Constrained tuning: Difference between revisions

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=== Comparison ===

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:

These tunings can be very different from each other.

==== Meantone ====

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>

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

==== Blackwood ====

There is a particularly large difference when looking at less-accurate tunings. Perhaps the clearest way to see the difference is to look at tunings where one prime has total freedom as a generator. In those situations, the free prime is tuned pure. Blackwood is a good example:

Note that the POTE tuning still has prime 5 tuned sharp, even though it could be tuned pure:

<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2799.594 \end{matrix} ]</math>

The CWE gives similar results, although tunes it a few cents flatter:

<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2795.127 \end{matrix} ]</math>

The CTE tuning, on the other hand, tunes prime 5 pure:

<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2786.314 \end{matrix} ]</math>

The thing is that although it seems nice on paper to have 5/1 tuned pure, as mentioned above, this leads to very lopsided behavior for chords like 4:5:6. Note that the tunings for KE and POTE distribute the error between 5/4 and 6/5 relatively evenly; both are very close to the delta-rational 0-397-720. The CTE tuning, on the other hand, has that chord tuned to 0-386-720, so that all of the error is on the 6/5 at about 18 cents sharp.

== Special constraint ==

@@ Line 277: / Line 277: @@
 === Comparison ===
-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:
+These tunings can be very different from each other.
+==== Meantone ====
+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>
@@ Line 288: / Line 292: @@
 <math>\langle \begin{matrix} 1200.000 & 1896.952 & 2787.809 & 3369.521 \end{matrix} ]</math>
+==== Blackwood ====
+There is a particularly large difference when looking at less-accurate tunings. Perhaps the clearest way to see the difference is to look at tunings where one prime has total freedom as a generator. In those situations, the free prime is tuned pure. Blackwood is a good example:
+Note that the POTE tuning still has prime 5 tuned sharp, even though it could be tuned pure:
+<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2799.594 \end{matrix} ]</math>
+The CWE gives similar results, although tunes it a few cents flatter:
+<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2795.127 \end{matrix} ]</math>
+The CTE tuning, on the other hand, tunes prime 5 pure:
+<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2786.314 \end{matrix} ]</math>
+The thing is that although it seems nice on paper to have 5/1 tuned pure, as mentioned above, this leads to very lopsided behavior for chords like 4:5:6. Note that the tunings for KE and POTE distribute the error between 5/4 and 6/5 relatively evenly; both are very close to the delta-rational 0-397-720. The CTE tuning, on the other hand, has that chord tuned to 0-386-720, so that all of the error is on the 6/5 at about 18 cents sharp.
 == Special constraint ==