18edo: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-26 00:35:14 UTC</tt>.<br>

: The original revision id was <tt>602810468</tt>.<br>

|| Degree || Cents ||= 5L3s Notation ||||||= [[Ups and Downs Notation|up/down ]][[Ups and Downs Notation|notation]]

sharp lower than flat || Nearest Ratio || Error ||&lt; 17-Limit Ratios* ||

|| 0 || 0 ||= C ||= C ||= unison ||= P1 || 1/1 || 0 ||&lt; 1/1 ||

|| 1 || 66.67 ||= Db ||= C^, Dv ||= up-unison,

downmajor 2nd ||= ^P1, vM2 || 27/26 || +1.329 ||&lt; 78/75, 75/72 ||

|| 2 || 133.33 ||= C# ||= D ||= major 2nd ||= M2 || 27/25 || +0.096 ||&lt; 51/55, 42/39 ||

|| 3 || 200 ||= D ||= D^, Ev ||= mid 2nd ||= ~2 || 9/8 || -3.910 ||&lt; 9/8 ||

|| 4 || 266.67 ||= Eb ||= Db, E ||= minor 2nd,

major 3rd ||= m2, M3 || 7/6 || -0.204 ||&lt; 75/64 ||

|| 5 || 333.33 ||= D# ||= E^ ||= mid 3rd ||= ~3 || 17/14 or 40/33 || -2.796 +0.293 ||&lt; 39/32 ||

|| 6 || 400 ||= E ||= Eb, F# ||= minor 3rd,

augmented 4th ||= m3, A4 || 5/4 or 44/35 || +13.686 +3.822 ||&lt; 64/55 ||

|| 7 || 466.67 ||= F ||= Fv ||= upminor 3rd,

down-fourth ||= ^m3, vP4 || 21/16 || -4.114 ||&lt; 21/16 ||

|| 8 || 533.33 ||= Gb ||= F ||= fourth ||= P4 || 15/11 || -3.617 ||&lt; 102/75 ||

|| 9 || 600 ||= F# ||= F^, Gv ||= upfourth,

downfifth ||= ^P4, vP5 || 17/12 or 24/17 || -3.000 +3.000 ||&lt; 17/12 ||

|| 10 || 666.67 ||= G ||= G ||= fifthj ||= P5 || 22/15 || +3.617 ||&lt; 75/51 ||

|| 11 || 733.33 ||= Hb ||= G^ ||= upfifth,

downmajor 6th ||= ^P5, vM6 || 32/21 || +4.114 ||&lt; 32/21 ||

|| 12 || 800 ||= G# ||= A ||= diminished 5th,

major 6th ||= d5, M6 || 8/5 or 35/22 || -13.686 -3.822 ||&lt; 51/32 ||

|| 13 || 866.67 ||= H ||= A^, Bv ||= mid 6th ||= ~6 || 28/17 or 33/20 || +2.796 -0.293 ||&lt; 64/39 ||

|| 14 || 933.33 ||= A ||= B ||= minor 6th,

major 7th ||= m6, M7 || 12/7 || +0.204 ||&lt; 55/32 ||

|| 15 || 1000 ||= Bb ||= B^ ||= mid 7th ||= ~7 || 16/9 || +3.910 ||&lt; 16/9 ||

|| 16 || 1066.67 ||= A# ||= Bb, C# ||= minor 7th ||= m7 || 50/27 || -0.096 ||&lt; 39/21 ||

|| 17 || 1133.33 ||= B ||= Cv ||= upminor 7th,

down-octave ||= ^m7, vP8 || 52/27 || -1.329 ||&lt; 75/39 ||

|| 18 || 1200 ||= C ||= C ||= octave ||= P8 || 2/1 || 0 ||&lt; 2/1** ||

18edo can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 10\18. This is only 4¢ worse that the best approximation. Using this 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo "on the fly".

||= 11 ||= 733 ||= up 5th ||= ^5, vm6 ||= A^, Bv ||= up fifth, downmajor 6th ||= ^5, vM6 ||= A^, Bv ||

||= 16 ||= 1067 ||= major 7th ||= M7 ||= C ||= minor 7th ||= m7, d8 ||= C ||

         &lt;td style="text-align: center;"&gt;5L3s Notation&lt;br /&gt;

sharp lower than flat&lt;br /&gt;

         &lt;td style="text-align: left;"&gt;17-Limit Ratios*&lt;br /&gt;

18edo can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 10\18. This is only 4¢ worse that the best approximation. Using this 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the &lt;u&gt;melodic&lt;/u&gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&lt;br /&gt;

The second approach preserves the &lt;u&gt;harmonic&lt;/u&gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo &amp;quot;on the fly&amp;quot;.&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;up 5th&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;m7, d8&lt;br /&gt;