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 ===

We can state a few helpful analogies:

'''(n-)A[Q]Sp:n-E[Q]Dp'''

An arithmetic sequence of some kind of quantity Q is analogous to an equal division. Both require an interval '''p''' to be specified. The key difference is that a sequence is potentially open-ended — proceeding forever without repeating (such as the overtone series) — so its parameter '''n''', for the total number of pitches, is optional.

'''

OS:AFS::OD:EFD

AS:APS::__:EPD

US:ALS::UD:ELD

'''

Each of these rows has the form rational sequence : irrational division :: rational division : irrational division. The first row is for frequency, the second for pitch, and the third for length.

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.

== Non-arithmetic tunings ==

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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 ===
+We can state a few helpful analogies:
+'''(n-)A[Q]Sp:n-E[Q]Dp'''
+An arithmetic sequence of some kind of quantity Q is analogous to an equal division. Both require an interval '''p''' to be specified. The key difference is that a sequence is potentially open-ended — proceeding forever without repeating (such as the overtone series) — so its parameter '''n''', for the total number of pitches, is optional.
+'''
+OS:AFS::OD:EFD
+AS:APS::__:EPD
+US:ALS::UD:ELD
+'''
+Each of these rows has the form rational sequence : irrational division :: rational division : irrational division. The first row is for frequency, the second for pitch, and the third for length.
+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.
 == Non-arithmetic tunings ==