Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 593748916 - Original comment: ** |
Wikispaces>TallKite **Imported revision 593964730 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2016-10-04 03:12:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593964730</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> | ||
| Line 88: | Line 88: | ||
=__**Other EDOs**__= | =__**Other EDOs**__= | ||
The up symbol means "sharpened by one EDO-step" in every single EDO. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47-edo. | |||
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: | ||
| Line 126: | Line 128: | ||
The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning. | The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning. | ||
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. | 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. To describe someone singing below pitch, instead of saying "you're singing flat", you would say "you're singing down". | ||
In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards: | 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 runs backwards: | ||
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. | ||
| Line 134: | Line 136: | ||
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. | 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. | ||
M2 + M2 | ||= initial question ||= reverse everything ||= do the math ||= reverse again || | ||
D to F# | ||= M2 + M2 ||= m2 + m2 ||= dim3 ||= aug3 || | ||
D to F | ||= D to F# ||= D to Fb ||= dim3 ||= aug3 || | ||
Eb + m3 | ||= D to F ||= D to F ||= m3 ||= M3 || | ||
Eb + P5 | ||= Eb + m3 ||= E# + M3 ||= G## ||= Gbb || | ||
||= Eb + P5 ||= E# + P5 ||= B# ||= Bb || | |||
||= C major ||= C minor ||= C Eb G ||= C E# G || | |||
||= Eb major ||= E# minor ||= E# G# B# ||= Eb Gb Db || | |||
||= G7 = M3 + P5 + m7 ||= m3 + P5 + M7 ||= G Bb D F# ||= G B# D Fb || | |||
Both approaches have their merit, but the first one will be used here | Whichever approach is used, __**up is always ascending**__. Both approaches have their merit, but the first one will be used here. | ||
| Line 1,406: | Line 1,412: | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Other EDOs</strong></u></h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Other EDOs</strong></u></h1> | ||
<br /> | <br /> | ||
The up symbol means &quot;sharpened by one EDO-step&quot; in every single EDO. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47-edo.<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;fifth-less&quot; EDOs, with fifths wider than 720¢<br /> | &quot;fifth-less&quot; EDOs, with fifths wider than 720¢<br /> | ||
| Line 1,415: | Line 1,423: | ||
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:3395:&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:3395 --><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 /> | ||
| Line 1,443: | Line 1,451: | ||
The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning.<br /> | The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning.<br /> | ||
<br /> | <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.<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. To describe someone singing below pitch, instead of saying &quot;you're singing flat&quot;, you would say &quot;you're singing down&quot;.<br /> | ||
<br /> | <br /> | ||
In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards:<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 runs backwards:<br /> | ||
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 /> | ||
| Line 1,451: | Line 1,459: | ||
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.<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.<br /> | ||
M2 + M2 - | |||
D to F# - | |||
D to F - | <table class="wiki_table"> | ||
Eb + m3 - | <tr> | ||
Eb + P5 --& | <td style="text-align: center;">initial question<br /> | ||
</td> | |||
<td style="text-align: center;">reverse everything<br /> | |||
</td> | |||
<td style="text-align: center;">do the math<br /> | |||
</td> | |||
<td style="text-align: center;">reverse again<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">M2 + M2<br /> | |||
</td> | |||
<td style="text-align: center;">m2 + m2<br /> | |||
</td> | |||
<td style="text-align: center;">dim3<br /> | |||
</td> | |||
<td style="text-align: center;">aug3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">D to F#<br /> | |||
</td> | |||
<td style="text-align: center;">D to Fb<br /> | |||
</td> | |||
<td style="text-align: center;">dim3<br /> | |||
</td> | |||
<td style="text-align: center;">aug3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">D to F<br /> | |||
</td> | |||
<td style="text-align: center;">D to F<br /> | |||
</td> | |||
<td style="text-align: center;">m3<br /> | |||
</td> | |||
<td style="text-align: center;">M3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">Eb + m3<br /> | |||
</td> | |||
<td style="text-align: center;">E# + M3<br /> | |||
</td> | |||
<td style="text-align: center;">G##<br /> | |||
</td> | |||
<td style="text-align: center;">Gbb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">Eb + P5<br /> | |||
</td> | |||
<td style="text-align: center;">E# + P5<br /> | |||
</td> | |||
<td style="text-align: center;">B#<br /> | |||
</td> | |||
<td style="text-align: center;">Bb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">C major<br /> | |||
</td> | |||
<td style="text-align: center;">C minor<br /> | |||
</td> | |||
<td style="text-align: center;">C Eb G<br /> | |||
</td> | |||
<td style="text-align: center;">C E# G<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">Eb major<br /> | |||
</td> | |||
<td style="text-align: center;">E# minor<br /> | |||
</td> | |||
<td style="text-align: center;">E# G# B#<br /> | |||
</td> | |||
<td style="text-align: center;">Eb Gb Db<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">G7 = M3 + P5 + m7<br /> | |||
</td> | |||
<td style="text-align: center;">m3 + P5 + M7<br /> | |||
</td> | |||
<td style="text-align: center;">G Bb D F#<br /> | |||
</td> | |||
<td style="text-align: center;">G B# D Fb<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | <br /> | ||
Both approaches have their merit, but the first one will be used here | Whichever approach is used, <u><strong>up is always ascending</strong></u>. Both approaches have their merit, but the first one will be used here.<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
| Line 1,632: | Line 1,730: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:3396:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1035px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:3396 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextLocalImageRule:3397:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 957px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:3397 --></h2> | ||
<br /> | <br /> | ||
<!-- 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> | ||