Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 594511442 - Original comment: ** |
Wikispaces>TallKite **Imported revision 594513846 - 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>2016-10-06 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 18:20:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>594513846</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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(8, 11b, 13 and 18) | (8, 11b, 13 and 18) | ||
There are three strategies for notating these EDOs. | There are three strategies for notating these EDOs. The best 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. | ||
The third approach is to use | The third approach is to use the natural generator, see the next section. | ||
__**Pentatonic notation for 8edo, 11b-edo, 13edo and 18edo**__ | __**Pentatonic notation for 8edo, 11b-edo, 13edo and 18edo**__ | ||
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requires learning octatonic interval arithmetic and staff notation | requires learning octatonic interval arithmetic and staff notation | ||
__**11edo**__ nonotonic narrow-fifth-based (3/2 maps to 6\11 = perfect 6th) | __**11edo**__ nonotonic narrow-fifth-based (3/2 maps to 6\11 = perfect 6th) | ||
nonotonic | nonotonic genchain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 | ||
A B C * D E F G * H J A | |||
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8 | P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8 | ||
requires learning nonotonic interval arithmetic and staff notation | requires learning nonotonic interval arithmetic and staff notation | ||
__**11b-edo**__ octatonic wide-fifth-based (3/2 maps to 7\11 = perfect 6th) | __**11b-edo**__ octatonic wide-fifth-based (3/2 maps to 7\11 = perfect 6th) | ||
octatonic | A B * C D * E F G * H A | ||
octatonic genchain of sixths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 | |||
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9 | P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9 | ||
requires learning octatonic interval arithmetic and notation | requires learning octatonic interval arithmetic and notation | ||
__**13edo**__ octatonic wide-fifth-based (3/2 maps to 8\13 = perfect 6th) | __**13edo**__ octatonic wide-fifth-based (3/2 maps to 8\13 = perfect 6th) | ||
octotonic | octotonic genchain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7 | ||
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9 | P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9 | ||
requires learning octatonic interval arithmetic and notation | requires learning octatonic interval arithmetic and notation | ||
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= | =__Natural Generators__= | ||
Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like. | Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like. | ||
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E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | ||
Natural generators can be used for other EDOs as well. For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72. | Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So except for 8-edo, this notation is recommended only as an alternate, composer-oriented notation. | ||
For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72. | |||
__**Chroma-2 edos:**__ | __**Chroma-2 edos:**__ | ||
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D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D | D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D | ||
P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5... | P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5... | ||
etc. | etc. | ||
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etc. | etc. | ||
__**Chroma-5 edos:**__ | |||
genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc. | |||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | |||
"Every Good Boy Deserves Fudge And Candy" | |||
= | __**25-edo**__: (generator = 7\25 = perfect 3rd) | ||
D * * * E * * F * * * G * * A * * * B * * C * * * D | |||
P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8 | |||
etc. | etc. | ||
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The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, "up" means "a major 3rd (~81/64) minus a quarter-octave". Using ~25/24 as the generator yields the same scale. | The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, "up" means "a major 3rd (~81/64) minus a quarter-octave". Using ~25/24 as the generator yields the same scale. | ||
[[Blackwood]] [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. "Up" means "2/5 of an octave minus ~5/4": | [[Blackwood]] [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. "Up" means "2/5 of an octave minus ~5/4": | ||
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=__Rank-2 Scales: Non-5th Generators__= | |||
= | |||
The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation. | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation. | ||
An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by | An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by the natural generator, the 2nd = 3\22. | ||
Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb | Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb | ||
Scale fragment: D D# Eb E | Scale fragment: D D# Eb E | ||
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||= 12 ||= -18 ||= G### ||= +4 ||= Ab || | ||= 12 ||= -18 ||= G### ||= +4 ||= Ab || | ||
||= 13 ||= -3 ||= A ||= ||= || | ||= 13 ||= -3 ||= A ||= ||= || | ||
||= 14 ||= -10 ||= | ||= 14 ||= -10 ||= A# ||= +12 ||= Bbb || | ||
||= 15 ||= -17 ||= | ||= 15 ||= -17 ||= Ax ||= +5 ||= Bb || | ||
||= 16 ||= -2 ||= B ||= ||= || | ||= 16 ||= -2 ||= B ||= ||= || | ||
||= 17 ||= -9 ||= | ||= 17 ||= -9 ||= B# ||= +13 ||= Cbb || | ||
||= 18 ||= -16 ||= | ||= 18 ||= -16 ||= Bx ||= +6 ||= Cb || | ||
||= 19 ||= -1 ||= C ||= ||= || | ||= 19 ||= -1 ||= C ||= ||= || | ||
||= 20 ||= -8 ||= | ||= 20 ||= -8 ||= C# ||= +14 ||= Dbb || | ||
||= 21 ||= -15 ||= | ||= 21 ||= -15 ||= Cx ||= +7 ||= Db || | ||
||= 22 ||= 0 ||= D ||= ||= ||</pre></div> | ||= 22 ||= 0 ||= D ||= ||= || | ||
=__Ups and downs solfege__= | |||
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down: | |||
The initial consonant remains as before: D, R, M, F, S, L and T | |||
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb | |||
The vowels are pronounced as in Spanish or Italian | |||
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u | |||
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv | |||
The 2nd vowel is separated from the first by either a glottal stop, an "h", a "w", or a "y" | |||
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo. | |||
This suffices for many but not all edos, as some require triple sharps or quadruple ups. | |||
Fixed-do solfege: | |||
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb | |||
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^ | |||
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^ | |||
etc. | |||
Moveable-do solfege: | |||
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval. | |||
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim | |||
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1 | |||
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1 | |||
etc. | |||
Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim | |||
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2 | |||
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2 | |||
etc.</pre></div> | |||
<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>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule:36:&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:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#A 22edo example">A 22edo example</a></div> | <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>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule:36:&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:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#A 22edo example">A 22edo example</a></div> | ||
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<!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Pentatonic&quot; EDOs">&quot;Pentatonic&quot; EDOs</a></div> | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Pentatonic&quot; EDOs">&quot;Pentatonic&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Supersharp&quot; EDOs">&quot;Supersharp&quot; EDOs</a></div> | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Supersharp&quot; EDOs">&quot;Supersharp&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 1em;"><a href="# | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 1em;"><a href="#Natural Generators">Natural Generators</a></div> | ||
<!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="# | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: 8ve Periods">Rank-2 Scales: 8ve Periods</a></div> | ||
<!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: 8ve Periods">Rank-2 Scales: 8ve Periods</a></div> | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: Non-8ve Periods">Rank-2 Scales: Non-8ve Periods</a></div> | ||
<!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: Non- | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: Non-5th Generators">Rank-2 Scales: Non-5th Generators</a></div> | ||
<!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 1em;"><a href="# | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div> | ||
<!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --></div> | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --></div> | ||
<!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="A 22edo example"></a><!-- ws:end:WikiTextHeadingRule:0 --><u>A 22edo example</u></h1> | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="A 22edo example"></a><!-- ws:end:WikiTextHeadingRule:0 --><u>A 22edo example</u></h1> | ||
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(8, 11b, 13 and 18)<br /> | (8, 11b, 13 and 18)<br /> | ||
<br /> | <br /> | ||
There are three strategies for notating these EDOs. | There are three strategies for notating these EDOs. The best 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 /> | ||
<br /> | <br /> | ||
The third approach is to use | The third approach is to use the natural generator, see the next section.<br /> | ||
<br /> | <br /> | ||
<u><strong>Pentatonic notation for 8edo, 11b-edo, 13edo and 18edo</strong></u><br /> | <u><strong>Pentatonic notation for 8edo, 11b-edo, 13edo and 18edo</strong></u><br /> | ||
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requires learning octatonic interval arithmetic and staff notation<br /> | requires learning octatonic interval arithmetic and staff notation<br /> | ||
<br /> | <br /> | ||
<u><strong>11edo</strong></u> nonotonic narrow-fifth-based (3/2 maps to 6\11 = perfect 6th) | <u><strong>11edo</strong></u> nonotonic narrow-fifth-based (3/2 maps to 6\11 = perfect 6th)<br /> | ||
nonotonic | nonotonic genchain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9<br /> | ||
A B C * D E F G * H J A<br /> | |||
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8<br /> | P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8<br /> | ||
requires learning nonotonic interval arithmetic and staff notation<br /> | requires learning nonotonic interval arithmetic and staff notation<br /> | ||
<br /> | <br /> | ||
<u><strong>11b-edo</strong></u> octatonic wide-fifth-based (3/2 maps to 7\11 = perfect 6th) | <u><strong>11b-edo</strong></u> octatonic wide-fifth-based (3/2 maps to 7\11 = perfect 6th)<br /> | ||
octatonic | A B * C D * E F G * H A<br /> | ||
octatonic genchain of sixths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7<br /> | |||
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9<br /> | P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9<br /> | ||
requires learning octatonic interval arithmetic and notation<br /> | requires learning octatonic interval arithmetic and notation<br /> | ||
<br /> | <br /> | ||
<u><strong>13edo</strong></u> octatonic wide-fifth-based (3/2 maps to 8\13 = perfect 6th)<br /> | <u><strong>13edo</strong></u> octatonic wide-fifth-based (3/2 maps to 8\13 = perfect 6th)<br /> | ||
octotonic | octotonic genchain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7<br /> | ||
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9<br /> | P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9<br /> | ||
requires learning octatonic interval arithmetic and notation<br /> | requires learning octatonic interval arithmetic and notation<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name=" | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Natural Generators"></a><!-- ws:end:WikiTextHeadingRule:26 --><u>Natural Generators</u></h1> | ||
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Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.<br /> | Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.<br /> | ||
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E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | ||
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Natural generators can be used for other EDOs as well. For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.<br /> | Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So except for 8-edo, this notation is recommended only as an alternate, composer-oriented notation.<br /> | ||
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For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 6, 7 or 8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.<br /> | |||
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<u><strong>Chroma-2 edos:</strong></u><br /> | <u><strong>Chroma-2 edos:</strong></u><br /> | ||
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D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D<br /> | D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D<br /> | ||
P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...<br /> | P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...<br /> | ||
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etc.<br /> | etc.<br /> | ||
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etc.<br /> | etc.<br /> | ||
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<u><strong>Chroma-5 edos:</strong></u><br /> | |||
genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.<br /> | |||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | |||
&quot;Every Good Boy Deserves Fudge And Candy&quot;<br /> | |||
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< | <u><strong>25-edo</strong></u>: (generator = 7\25 = perfect 3rd)<br /> | ||
D * * * E * * F * * * G * * A * * * B * * C * * * D<br /> | |||
P1 - A1 - d2 - m2 - M2 - A2 - d3 - P3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 - d6 - P6 - A6 - d7 - m7 - M7 - A7 - d8 - P8<br /> | |||
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etc.<br /> | etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Rank-2 Scales: 8ve Periods"></a><!-- ws:end:WikiTextHeadingRule:28 --><u>Rank-2 Scales: 8ve Periods</u></h1> | ||
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Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about <strong>frameworks</strong> like 12-tone. The generator chain is called a <strong>genchain</strong>. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (chroma 1 or -1):<br /> | Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about <strong>frameworks</strong> like 12-tone. The generator chain is called a <strong>genchain</strong>. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (chroma 1 or -1):<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Rank-2 Scales: Non-8ve Periods"></a><!-- ws:end:WikiTextHeadingRule:30 --><u>Rank-2 Scales: Non-8ve Periods</u></h1> | ||
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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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The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, &quot;up&quot; means &quot;a major 3rd (~81/64) minus a quarter-octave&quot;. Using ~25/24 as the generator yields the same scale.<br /> | The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, &quot;up&quot; means &quot;a major 3rd (~81/64) minus a quarter-octave&quot;. Using ~25/24 as the generator yields the same scale.<br /> | ||
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<a class="wiki_link" href="/Blackwood">Blackwood</a> [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. &quot;Up&quot; means &quot;2/5 of an octave minus ~5/4&quot;:<br /> | <a class="wiki_link" href="/Blackwood">Blackwood</a> [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. &quot;Up&quot; means &quot;2/5 of an octave minus ~5/4&quot;:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Rank-2 Scales: Non-5th Generators"></a><!-- ws:end:WikiTextHeadingRule:32 --><u>Rank-2 Scales: Non-5th Generators</u></h1> | |||
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The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | ||
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An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by | An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by the natural generator, the 2nd = 3\22.<br /> | ||
Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb<br /> | Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb<br /> | ||
Scale fragment: D D# Eb E<br /> | Scale fragment: D D# Eb E<br /> | ||
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<td style="text-align: center;">-10<br /> | <td style="text-align: center;">-10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">A#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+12<br /> | <td style="text-align: center;">+12<br /> | ||
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<td style="text-align: center;">-17<br /> | <td style="text-align: center;">-17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Ax<br /> | ||
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<td style="text-align: center;">+5<br /> | <td style="text-align: center;">+5<br /> | ||
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<td style="text-align: center;">-9<br /> | <td style="text-align: center;">-9<br /> | ||
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<td style="text-align: center;"> | <td style="text-align: center;">B#<br /> | ||
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<td style="text-align: center;">+13<br /> | <td style="text-align: center;">+13<br /> | ||
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<td style="text-align: center;">-16<br /> | <td style="text-align: center;">-16<br /> | ||
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<td style="text-align: center;"> | <td style="text-align: center;">Bx<br /> | ||
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<td style="text-align: center;">+6<br /> | <td style="text-align: center;">+6<br /> | ||
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<td style="text-align: center;">-8<br /> | <td style="text-align: center;">-8<br /> | ||
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<td style="text-align: center;"> | <td style="text-align: center;">C#<br /> | ||
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<td style="text-align: center;">+14<br /> | <td style="text-align: center;">+14<br /> | ||
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<td style="text-align: center;">-15<br /> | <td style="text-align: center;">-15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Cx<br /> | ||
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<td style="text-align: center;">+7<br /> | <td style="text-align: center;">+7<br /> | ||
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</table> | </table> | ||
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<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Ups and downs solfege"></a><!-- ws:end:WikiTextHeadingRule:34 --><u>Ups and downs solfege</u></h1> | |||
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Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | |||
The initial consonant remains as before: D, R, M, F, S, L and T<br /> | |||
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb<br /> | |||
The vowels are pronounced as in Spanish or Italian<br /> | |||
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u<br /> | |||
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv<br /> | |||
The 2nd vowel is separated from the first by either a glottal stop, an &quot;h&quot;, a &quot;w&quot;, or a &quot;y&quot;<br /> | |||
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo.<br /> | |||
This suffices for many but not all edos, as some require triple sharps or quadruple ups.<br /> | |||
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Fixed-do solfege:<br /> | |||
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb<br /> | |||
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^<br /> | |||
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^<br /> | |||
etc.<br /> | |||
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Moveable-do solfege:<br /> | |||
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.<br /> | |||
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim<br /> | |||
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1<br /> | |||
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1<br /> | |||
etc.<br /> | |||
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Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim<br /> | |||
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2<br /> | |||
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2<br /> | |||
etc.</body></html></pre></div> | |||