Kite's ups and downs notation: Difference between revisions
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<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;white-space: pre-wrap ! important" class="old-revision-html">="Ups and Downs" Notation= | <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;white-space: pre-wrap ! important" class="old-revision-html">="Ups and Downs" Notation= | ||
Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the mid symbol "~" which undoes ups and downs. | Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs (see the Cancelling section). | ||
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths adds up to one EDO-step. So C# is right next to C, and your keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes. | To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths adds up to one EDO-step. So C# is right next to C, and your keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes. | ||
| Line 100: | Line 100: | ||
This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | ||
This section will cover sweet EDOs and the other categories will be covered in | 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 "generation" 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. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others. | ||
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. Likewise the pentatonic kite, minus the spine, contains the EDOs that can be notated pentatonically without ups and downs. | |||
The diagram only shows part of the full Stern-Brocot tree. The cents at the top extend from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add 7 fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds. Every EDO will appear on two of these six kites. This means that every edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later. | |||
This section will cover sweet EDOs and the other categories will be covered in later sections. | |||
As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | ||
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JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber. | JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber. | ||
24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one: | 24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth (generator) isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one: | ||
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G# | Eb-Bb-F-C-G-D-A-E-B-F#-C#-G# | ||
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^ | Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^ | ||
Just as G# could be written as Ab, all the up notes could be written as down notes. | Just as G# could alternatively be written as Ab, all the up notes could alternatively be written as down notes. | ||
In open EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs: | In open EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs: | ||
| Line 751: | Line 757: | ||
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2 | 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 | Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2 | ||
etc.</pre></div> | etc. | ||
==__Rank-2 Notation__== | |||
Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See part V of my book for more on frameworks.) | |||
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime.</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:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | <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:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | ||
<br /> | <br /> | ||
Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the mid symbol &quot;~&quot; which undoes ups and downs.<br /> | Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the optional mid symbol &quot;~&quot; which undoes ups and downs (see the Cancelling section).<br /> | ||
<br /> | <br /> | ||
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths adds up to one EDO-step. So C# is right next to C, and your keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes.<br /> | To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths adds up to one EDO-step. So C# is right next to C, and your keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes.<br /> | ||
| Line 843: | Line 855: | ||
fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br /> | fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1384:&lt;img src=&quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1035px; width: 800px;&quot; /&gt; --><img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:1384 --><br /> | ||
<br /> | <br /> | ||
This is in addition to the trivial EDOs, 1, 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, 1, 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 /> | ||
This section will cover sweet EDOs and the other categories will be covered in | 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. The colored regions of the tree are what I call <strong>kites</strong>. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.<br /> | ||
<br /> | |||
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. Likewise the pentatonic kite, minus the spine, contains the EDOs that can be notated pentatonically without ups and downs.<br /> | |||
<br /> | |||
The diagram only shows part of the full Stern-Brocot tree. The cents at the top extend from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add 7 fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds. Every EDO will appear on two of these six kites. This means that every edo has a &quot;natural&quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.<br /> | |||
<br /> | |||
This section will cover sweet EDOs and the other categories will be covered in later sections.<br /> | |||
<br /> | <br /> | ||
As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | ||
| Line 868: | Line 886: | ||
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.<br /> | JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.<br /> | ||
<br /> | <br /> | ||
24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:<br /> | 24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth (generator) isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:<br /> | ||
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#<br /> | Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#<br /> | ||
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^<br /> | Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^<br /> | ||
Just as G# could be written as Ab, all the up notes could be written as down notes.<br /> | Just as G# could alternatively be written as Ab, all the up notes could alternatively be written as down notes.<br /> | ||
<br /> | <br /> | ||
In open EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs:<br /> | In open EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs:<br /> | ||
| Line 1,023: | Line 1,041: | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1385:&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:1385 --><br /> | ||
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| Line 2,799: | Line 2,817: | ||
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2<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 /> | Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2<br /> | ||
etc.</body></html></pre></div> | etc.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Summary of EDO notation-Rank-2 Notation"></a><!-- ws:end:WikiTextHeadingRule:36 --><u>Rank-2 Notation</u></h2> | |||
<br /> | |||
Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See part V of my book for more on frameworks.) <br /> | |||
<br /> | |||
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime.</body></html></pre></div> | |||