Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 594434572 - Original comment: ** |
Wikispaces>TallKite **Imported revision 594507750 - 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 16:38:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>594507750</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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In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G. Sharp is flatter than natural. __**Up is still ascending in pitch**__. If someone's singing above pitch, instead of saying "you're singing sharp", you would say "you're singing up". | In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G. Sharp is flatter than natural. __**Up is still ascending in pitch**__. If someone's singing above pitch, instead of saying "you're singing sharp", you would say "you're singing up". | ||
In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. Sharp is still sharper than natural. The chain of fifths | In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. Sharp is still sharper than natural. The chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc. | ||
M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc. | |||
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc. | F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc. | ||
16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8 | 16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8 | ||
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C | 16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C | ||
Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim and sharp for flat. Perfect and natural are unaffected. Examples: | Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples: | ||
||= initial question ||= reverse everything ||= do the math ||= reverse again || | ||= initial question ||= reverse everything ||= do the math ||= reverse again || | ||
||= M2 + M2 ||= m2 + m2 ||= dim3 ||= aug3 || | ||= M2 + M2 ||= m2 + m2 ||= dim3 ||= aug3 || | ||
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||= Eb + m3 ||= E# + M3 ||= G## ||= Gbb || | ||= Eb + m3 ||= E# + M3 ||= G## ||= Gbb || | ||
||= Eb + P5 ||= E# + P5 ||= B# ||= Bb || | ||= Eb + P5 ||= E# + P5 ||= B# ||= Bb || | ||
||= A minor ||= A major ||= A C# E ||= A Cb E || | ||= A minor chord ||= A major ||= A C# E ||= A Cb E || | ||
||= Eb major ||= E# minor ||= E# G# B# ||= Eb Gb Db || | ||= Eb major chord ||= E# minor ||= E# G# B# ||= Eb Gb Db || | ||
||= G7 = M3 + P5 + m7 ||= m3 + P5 + M7 ||= G Bb D F# ||= G B# D Fb || | ||= G7 = G + M3 + P5 + m7 ||= G + m3 + P5 + M7 ||= G Bb D F# ||= G B# D Fb || | ||
||= Ab aug = Ab + M3 + A5 ||= A# + m3 + d5 ||= A# C# E ||= Ab Cb E || | |||
||= C major scale = C + M2 + M3 | |||
+ P4 + P5 + M6 + M7 + P8 ||= C + m2 + m3 + P4 | |||
+ P5 + m6 + m7 + P8 ||= C Db Eb F | |||
G Ab Bb C ||= C D# E# F | |||
G A# B# C || | |||
||= C# minor scale = C# + M2 + m3 | |||
+ P4 + P5 + m6 + m7 + P8 ||= Cb + m2 + M3 + P4 | |||
+ P5 + M6 + M7 + P8 ||= Cb Dbb Eb Fb | |||
Gb Ab Bb Cb ||= C# D## E# F# | |||
G# A# B# C# || | |||
Both approaches have their merit, but the first one will be used here. | Both approaches have their merit, but the first one will be used here. | ||
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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. | 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. | ||
Every EDO has a "natural" heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means "sharpened by one EDO-step", and ups and downs aren't needed. The generator is always perfect, so there's a perfect 2nd, a major & minor 3rd, 4th, 5th and 6th, and a perfect 7th. | Every non-perfect EDO has a "natural" heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means "sharpened by one EDO-step", and ups and downs aren't needed. The generator is always perfect, so there's a perfect 2nd, a major & minor 3rd, 4th, 5th and 6th, and a perfect 7th. | ||
__**Natural generators for 8edo, 11edo, 13edo and 18edo**__ | __**Natural generators for 8edo, 11edo, 13edo and 18edo**__ | ||
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D - E - F - G - G#/Ab - A -B - C - D | D - E - F - G - G#/Ab - A -B - C - D | ||
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8 | P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8 | ||
genchain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc. | |||
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc. | A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc. | ||
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D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D | D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D | ||
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8 | P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8 | ||
genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | |||
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 | ||
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D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D | D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D | ||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | ||
genchain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc. | |||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | ||
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D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D | D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - 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 | 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 | ||
genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | |||
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, | 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. | ||
__**Chroma-2 and chroma-5 edos:**__ | |||
genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - d6 etc. | |||
Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C | |||
"Every Good Boy Deserves Fudge And Candy" | |||
__**10-edo**__: (generator = 3\10 = perfect 3rd) | |||
D E * F G * A B * C D | |||
P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8 | |||
__**17-edo**__: (generator = 3\17 = perfect 3rd) | |||
D * E * * F * G * * A * B * * C * D | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8 | |||
__**18-edo**__: see above | |||
__**17-edo**__: (generator = 3\17 = perfect 3rd) | |||
D * E * * F * G * * A * B * * C * D | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8 | |||
For 22-edo, it's a 3\22 2nd | For 22-edo, it's a 3\22 2nd | ||
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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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:3428:&lt;img src=&quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1002px; width: 800px;&quot; /&gt; --><img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:3428 --><br /> | ||
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | ||
<br /> | <br /> | ||
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In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G. Sharp is flatter than natural. <u><strong>Up is still ascending in pitch</strong></u>. If someone's singing above pitch, instead of saying &quot;you're singing sharp&quot;, you would say &quot;you're singing up&quot;.<br /> | In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G. Sharp is flatter than natural. <u><strong>Up is still ascending in pitch</strong></u>. If someone's singing above pitch, instead of saying &quot;you're singing sharp&quot;, you would say &quot;you're singing up&quot;.<br /> | ||
<br /> | <br /> | ||
In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. Sharp is still sharper than natural. The chain of fifths | In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. Sharp is still sharper than natural. The chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.<br /> | ||
M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.<br /> | |||
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.<br /> | F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.<br /> | ||
16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br /> | 16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br /> | ||
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C<br /> | 16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C<br /> | ||
<br /> | <br /> | ||
Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim and sharp for flat. Perfect and natural are unaffected. Examples:<br /> | Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:<br /> | ||
| Line 1,546: | Line 1,576: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">A minor<br /> | <td style="text-align: center;">A minor chord<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">A major<br /> | <td style="text-align: center;">A major<br /> | ||
| Line 1,556: | Line 1,586: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">Eb major<br /> | <td style="text-align: center;">Eb major chord<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">E# minor<br /> | <td style="text-align: center;">E# minor<br /> | ||
| Line 1,566: | Line 1,596: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">G7 = M3 + P5 + m7<br /> | <td style="text-align: center;">G7 = G + M3 + P5 + m7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">m3 + P5 + M7<br /> | <td style="text-align: center;">G + m3 + P5 + M7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">G Bb D F#<br /> | <td style="text-align: center;">G Bb D F#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">G B# D Fb<br /> | <td style="text-align: center;">G B# D Fb<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">Ab aug = Ab + M3 + A5<br /> | |||
</td> | |||
<td style="text-align: center;">A# + m3 + d5<br /> | |||
</td> | |||
<td style="text-align: center;">A# C# E<br /> | |||
</td> | |||
<td style="text-align: center;">Ab Cb E<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">C major scale = C + M2 + M3<br /> | |||
+ P4 + P5 + M6 + M7 + P8<br /> | |||
</td> | |||
<td style="text-align: center;">C + m2 + m3 + P4<br /> | |||
+ P5 + m6 + m7 + P8<br /> | |||
</td> | |||
<td style="text-align: center;">C Db Eb F<br /> | |||
G Ab Bb C<br /> | |||
</td> | |||
<td style="text-align: center;">C D# E# F<br /> | |||
G A# B# C<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">C# minor scale = C# + M2 + m3<br /> | |||
+ P4 + P5 + m6 + m7 + P8<br /> | |||
</td> | |||
<td style="text-align: center;">Cb + m2 + M3 + P4 <br /> | |||
+ P5 + M6 + M7 + P8<br /> | |||
</td> | |||
<td style="text-align: center;">Cb Dbb Eb Fb<br /> | |||
Gb Ab Bb Cb<br /> | |||
</td> | |||
<td style="text-align: center;">C# D## E# F#<br /> | |||
G# A# B# C#<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule:8 --><u>Chord names in other EDOs</u></h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule:8 --><u>Chord names in other EDOs</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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Every EDO has a &quot;natural&quot; heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means &quot;sharpened by one EDO-step&quot;, and ups and downs aren't needed. The generator is always perfect, so there's a perfect 2nd, a major &amp; minor 3rd, 4th, 5th and 6th, and a perfect 7th.<br /> | Every non-perfect EDO has a &quot;natural&quot; heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means &quot;sharpened by one EDO-step&quot;, and ups and downs aren't needed. The generator is always perfect, so there's a perfect 2nd, a major &amp; minor 3rd, 4th, 5th and 6th, and a perfect 7th.<br /> | ||
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<u><strong>Natural generators for 8edo, 11edo, 13edo and 18edo</strong></u><br /> | <u><strong>Natural generators for 8edo, 11edo, 13edo and 18edo</strong></u><br /> | ||
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D - E - F - G - G#/Ab - A -B - C - D<br /> | D - E - F - G - G#/Ab - A -B - C - D<br /> | ||
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br /> | P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br /> | ||
genchain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.<br /> | |||
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.<br /> | A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.<br /> | ||
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D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D<br /> | D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D<br /> | ||
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8<br /> | P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8<br /> | ||
genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | |||
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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D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D<br /> | D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D<br /> | ||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | ||
genchain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br /> | |||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | ||
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D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D<br /> | D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - 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 /> | 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 /> | ||
genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | |||
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, | 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 /> | ||
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<u><strong>Chroma-2 and chroma-5 edos:</strong></u><br /> | |||
genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - d6 etc.<br /> | |||
Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C<br /> | |||
&quot;Every Good Boy Deserves Fudge And Candy&quot; <br /> | |||
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<u><strong>10-edo</strong></u>: (generator = 3\10 = perfect 3rd)<br /> | |||
D E * F G * A B * C D<br /> | |||
P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8<br /> | |||
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<u><strong>17-edo</strong></u>: (generator = 3\17 = perfect 3rd)<br /> | |||
D * E * * F * G * * A * B * * C * D<br /> | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8<br /> | |||
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<u><strong>18-edo</strong></u>: see above<br /> | |||
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<u><strong>17-edo</strong></u>: (generator = 3\17 = perfect 3rd)<br /> | |||
D * E * * F * G * * A * B * * C * D<br /> | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8<br /> | |||
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For 22-edo, it's a 3\22 2nd<br /> | For 22-edo, it's a 3\22 2nd<br /> | ||