Harmonotonic tuning: Difference between revisions

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But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.

=== Comparing ~~Sequences and Divisions~~ ===

=== Comparing Arithmetic Tunings ===

We can state a few helpful analogies:

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We haven't looked in detail at the middle row, for pitch. EPD, again, is long for simply ED. AS stands for ambitonal sequence; these are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. There is one blank space in the system of analogies for rational divisions of pitch; these are theoretically impossible.

Every rational arithmetic tuning is a subtype of its corresponding irrational arithmetic tuning:

* An OS is a specific (rational) type of AFS.

* An OD is a specific (rational) type of EFD.

* An AS is a specific (rational) type of APS.

* A US is a specific (rational) type of ALS.

* A UD is a specific (rational) type of ELD.

== Non-arithmetic monotonic tunings ==

@@ Line 54: / Line 54: @@
 But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.
-=== Comparing Sequences and Divisions ===
+=== Comparing Arithmetic Tunings ===
 We can state a few helpful analogies:
@@ Line 71: / Line 71: @@
 We haven't looked in detail at the middle row, for pitch. EPD, again, is long for simply ED. AS stands for ambitonal sequence; these are sequences which are rational but ambiguous between otonality and utonality, such as a chain of the same JI pitch. There is one blank space in the system of analogies for rational divisions of pitch; these are theoretically impossible.
+Every rational arithmetic tuning is a subtype of its corresponding irrational arithmetic tuning:
+* An OS is a specific (rational) type of AFS.
+* An OD is a specific (rational) type of EFD.
+* An AS is a specific (rational) type of APS.
+* A US is a specific (rational) type of ALS.
+* A UD is a specific (rational) type of ELD.
 == Non-arithmetic monotonic tunings ==