Kite's ups and downs notation: Difference between revisions
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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>2016-09- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-30 19:41:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593739606</tt>.<br> | ||
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Ups and downs can be used to notate 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. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.) | Ups and downs can be used to notate 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. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.) | ||
There are two ways ups and downs can be used. One is for | |||
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain: | Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain: | ||
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Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different "height"; one is up, another is down, etc. See [[xenharmonic/Naming Rank-2 Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods|xenharmonic.wikispaces.com/Naming+Rank-2+Scales]] | Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different "height"; one is up, another is down, etc. See [[xenharmonic/Naming Rank-2 Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods|xenharmonic.wikispaces.com/Naming+Rank-2+Scales]] | ||
== == | |||
==__Generators other than a fifth__ | |||
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. | |||
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5 | |||
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4 | |||
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3 | |||
When the period is an octave, the genweb octave-reduces to a single horizontal genchain: | |||
F --- C --- G --- D --- A --- E --- B | |||
But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, [[Diaschismic family|Srutal]] [10] might look like this: | |||
F^3 --- C^4 --- G^4 --- D^5 --- A^5 | |||
C3 ---- G3 ----- D4 ---- A4 ---- E5 | |||
F^2 --- C^3 --- G^3 --- D^4 --- A^4 | |||
C2 ---- G2 ----- D3 ---- A3 ---- E3 | |||
F^1 --- C^2 --- G^2 --- D^3 --- A^3 | |||
C1 ---- G1 ----- D2 ---- A2 ---- E2 | |||
which octave-reduces to two genchains: | |||
F^ --- C^ --- G^ --- D^ --- A^ | |||
C ---- G ----- D ---- A ---- E | |||
Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of "up" here is "a half-octave minus a fourth", with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave. | |||
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth. | |||
Gv --- Dv --- Av --- Ev --- Bv | |||
C ----- G ----- D ---- A ---- E | |||
It would also be valid to exchange the two rows: | |||
C ----- G ----- D ---- A ---- E | |||
Gv --- Dv --- Av --- Ev --- Bv | |||
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8. | |||
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete. | |||
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names: | |||
F^ -- G --- G^ -- A --- A^ | |||
C --- C^ -- D --- D^ -- E | |||
Another alternative is to use [[Kite's color notation|color notation]]. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]: | |||
yF# --- yC# --- yG# --- yD# --- yA# | |||
wC ---- wG ---- wD ---- wA ---- wE | |||
As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period: | |||
wC ---- wG ---- wD ----- wA ---- wE | |||
gGb --- gDb --- gAb --- gEb --- gBb | |||
All five Srutal [10] modes, using ups and downs. Every other scale note has an up. | |||
|| scale name || sL pattern || example in C || 1st genchain || 2nd genchain || | |||
|| 1st Srutal [10] || ssssL-ssssL || C C^ D D^ E F^ G G^ A A^ C || __**C**__ G D A E || F^ C^ G^ D^ A^ || | |||
|| 2nd Srutal [10] || sssLs-sssLs || C C^ D D^ F F^ G G^ A Bb^ C || F __**C**__ G D A || Bb^ F^ C^ G^ D^ || | |||
|| 3rd Srutal [10] || ssLss-ssLss || C C^ D Eb^ F F^ G G^ Bb Bb^ C || Bb F __**C**__ G D || Eb^ Bb^ F^ C^ G^ || | |||
|| 4th Srutal [10] || sLsss-sLsss || C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C || Eb Bb F __**C**__ G || Ab^ Eb^ Bb^ F^ C^ || | |||
|| 5th Srutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ || | |||
The [[Octatonic scale|Diminished]] [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains. | |||
Gb^^ ----- Db^^ | |||
Eb^ ------- Bb^ | |||
C ---------- G | |||
Av --------- Ev | |||
The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. "Up" means "a quarter-octave minus a ~32/27". | |||
Using ~25/24 as the generator yields the same scales and mode numbers: | |||
Gb^^ ----- G | |||
Eb^ ------- Ev | |||
C ---------- Db^^ | |||
Av --------- Bb^ | |||
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8]. | |||
ggGb ----- ggDb | |||
gEb ------- gBb | |||
wC -------- wG | |||
yA --------- yE | |||
Both Diminished [8] modes, using ups and downs: | |||
|| scale name || sL pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || | |||
|| 1st Diminished[ 8] || sLsL sLsL || C Db^^ Eb^ Ev Gb^^ G Av Bb^ C ||= __**C**__ G || Eb^ Bb^ || Gb^^ Db^^ || Av Ev || | |||
|| 2nd Diminished [8] || LsLs LsLs || C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C ||= F __**C**__ || Ab^ Eb^ || Cb^^ Gb^^ || Dv Av || | |||
There are only two [[Blackwood]] [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height": | |||
E^^ ------- G#^^ | |||
D^ -------- F#^ | |||
C ---------- E | |||
Bbv ------- Fv | |||
Gvv ------- Dvv | |||
Ups and downs could indicate the generator instead of the period: | |||
F ------ Av | |||
D ------ F#v | |||
C ------ Ev | |||
A ------ C#v | |||
G ------ Bv | |||
Assuming octave equivalence, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2. | |||
F ------ Av | |||
C ------ Ev | |||
G ------ Bv | |||
D ------ F#v | |||
A ------ C#v | |||
In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y. | |||
wF ------ yA | |||
wC ------ yE | |||
wG ------ yB | |||
wD ------ yF# | |||
wA ------ yC# | |||
Both Blackwood modes, using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4": | |||
|| scale name || sL pattern || example in C || genchains || | |||
|| 1st Blackwood [10] || Ls-Ls-Ls-Ls-Ls || C C#v D Ev F F#v G Av A Bv C ||= __**C**__-Ev, D-F#v, F-Av, G-Bv, A-C#v || | |||
|| 2nd Blackwood [10] || sL-sL-sL-sL-sL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^-__**C**__, Bb^-D, C^-E, Eb^-G, F^-A || | |||
=__Generators other than a fifth__= | |||
The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation. | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation. | ||
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||= 22 ||= 0 ||= D ||= ||= ||</pre></div> | ||= 22 ||= 0 ||= D ||= ||= ||</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule:32:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#x&quot;Ups and downs&quot; for 22edo">&quot;Ups and downs&quot; for 22edo</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Other EDOs">Other EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 1em;"><a href="#x22edo Chord Names">22edo Chord Names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#toc3"></a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#Chord names in other EDOs">Chord names in other EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 1em;"><a href="#Cross-EDO considerations">Cross-EDO considerations</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Scale Fragments">Scale Fragments</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Summary of EDO notation">Summary of EDO notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Regular&quot; EDOs">&quot;Regular&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Perfect&quot; EDOs">&quot;Perfect&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Fourthward&quot; EDOs">&quot;Fourthward&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Pentatonic&quot; EDOs">&quot;Pentatonic&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-&quot;Fifth-less&quot; EDOs">&quot;Fifth-less&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 1em;"><a href="#Rank-2 Notation">Rank-2 Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Generators other than a fifth">Generators other than a fifth</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --></div> | ||
<!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and downs&quot; for 22edo"></a><!-- ws:end:WikiTextHeadingRule:0 --><u>&quot;Ups and downs&quot; for 22edo</u></h1> | |||
<!-- ws:end:WikiTextTocRule: | |||
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Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works 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;.<br /> | Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works 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;.<br /> | ||
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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 /> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4254:&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:4254 --><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 /> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4255:&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:4255 --><br /> | ||
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Ups and downs can be used to notate 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. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | Ups and downs can be used to notate 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. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | ||
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There are two ways ups and downs can be used. One is for<br /> | |||
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Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:<br /> | Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:<br /> | ||
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Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different &quot;height&quot;; one is up, another is down, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods">xenharmonic.wikispaces.com/Naming+Rank-2+Scales</a><br /> | Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different &quot;height&quot;; one is up, another is down, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods">xenharmonic.wikispaces.com/Naming+Rank-2+Scales</a><br /> | ||
<br /> | <br /> | ||
< | <br /> | ||
<br /> | |||
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.<br /> | |||
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5<br /> | |||
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4<br /> | |||
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3<br /> | |||
<br /> | |||
When the period is an octave, the genweb octave-reduces to a single horizontal genchain:<br /> | |||
F --- C --- G --- D --- A --- E --- B<br /> | |||
<br /> | |||
But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, <a class="wiki_link" href="/Diaschismic%20family">Srutal</a> [10] might look like this:<br /> | |||
F^3 --- C^4 --- G^4 --- D^5 --- A^5<br /> | |||
C3 ---- G3 ----- D4 ---- A4 ---- E5<br /> | |||
F^2 --- C^3 --- G^3 --- D^4 --- A^4<br /> | |||
C2 ---- G2 ----- D3 ---- A3 ---- E3<br /> | |||
F^1 --- C^2 --- G^2 --- D^3 --- A^3<br /> | |||
C1 ---- G1 ----- D2 ---- A2 ---- E2<br /> | |||
<br /> | |||
which octave-reduces to two genchains:<br /> | |||
F^ --- C^ --- G^ --- D^ --- A^<br /> | |||
C ---- G ----- D ---- A ---- E<br /> | |||
<br /> | |||
Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of &quot;up&quot; here is &quot;a half-octave minus a fourth&quot;, with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.<br /> | |||
<br /> | |||
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.<br /> | |||
Gv --- Dv --- Av --- Ev --- Bv<br /> | |||
C ----- G ----- D ---- A ---- E<br /> | |||
<br /> | |||
It would also be valid to exchange the two rows:<br /> | |||
C ----- G ----- D ---- A ---- E<br /> | |||
Gv --- Dv --- Av --- Ev --- Bv<br /> | |||
<br /> | |||
Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.<br /> | |||
<br /> | |||
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.<br /> | |||
<br /> | |||
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:<br /> | |||
F^ -- G --- G^ -- A --- A^<br /> | |||
C --- C^ -- D --- D^ -- E<br /> | |||
<br /> | |||
Another alternative is to use <a class="wiki_link" href="/Kite%27s%20color%20notation">color notation</a>. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:<br /> | |||
<br /> | |||
yF# --- yC# --- yG# --- yD# --- yA#<br /> | |||
wC ---- wG ---- wD ---- wA ---- wE<br /> | |||
<br /> | |||
As always, y means &quot;81/80 below w&quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:<br /> | |||
wC ---- wG ---- wD ----- wA ---- wE<br /> | |||
gGb --- gDb --- gAb --- gEb --- gBb<br /> | |||
<br /> | |||
All five Srutal [10] modes, using ups and downs. Every other scale note has an up.<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>scale name<br /> | |||
</td> | |||
<td>sL pattern<br /> | |||
</td> | |||
<td>example in C<br /> | |||
</td> | |||
<td>1st genchain<br /> | |||
</td> | |||
<td>2nd genchain<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1st Srutal [10]<br /> | |||
</td> | |||
<td>ssssL-ssssL<br /> | |||
</td> | |||
<td>C C^ D D^ E F^ G G^ A A^ C<br /> | |||
</td> | |||
<td><u><strong>C</strong></u> G D A E<br /> | |||
</td> | |||
<td>F^ C^ G^ D^ A^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2nd Srutal [10]<br /> | |||
</td> | |||
<td>sssLs-sssLs<br /> | |||
</td> | |||
<td>C C^ D D^ F F^ G G^ A Bb^ C<br /> | |||
</td> | |||
<td>F <u><strong>C</strong></u> G D A<br /> | |||
</td> | |||
<td>Bb^ F^ C^ G^ D^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>3rd Srutal [10]<br /> | |||
</td> | |||
<td>ssLss-ssLss<br /> | |||
</td> | |||
<td>C C^ D Eb^ F F^ G G^ Bb Bb^ C<br /> | |||
</td> | |||
<td>Bb F <u><strong>C</strong></u> G D<br /> | |||
</td> | |||
<td>Eb^ Bb^ F^ C^ G^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4th Srutal [10]<br /> | |||
</td> | |||
<td>sLsss-sLsss<br /> | |||
</td> | |||
<td>C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C<br /> | |||
</td> | |||
<td>Eb Bb F <u><strong>C</strong></u> G<br /> | |||
</td> | |||
<td>Ab^ Eb^ Bb^ F^ C^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5th Srutal [10]<br /> | |||
</td> | |||
<td>Lssss-Lssss<br /> | |||
</td> | |||
<td>C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C<br /> | |||
</td> | |||
<td>Ab Eb Bb F <u><strong>C</strong></u><br /> | |||
</td> | |||
<td>Db^ Ab^ Eb^ Bb^ F^<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<br /> | |||
The <a class="wiki_link" href="/Octatonic%20scale">Diminished</a> [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.<br /> | |||
Gb^^ ----- Db^^<br /> | |||
Eb^ ------- Bb^<br /> | |||
C ---------- G<br /> | |||
Av --------- Ev<br /> | |||
The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. &quot;Up&quot; means &quot;a quarter-octave minus a ~32/27&quot;.<br /> | |||
<br /> | |||
Using ~25/24 as the generator yields the same scales and mode numbers:<br /> | |||
Gb^^ ----- G<br /> | |||
Eb^ ------- Ev<br /> | |||
C ---------- Db^^<br /> | |||
Av --------- Bb^<br /> | |||
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8].<br /> | |||
ggGb ----- ggDb<br /> | |||
gEb ------- gBb<br /> | |||
wC -------- wG<br /> | |||
yA --------- yE<br /> | |||
<br /> | |||
Both Diminished [8] modes, using ups and downs:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>scale name<br /> | |||
</td> | |||
<td>sL pattern<br /> | |||
</td> | |||
<td>example in C<br /> | |||
</td> | |||
<td>1st chain<br /> | |||
</td> | |||
<td>2nd chain<br /> | |||
</td> | |||
<td>3rd chain<br /> | |||
</td> | |||
<td>4th chain<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1st Diminished[ 8]<br /> | |||
</td> | |||
<td>sLsL sLsL<br /> | |||
</td> | |||
<td>C Db^^ Eb^ Ev Gb^^ G Av Bb^ C<br /> | |||
</td> | |||
<td style="text-align: center;"><u><strong>C</strong></u> G<br /> | |||
</td> | |||
<td>Eb^ Bb^<br /> | |||
</td> | |||
<td>Gb^^ Db^^<br /> | |||
</td> | |||
<td>Av Ev<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2nd Diminished [8]<br /> | |||
</td> | |||
<td>LsLs LsLs<br /> | |||
</td> | |||
<td>C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C<br /> | |||
</td> | |||
<td style="text-align: center;">F <u><strong>C</strong></u><br /> | |||
</td> | |||
<td>Ab^ Eb^<br /> | |||
</td> | |||
<td>Cb^^ Gb^^<br /> | |||
</td> | |||
<td>Dv Av<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<br /> | |||
There are only two <a class="wiki_link" href="/Blackwood">Blackwood</a> [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different &quot;height&quot;:<br /> | |||
E^^ ------- G#^^<br /> | |||
D^ -------- F#^<br /> | |||
C ---------- E<br /> | |||
Bbv ------- Fv<br /> | |||
Gvv ------- Dvv<br /> | |||
<br /> | |||
Ups and downs could indicate the generator instead of the period:<br /> | |||
F ------ Av<br /> | |||
D ------ F#v<br /> | |||
C ------ Ev<br /> | |||
A ------ C#v<br /> | |||
G ------ Bv<br /> | |||
<br /> | |||
Assuming octave equivalence, the lattice rows can be reordered to make a &quot;pseudo-period&quot; of 3\5 = ~3/2.<br /> | |||
F ------ Av<br /> | |||
C ------ Ev<br /> | |||
G ------ Bv<br /> | |||
D ------ F#v<br /> | |||
A ------ C#v<br /> | |||
<br /> | |||
In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y.<br /> | |||
wF ------ yA<br /> | |||
wC ------ yE<br /> | |||
wG ------ yB<br /> | |||
wD ------ yF#<br /> | |||
wA ------ yC#<br /> | |||
<br /> | |||
Both Blackwood modes, using ups and downs to mean &quot;raised/lowered by 2/5 of an octave minus ~5/4&quot;:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>scale name<br /> | |||
</td> | |||
<td>sL pattern<br /> | |||
</td> | |||
<td>example in C<br /> | |||
</td> | |||
<td>genchains<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1st Blackwood [10]<br /> | |||
</td> | |||
<td>Ls-Ls-Ls-Ls-Ls<br /> | |||
</td> | |||
<td>C C#v D Ev F F#v G Av A Bv C<br /> | |||
</td> | |||
<td style="text-align: center;"><u><strong>C</strong></u>-Ev, D-F#v, F-Av, G-Bv, A-C#v<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2nd Blackwood [10]<br /> | |||
</td> | |||
<td>sL-sL-sL-sL-sL<br /> | |||
</td> | |||
<td>C C^ D Eb^ E F^ G Ab^ A Bb^ C<br /> | |||
</td> | |||
<td style="text-align: center;">Ab^-<u><strong>C</strong></u>, Bb^-D, C^-E, Eb^-G, F^-A<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Generators other than a fifth"></a><!-- ws:end:WikiTextHeadingRule:30 --><u>Generators other than a fifth</u></h1> | |||
<br /> | <br /> | ||
The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | ||