Dave Keenan wrote: ↑Wed May 24, 2023 11:10 am
I get 700.3106 ¢ for the 12-generator fifth. That's (12 ×
n(1\2)) mod 1200 ¢ = ((12 × n(1/2)) mod 1)×1200 ¢ = (12/ϕ² mod 1)×1200 ¢.
That's correct. I don't know where I got the other number from.
Dave Keenan wrote: ↑Wed May 24, 2023 11:10 am
I would expect a chain of 12 generators to give 13 notes, so I think you are missing a final G.
I meant a generator chain of 12 notes, not a chain of 12 generators. It just makes it easier to see how we notate an interval a certain number of generators away from every note.
Dave Keenan wrote: ↑Wed May 24, 2023 11:10 am
I agree we need not be constrained by the Trojan capture zones, and should avoid
.
I don't know what made you think that I don't like the Trojan capture zones. I think they are quite handy and a valid consideration when the fifth is between 699.1831¢ and 700.7552¢, which indeed holds in this case, and all of my proposals except one agreed with them. I however understand that you dislike the use of
here and therefore all of the proposals in the previous comment tried to avoid it.
Why did you choose to use
for -68 steps and not for +76? It's indeed a smaller number in absolute value, but +76 gives you an apotome complement of +8 generators, which is much better than the +152 generators that -68 gives you. Also, having
as an accidental for -68 and not an accidental for +76 gives you the problem in the beginning of my last post, where D
is below C
and the distance between them is only 8 generators.
I strongly prefer having an accidental for +76 generators, so if you insist on using the accidentals for 144edo, the notation will be
Acc Gens Cents
+55 9.76
-34 15.79
+21 25.54
+76 35.3
-13 41.33
+42 51.09
+84 102.17
D:A +12 700.31
However, I think this can be improved. The Trojan symbols that fits 35.3¢ (or 0.3455A) is actually
, and I don't see a reason to not use it here, other than lack of compatibility with 144edo, so I think a better notation will be
Acc Gens Cents
+55 9.76
-34 15.79
+21 25.54
+76 35.3
-13 41.33
+42 51.09
+84 102.17
D:A +12 700.31
There is also a problem of
notating an interval of 9.76¢ (0.0955A) which is in the capture zone of
. Now, I won't suggest to replace the symbols, but I think that making
notate -89 generators is a solution of this discrepancy that is worth considering. It gives you a quite handy accidental of
for -5 generators, and keeps both +8 generators and -13 generators from every note fine in regards to nominal crossings. Therefore an alternative suggestion (which was also the last one on my previous post) is
Acc Gens Cents
-89 6.03
-34 15.79
+21 25.54
+76 35.3
-13 41.33
+42 51.09
+84 102.17
D:A +12 700.31
This also has the advantage that it can be extended, if needed, to the system I introduced in my first post on this thread.