Comparison of mode notation systems: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

|| 8th Sensi [8] || Ls Lss Lss || C D Eb F Gb G# A# B C || F A# D Gb B Eb G# __**C**__ ||

The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:  

They would follow a more regular pattern if using octotonic fourth-based notation:

1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.

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.

F^ -- G --- G^ -- A --- A^

C --- C^ -- D --- D^ -- E

yF# --- yC# --- yG# --- yD# --- yA#

wC ---- wG ---- wD ---- wA ---- wE

wC ---- wG ---- wD ---- wA ----- wE

gGb --- gDb --- gAb --- gEb --- gBb

Gb^^ ----- Db^^

Eb^ ------- Bb^

C ---------- Db^^

Av --------- Bb^

In color notation, the diminished comma 648/625 is g&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2. The period is ~6/5 = Tg3.

ggGb ----- ggDb

wC -------- wG

yA --------- yE

|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain ||

|| scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || 5th chain ||

Other problems with UDP are more of a taste issue. The most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Shrutal [10] 6|2(2), use four numbers where only two are needed. Also, as noted above, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.

=Jake Freivald method=  

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Naming Rank-2 Scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; &lt;/h1&gt;

  &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Kite Giedraitis method-Proposed method of naming all possible rank-2 scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;u&gt;&lt;span style="font-size: 1.3em; line-height: 1.5;"&gt;Proposed method of naming all possible rank-2 scales&lt;/span&gt;&lt;/u&gt;&lt;/h2&gt;

  &lt;br /&gt;

The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation: &lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

To find a &lt;a class="wiki_link" href="/MODMOS%20Scales"&gt;MODMOS&lt;/a&gt; scale's name, apply chromatic alterations to the MOS scale, using scale degrees, similar to UDP notation. &amp;quot;#&amp;quot; means raised by L-s, and for &lt;em&gt;some-temperament-name&lt;/em&gt;[N], &amp;quot;#&amp;quot; means moved N steps on the genchain, either forwards or backwards.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:&lt;br /&gt;

F^ -- C^ -- G^ -- D^ -- A^&lt;br /&gt;

Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

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.&lt;br /&gt;

&lt;br /&gt;

F^ -- G --- G^ -- A --- A^&lt;br /&gt;

C --- C^ -- D --- D^ -- E&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

yF# --- yC# --- yG# --- yD# --- yA#&lt;br /&gt;

wC ---- wG ---- wD ---- wA ---- wE&lt;br /&gt;

&lt;br /&gt;

wC ---- wG ---- wD ---- wA ----- wE&lt;br /&gt;

gGb --- gDb --- gAb --- gEb --- gBb&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

Gb^^ ----- Db^^&lt;br /&gt;

Eb^ ------- Bb^&lt;br /&gt;

C ---------- Db^^&lt;br /&gt;

Av --------- Bb^&lt;br /&gt;

In color notation, the diminished comma 648/625 is g&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2. The period is ~6/5 = Tg3.&lt;br /&gt;

ggGb ----- ggDb&lt;br /&gt;

wC -------- wG&lt;br /&gt;

yA --------- yE&lt;br /&gt;

     &lt;/tr&gt;

     &lt;tr&gt;

&lt;/td&gt;

         &lt;td&gt;sLsL sLsL&lt;br /&gt;

     &lt;/tr&gt;

     &lt;tr&gt;

&lt;/td&gt;

         &lt;td&gt;LsLs LsLs&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

         &lt;td style="text-align: center;"&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

         &lt;td&gt;B F#&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

     &lt;/tr&gt;

     &lt;tr&gt;

&lt;/td&gt;

         &lt;td&gt;LsLsLs LsLs&lt;br /&gt;

     &lt;/tr&gt;

     &lt;tr&gt;

&lt;/td&gt;

         &lt;td&gt;sLsLsL sLsL&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

&lt;strong&gt;&lt;u&gt;1st method&lt;/u&gt;:&lt;/strong&gt;&lt;br /&gt;

G A C D E F# G, with genchain C &lt;u&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;/u&gt; D A E * F# = G 3rd Meantone [6] #7&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

  &lt;br /&gt;

Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.&lt;br /&gt;

The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show (emphasis mine):&lt;br /&gt;

&amp;quot;A foolish consistency is the hobgoblin of little minds&amp;quot;. To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a &lt;u&gt;wise&lt;/u&gt; consistency, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

See below.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

One problem with &lt;a class="wiki_link" href="/Modal%20UDP%20Notation"&gt;UDP&lt;/a&gt; is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.&lt;br /&gt;

Other problems with UDP are more of a taste issue. The most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Shrutal [10] 6|2(2), use four numbers where only two are needed. Also, as noted above, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow"&gt;zero-based counting&lt;/a&gt; and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.&lt;br /&gt;

&lt;br /&gt;

  My goals for numbering the modes are to make it as simple as possible for people to identify and use the modes they're talking about. As such, desired characteristics include&lt;br /&gt;

(1) as little knowledge needed as possible, to help the less-sophisticated user,&lt;br /&gt;

I also have built-in checks: I know that if I start and end with the same step size that I'm doing something wrong, and using the technique for meantone[7] gives me the diatonic major scale LLsLLLs, or CDEFGABC.&lt;br /&gt;

&lt;br /&gt;

  &lt;span style="line-height: 1.5;"&gt;My suggestion is that (a) you still start with the step size that occurs the largest number of times, and (b) you still push the largest cluster of that as far out as possible. &lt;/span&gt;&lt;br /&gt;

&lt;br /&gt;

NOTE: NO collapsing genchains. NO generator knowledge needed. No mapping knowledge (or indeed mapping at all) required. Extensible to higher ranks without problems. It doesn't matter whether the scale is a temperament at all.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

&lt;span style="display: block; height: 1px; left: 0px; overflow: hidden; position: absolute; top: 4019.5px; width: 1px;"&gt;&lt;span style="background-color: #f6f7f8; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 12px; line-height: 16.08px;"&gt;&lt;strong&gt;&lt;span style="color: #3b5998;"&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/kite.giedraitis?fref=ufi" rel="nofollow"&gt;Giedraitis&lt;/a&gt;&lt;/span&gt;&lt;/strong&gt; &lt;span class="UFICommentBody"&gt;Been&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>