Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 18:20:01 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 18:45:53 UTC</tt>.

: The original revision id was <tt>~~594513846~~</tt>.

: The original revision id was <tt>594514874</tt>.

: The revision comment was: <tt></tt>

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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.

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.

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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

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.

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 __**not recommended**__ except 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.

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12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C

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, i.e. 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 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:

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||= 4 ||= -10 ||= Dv ||= +19 ||= C#^ = Db^^ ||

||= 5 ||= +2 ||= D ||= ||= ||

||= etc. || || || || ||

The value of i equals the stepspan of the up interval. 39-tone and 49-tone are problematic:

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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.

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.

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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.

<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;" />

<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;" />

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 kites, 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 spinal node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.

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<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;" />

<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;" />

<h2 id="toc3"><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;" /></h2>

<h2 id="toc3"><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;" /></h2>

<h1 id="toc4"><a name="Chord names in other EDOs"></a>Chord names in other EDOs</h1>

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<h1 id="toc13"><a name="Natural Generators"></a>Natural Generators</h1>

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.

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.

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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

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.

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 not recommended except 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.

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12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C

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, i.e. 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 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:

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</td>

</td>

</tr>

<tr>

<td style="text-align: center;">etc.

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

</tr>

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<h1 id="toc16"><a name="Rank-2 Scales: Non-5th Generators"></a>Rank-2 Scales: Non-5th Generators</h1>

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 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

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 18:20:01 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-06 18:45:53 UTC</tt>.<br>
-: The original revision id was <tt>594513846</tt>.<br>
+: The original revision id was <tt>594514874</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>
@@ Line 941: / Line 941: @@
 =__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.
 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 &amp; minor 3rd, 4th, 5th and 6th, and a perfect 7th.
@@ Line 975: / Line 975: @@
 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 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.
+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 __**not recommended**__ except 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.
@@ Line 1,054: / Line 1,054: @@
 -tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C
-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, i.e. 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 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:
@@ Line 1,203: / Line 1,203: @@
 ||= 4 ||= -10 ||= Dv ||= +19 ||= C#^ = Db^^ ||
 ||= 5 ||= +2 ||= D ||=   ||=   ||
+||= etc. ||   ||   ||   ||   ||
 The value of i equals the stepspan of the up interval. 39-tone and 49-tone are problematic:
@@ Line 1,283: / Line 1,284: @@
 =__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.
 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.
@@ Line 1,500: / Line 1,499: @@
 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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3428:&amp;lt;img src=&amp;quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1002px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3428 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3440:&amp;lt;img src=&amp;quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1002px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3440 --&gt;&lt;br /&gt;
 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 &amp;quot;generation&amp;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 &lt;strong&gt;kites&lt;/strong&gt;, 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 &lt;strong&gt;spinal&lt;/strong&gt; node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;
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@@ Line 1,844: / Line 1,843: @@
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-&lt;!-- ws:start:WikiTextLocalImageRule:3429:&amp;lt;img src=&amp;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&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3429 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3441:&amp;lt;img src=&amp;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&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3441 --&gt;&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:3442:&amp;lt;img src=&amp;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&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3442 --&gt;&lt;/h2&gt;
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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.&lt;br /&gt;
+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.&lt;br /&gt;
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 Every non-perfect EDO has a &amp;quot;natural&amp;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 &amp;quot;sharpened by one EDO-step&amp;quot;, and ups and downs aren't needed. The generator is always perfect, so there's a perfect 2nd, a major &amp;amp; minor 3rd, 4th, 5th and 6th, and a perfect 7th.&lt;br /&gt;
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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&lt;br /&gt;
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-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.&lt;br /&gt;
+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 &lt;u&gt;&lt;strong&gt;not recommended&lt;/strong&gt;&lt;/u&gt; except as an alternate, composer-oriented notation.&lt;br /&gt;
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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.&lt;br /&gt;
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 -tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C&lt;br /&gt;
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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.&lt;br /&gt;
+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, i.e. 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.&lt;br /&gt;
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 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 &amp;gt; 1 or &amp;lt; -1. If these are notated without ups and downs, the notes run out of order:&lt;br /&gt;
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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.&lt;br /&gt;
-&lt;br /&gt;
 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.&lt;br /&gt;
 Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb&lt;br /&gt;