Constrained tuning: Difference between revisions

Line 159:

</pre>

== Versus POTE tuning ==

== Versus CTWE and POTE tuning ==

~~The~~ pure~~-octave~~ CTE ~~tuning~~ can be very different from ~~[[POTE tuning]]~~. Take 7-limit meantone as an example~~, the~~ POTE [[tuning map]]:

Consider the fact that TE tuning does not treat divisive ratios as more important than multiplicative ratios – 5/3 and 15/1 are taken as equally important, for example. To address that, a skew on the space may be introduced, resulting in TWE tuning. Constraining the equave to pure on top of TWE gives CTWE aka KE tuning. POTE works as a quick approximation to CTWE. As POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus sacrifices multiplicative ratios for divisive ratios. On the contrary, CTE sticks to the original design book of TE as its result remains TE optimal.

These tunings can be very different from each other. Take 7-limit meantone as an example. The POTE [[tuning map]] is a little bit flatter than [[quarter-comma meantone]], with all the primes tuned flat:

<math>\langle \begin{matrix} 1200.000 & 1896.495 & 2785.980 & 3364.949 \end{matrix} ]</math>

~~This~~ is a little bit ~~flatter~~ than [[quarter-comma meantone]], with ~~all the primes~~ tuned flat.

The CTWE tuning map is a little bit sharper than quarter-comma meantone, with 5 tuned sharp and 3 and 7 flat:

<math>\langle \begin{matrix} 1200.000 & 1896.656 & 2786.625 & 3366.562 \end{matrix} ]</math>

The ~~pure-octave~~ CTE tuning map:

The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp.

<math>\langle \begin{matrix} 1200.000 & 1896.952 & 2787.809 & 3369.521 \end{matrix} ]</math>

~~This is a little bit sharper than quarter-comma meantone, with prime 3 tuned flat and 5 and 7 sharp.~~

Consider the fact that TE tuning does not treat divisive ratios as more important than multiplicative ratios – 5/3 and 15/1 are taken as equally important, for example. As POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus sacrifices multiplicative ratios for divisive ratios. On the contrary, CTE sticks to the original design book of TE as its result remains TE optimal.

== Special constraint ==

@@ Line 159: / Line 159: @@
 </pre>
-== Versus POTE tuning ==
+== Versus CTWE and POTE tuning ==
-The pure-octave CTE tuning can be very different from [[POTE tuning]]. Take 7-limit meantone as an example, the POTE [[tuning map]]:
+Consider the fact that TE tuning does not treat divisive ratios as more important than multiplicative ratios – 5/3 and 15/1 are taken as equally important, for example. To address that, a skew on the space may be introduced, resulting in TWE tuning. Constraining the equave to pure on top of TWE gives CTWE aka KE tuning. POTE works as a quick approximation to CTWE. As POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus sacrifices multiplicative ratios for divisive ratios. On the contrary, CTE sticks to the original design book of TE as its result remains TE optimal.
+These tunings can be very different from each other. Take 7-limit meantone as an example. The POTE [[tuning map]] is a little bit flatter than [[quarter-comma meantone]], with all the primes tuned flat:
 <math>\langle \begin{matrix} 1200.000 & 1896.495 & 2785.980 & 3364.949 \end{matrix} ]</math>
-This is a little bit flatter than [[quarter-comma meantone]], with all the primes tuned flat.
+The CTWE tuning map is a little bit sharper than quarter-comma meantone, with 5 tuned sharp and 3 and 7 flat:
+<math>\langle \begin{matrix} 1200.000 & 1896.656 & 2786.625 & 3366.562 \end{matrix} ]</math>
-The pure-octave CTE tuning map:
+The CTE tuning map is even sharper, with 3 tuned flat and 5 and 7 sharp.
 <math>\langle \begin{matrix} 1200.000 & 1896.952 & 2787.809 & 3369.521 \end{matrix} ]</math>
-This is a little bit sharper than quarter-comma meantone, with prime 3 tuned flat and 5 and 7 sharp.
-Consider the fact that TE tuning does not treat divisive ratios as more important than multiplicative ratios – 5/3 and 15/1 are taken as equally important, for example. As POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus sacrifices multiplicative ratios for divisive ratios. On the contrary, CTE sticks to the original design book of TE as its result remains TE optimal.
 == Special constraint ==