Pergen names: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 00:08:40 UTC</tt>.<br>

: The original revision id was <tt>622159641</tt>.<br>

[//This 13b part needs clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.//]

||= " ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C^^ = D ||= C - F^=Gv - C ||= 128/121, j4=P4 ||= " ||

||= {P8, P4/2} ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= C^^ = Db ||= C - D^=Ebv - F ||= semaphore ||= 14, 15*, 18b*, 19, 20*,

||= {P8, P5/4} ||= P5/4 = vM2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` C# ||= C Dv Evv F^ G^ ||=  ||=  ||

[Question: how to find all possible pergens?//]//

//Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.//

//[//Question: What to do if the edo's 5th falls in the sweet spot? Example?//]

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.

P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2

[&lt;em&gt;This 13b part needs clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.&lt;/em&gt;]&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;128/121, j4=P4&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;C Dv Evv F^ G^&lt;br /&gt;

[Question: how to find all possible pergens?&lt;em&gt;]&lt;/em&gt;&lt;br /&gt;

&lt;em&gt;Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;/em&gt;&lt;br /&gt;

&lt;em&gt;[&lt;/em&gt;Question: What to do if the edo's 5th falls in the sweet spot? Example?//]&lt;br /&gt;

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.&lt;br /&gt;

P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2&lt;br /&gt;