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= || | ||= 25edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= || | ||
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==__" | ==__"Superflat" EDOs__== | ||
(9, 11, 13b, 16, 18b and 23) | (9, 11, 13b, 16, 18b and 23) | ||
All | All superflat EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | ||
Edos 11 and 13 and problematic. See " | Edos 11 and 13 and problematic. See "Supersharp EDOs" below for alternate notations for them. | ||
**__9edo__:** C/D# Cb/D (# = v) | **__9edo__:** C/D# Cb/D (# = v) | ||
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==__**" | ==__**"Supersharp" EDOs**__== | ||
(8, 11b, 13 and 18) | (8, 11b, 13 and 18) | ||
There are three strategies for notating these EDOs. One is to convert them to | There are three strategies for notating these EDOs. One is to convert them to superflat EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo. | ||
Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo. | Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo. | ||
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requires learning octatonic interval arithmetic and notation | requires learning octatonic interval arithmetic and notation | ||
__**13b-edo**__ undecatonic narrow-fifth-based, | __**13b-edo**__ undecatonic narrow-fifth-based, superflat, 3/2 maps to 7\13 = perfect 7th | ||
undecatonic sixthwards chain of sevenths: | undecatonic sixthwards chain of sevenths: | ||
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11 | M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11 | ||
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For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't. | For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't. | ||
All | All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with chroma > 1 or < -1. If these are notated without ups and downs, the notes run out of order: | ||
17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C | 17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C | ||
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<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Regular&quot; EDOs">&quot;Regular&quot; EDOs</a></div> | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Regular&quot; EDOs">&quot;Regular&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Perfect&quot; EDOs">&quot;Perfect&quot; EDOs</a></div> | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Perfect&quot; EDOs">&quot;Perfect&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot; | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Superflat&quot; EDOs">&quot;Superflat&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Pentatonic&quot; EDOs">&quot;Pentatonic&quot; EDOs</a></div> | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Pentatonic&quot; EDOs">&quot;Pentatonic&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot; | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Supersharp&quot; EDOs">&quot;Supersharp&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div> | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div> | ||
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: 8ve Periods">Rank-2 Scales: 8ve Periods</a></div> | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: 8ve Periods">Rank-2 Scales: 8ve Periods</a></div> | ||
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<br /> | <br /> | ||
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | ||
&quot; | &quot;supersharp&quot; EDOs, with fifths wider than 720¢<br /> | ||
&quot;pentatonic&quot; EDOs, with a fifth = 720¢<br /> | &quot;pentatonic&quot; EDOs, with a fifth = 720¢<br /> | ||
&quot;regular&quot; EDOs, with a fifth that hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | &quot;regular&quot; EDOs, with a fifth that hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | ||
&quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | &quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | ||
&quot; | &quot;superflat&quot; EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | ||
| Line 1,650: | Line 1,650: | ||
0-7-13-20 = Cv Evv Gv Bvv is &quot;Cv.vM7&quot;, &quot;C down, downmajor seven&quot;.<br /> | 0-7-13-20 = Cv Evv Gv Bvv is &quot;Cv.vM7&quot;, &quot;C down, downmajor seven&quot;.<br /> | ||
<br /> | <br /> | ||
Sus chords: as in conventional notation, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the | Sus chords: as in conventional notation, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the superflat EDOs below for an exception.<br /> | ||
<br /> | <br /> | ||
&quot;Aug&quot; and &quot;dim&quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = &quot;aug three chord&quot; (not &quot;aug chord&quot;, because that refers to the conventional aug chord M3,A5). Likewise d3,P5 is a &quot;dim three chord&quot;, and m3,d5 is a &quot;dim&quot; chord.<br /> | &quot;Aug&quot; and &quot;dim&quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = &quot;aug three chord&quot; (not &quot;aug chord&quot;, because that refers to the conventional aug chord M3,A5). Likewise d3,P5 is a &quot;dim three chord&quot;, and m3,d5 is a &quot;dim&quot; chord.<br /> | ||
| Line 2,057: | Line 2,057: | ||
C _ C# Db _ D (31edo)<br /> | C _ C# Db _ D (31edo)<br /> | ||
<br /> | <br /> | ||
The scale fragment concisely conveys the &quot;flavor&quot; of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For | The scale fragment concisely conveys the &quot;flavor&quot; of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For superflat EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For supersharp EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.<br /> | ||
<br /> | <br /> | ||
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &quot;wide-fifth&quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &quot;narrow-fifth&quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and | Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &quot;wide-fifth&quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &quot;narrow-fifth&quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and superflat EDOs, substitute M2 for m2.)<br /> | ||
<br /> | <br /> | ||
| Line 2,095: | Line 2,095: | ||
<td style="text-align: center;">6edo<br /> | <td style="text-align: center;">6edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">supersharp<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,151: | Line 2,151: | ||
<td style="text-align: center;">8edo<br /> | <td style="text-align: center;">8edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">supersharp<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,179: | Line 2,179: | ||
<td style="text-align: center;">9edo<br /> | <td style="text-align: center;">9edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">superflat<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,235: | Line 2,235: | ||
<td style="text-align: center;">11edo<br /> | <td style="text-align: center;">11edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">superflat<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,291: | Line 2,291: | ||
<td style="text-align: center;">13b-edo<br /> | <td style="text-align: center;">13b-edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">superflat<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,375: | Line 2,375: | ||
<td style="text-align: center;">16edo<br /> | <td style="text-align: center;">16edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">superflat<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,431: | Line 2,431: | ||
<td style="text-align: center;">18b-edo<br /> | <td style="text-align: center;">18b-edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">superflat<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,571: | Line 2,571: | ||
<td style="text-align: center;">23edo<br /> | <td style="text-align: center;">23edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">superflat<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,506: | Line 3,506: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Summary of EDO notation-&quot; | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Summary of EDO notation-&quot;Superflat&quot; EDOs"></a><!-- ws:end:WikiTextHeadingRule:20 --><u>&quot;Superflat&quot; EDOs</u></h2> | ||
(9, 11, 13b, 16, 18b and 23)<br /> | (9, 11, 13b, 16, 18b and 23)<br /> | ||
All | All superflat EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | ||
Edos 11 and 13 and problematic. See &quot; | Edos 11 and 13 and problematic. See &quot;Supersharp EDOs&quot; below for alternate notations for them.<br /> | ||
<br /> | <br /> | ||
<strong><u>9edo</u>:</strong> C/D# Cb/D (# = v)<br /> | <strong><u>9edo</u>:</strong> C/D# Cb/D (# = v)<br /> | ||
| Line 3,603: | Line 3,603: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Summary of EDO notation-&quot; | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Summary of EDO notation-&quot;Supersharp&quot; EDOs"></a><!-- ws:end:WikiTextHeadingRule:24 --><u><strong>&quot;Supersharp&quot; EDOs</strong></u></h2> | ||
(8, 11b, 13 and 18)<br /> | (8, 11b, 13 and 18)<br /> | ||
<br /> | <br /> | ||
There are three strategies for notating these EDOs. One is to convert them to | There are three strategies for notating these EDOs. One is to convert them to superflat EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.<br /> | ||
<br /> | <br /> | ||
Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.<br /> | Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.<br /> | ||
| Line 3,654: | Line 3,654: | ||
requires learning octatonic interval arithmetic and notation<br /> | requires learning octatonic interval arithmetic and notation<br /> | ||
<br /> | <br /> | ||
<u><strong>13b-edo</strong></u> undecatonic narrow-fifth-based, | <u><strong>13b-edo</strong></u> undecatonic narrow-fifth-based, superflat, 3/2 maps to 7\13 = perfect 7th<br /> | ||
undecatonic sixthwards chain of sevenths:<br /> | undecatonic sixthwards chain of sevenths:<br /> | ||
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11<br /> | M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11<br /> | ||
| Line 3,746: | Line 3,746: | ||
For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.<br /> | For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.<br /> | ||
<br /> | <br /> | ||
All | All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with chroma &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:<br /> | ||
<br /> | <br /> | ||
17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C<br /> | 17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C<br /> | ||