Constrained tuning: Difference between revisions

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</pre>

== ~~CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning~~ ==

== Comparison of tunings ==

{{Todo|inline=1| rework |comment=More properly and concisely summarize each side's POV. }}

=== Criticism of CTE ===

People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle.

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So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap|k {{=}} 0}} is the TE norm, {{nowrap|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE.

=== ~~Comparison~~ ===

=== Examples ===

These tunings can be very different from each other.

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 </pre>
-== CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning ==
+== Comparison of tunings ==
+{{Todo|inline=1| rework |comment=More properly and concisely summarize each side's POV. }}
 === Criticism of CTE ===
 People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle.
@@ Line 382: / Line 384: @@
 So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap|k {{=}} 0}} is the TE norm, {{nowrap|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE.
-=== Comparison ===
+=== Examples ===
 These tunings can be very different from each other.