Dave Keenan wrote:Well George, we've slept on it for 9 months now, and there has recently been a request on one of the facebook groups, for someone to add the corresponding Sagittal notation(s) to every EDO entry in the Xenharmonic Wiki. So we really ought to decide whether these will become the new standard notations for these poor-fifth EDOs, and update figures 8 and 9 on pages 16 and 17 of

Sagittal.pdf (the updated Xenharmonikon journal article) accordingly.

Since no one else is arguing, I suggest that we both attempt to come up with reasons why the existing notations for these EDOs should not be changed, or should be changed in ways different from this proposal. i.e. play devil's advocate. For this purpose, it is useful to repost this diagram.

There are eleven existing native-fifth notations that would change under this proposal. These can be grouped as follows. You should locate each group on the above diagram.

Near-superpythagorean (amber): 27, 49 (also includes 54 (2x27) and 71, which don't presently have native fifth notations)

Near-meantone: (red) 26, 45, 64 (also includes 52 (2x26), which doesn't presently have a native fifth notation)

Narrow fifths with one step per apotome (red): 33, 40, 47

Mavila, -1 step per apotome (red): 9, 16, 23

I note that 27 is

*not *simplified by this proposal, since 1\27 changes from the spartan

to the non-spartan

. Nor is 26 simplified, as it goes from being notated only with sharps and flats (apotomes), to requiring spartan symbols (for limma fractions).

One could argue that the blue area on the diagram (JI-based notations) should be expanded to include the first two categories above. This would change the boundaries, in fifth sizes, from +-7.5 c of just, to +-10 c of just. We might continue to show how apotome and limma fraction notations can be

*defined *for those with fifth errors between 7.5 c and 10 c, but we need not list them as the

*standard *native-fifth notations for those EDOs (the first two categories above).

This is the first installment of my response, which will cover all of the near-superpythagorean divisions (27, 49, 54, and 71) and some additional adjoining divisions (32, 37, 42, and 59). In evaluating these (one by one), I have examined the fractional apotome bad-5ths notation for each one to determine whether it could be replaced by a simpler JI-based notation. For reference, I have also tabulated the prime 3 error both in absolute (cents) and relative (% of a degree) terms, as well as the prime-limit consistency. In additional, I have listened to some of these in Scala to judge whether a JI-based simplification is worthwhile.

Here goes:

27 is 9-limit consistent (prime 3 has 9.156 cents or 20.60% error) and has a valid 5-comma (and 11M-diesis) of 1 degree and valid 13L-diesis (and 13M-diesis) of 2 degrees (7C vanishes). I recommend a simplification of the bad-5ths notation:

27:

(bad-5ths proposal)

replacing it with the following JI notation:

27:

(JI notation)

This is not the JI notation that we formerly agreed on, since I have replaced 13M

with 13L

, which directly notates 13/8.

49 is 7-limit consistent (prime 3 has 8.249 cents or 33.68% error) and has a valid 5-comma of 2 degrees and valid 11M-diesis of 3 degrees (7C vanishes). I recommend keeping the original 11-limit notation,

49:

(JI notation)

which is simpler than the bad-5ths notation:

49:

(bad-5ths proposal)

54 is 5-limit inconsistent (prime 3 has 9.156 cents or 41.20% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 3 degrees. It does not appear that the bad-5ths proposal can be simplified.

54:

(bad-5ths proposal)

or as subset of 108

71 is 5-limit consistent (prime 3 has 7.904 cents or 46.77% error) and has a valid 5-comma of 3 degrees, 11M-diesis of 4 degrees, and 13M-diesis of 5 degrees. However, although it has a valid 7-comma of 1 degree, it is severely 7-limit inconsistent, making it impractical to use the 7-comma symbol in the notation. Therefore, I recommend that the bad-5ths proposal be used.

71:

(bad-5ths proposal)

or as subset of 142

32 is 5-limit inconsistent (prime 3 has 10.545 cents or 28.12% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 2 degrees. It does not appear that the bad-5ths proposal can be simplified.

32:

(bad-5ths proposal)

or as subset of 96

37 is 7-limit consistent (prime 3 has 11.559 cents or 35.64% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 2 degrees. It does not appear that the bad-5ths proposal can be simplified.

37:

(bad-5ths proposal)

42 is 7-limit consistent (prime 3 has 12.331 cents or 43,16% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 2 degrees. It does not appear that the bad-5ths proposal can be simplified.

42:

(bad-5ths proposal)

or as subset of 84

59 is 7-limit consistent (prime 3 has 9.909 cents or 48.72% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 3 degrees. It does not appear that the bad-5ths proposal can be simplified.

59:

(bad-5ths proposal)

or as subset of 118