## A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

George Secor wrote:
Dave Keenan wrote:66 is 1:3:17 consistent. It is not 1:3:p consistent for any other prime p up to 19 (no higher primes were checked). So no useful JI-based notation is possible for it. But the following apotome-fraction notation is acceptable. Although is not valid as 2 degrees, it's close.
66: AF (same as 59)
I agree -- with one exception: according to my spreadsheet 143C is indeed valid as 2 degrees of 66-EDO!
It's good that we agree on the notation, but it's strange that our spreadsheets disagree on this.

I've attached below, the spreadsheet I built last week for this purpose. It shows on a chart, how the size of each symbol's comma varies, as a fraction of the apotome, as the size of the fifth changes.

EDOs are listed under their fifth-size to the nearest cent. A table below the chart tells you how many steps-per-apotome each EDO has. You can then calculate the midpoints between steps, as decimal fractions of the apotome, and determine which symbols fall within the capture zone for each step. e.g. For an EDO with 2 steps to the apotome, a symbol must fall between 0.25 and 0.75 apotomes in order to correspond to 1 step.
comma apotome fraction vs fifth size.xlsx
comma apotome fraction vs fifth size.gif

Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

I suggest that, to reduce the total number of symbols required to cover all the EDOs up to 72, we adopt the alternative notation for 58-edo described in the note at the bottom of page 15 of the XH article. i.e.

Change it from
58:
to
58:

George Secor
Posts: 31
Joined: Tue Sep 01, 2015 11:36 pm
Location: Godfrey, Illinois, US

### Re: A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan wrote:

I suggest that, to reduce the total number of symbols required to cover all the EDOs up to 72, we adopt the alternative notation for 58-edo described in the note at the bottom of page 15 of the XH article. i.e.

Change it from
58:
to
58:

I agree.

George Secor
Posts: 31
Joined: Tue Sep 01, 2015 11:36 pm
Location: Godfrey, Illinois, US

### Re: A proposal to simplify the notation of EDOs with bad fifths

I previously wrote:

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.
or as subset of 108

Instead of the above, I am now submitting the following proposal.

54-edo 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. I recommend that although (as 2deg54) does not give the best 5/4 (due to 5-limit inconsistency), it does give the best 5/3 and 7/5 and results in usable 7-limit chords. becomes 1deg as 143C:
54:

George Secor
Posts: 31
Joined: Tue Sep 01, 2015 11:36 pm
Location: Godfrey, Illinois, US

### Re: A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan wrote:
Mon Jul 23, 2018 6:40 pm
Here is the diagram showing my preferred notations for EDOs from 5 to 72, with symbols up to the 1/2-apotome or 3/4-limma.


45

I am submitting the following counter-proposal.

45-edo is 7-limit consistent (prime 3 has -8.622 cents or -32.33% error) and has a valid apotome of 2 degrees. I recommend that although (as 1deg45) does not give the best 11/8 (due to 11-limit inconsistency), it does give the best 11/6, 11/7, and 11/9 and results in good 11-limit (and 13-limit) chords:

George Secor
Posts: 31
Joined: Tue Sep 01, 2015 11:36 pm
Location: Godfrey, Illinois, US

### Re: A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan wrote:
Mon Jul 23, 2018 6:40 pm
Here is the diagram showing my preferred notations for EDOs from 5 to 72, with symbols up to the 1/2-apotome or 3/4-limma.

42
This should be:
42

Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

Great to hear from you, George.

It's good that we agree on 58, and thanks for picking up my error in 42. It must of course agree with the unified apotome-fraction notation (given here).

I will respond regarding 45 and 52 when I've had a chance to get back up to speed on this stuff and compare your proposals with the alternatives.

Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

George Secor wrote:
Thu Nov 14, 2019 7:56 am
I am now submitting the following proposal.

54-edo 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. I recommend that although (as 2deg54) does not give the best 5/4 (due to 5-limit inconsistency), it does give the best 5/3 and 7/5 and results in usable 7-limit chords. becomes 1deg as 143C:
54:
I have compared this to the alternatives and I agree it is the best choice. It also correctly notates 27-edo as a subset. Or putting it another way, it gives a consistent apotome-fraction notation for EDOs with fifths having an error of between +8.3 and +9.8 cents. Namely 27 and 54. Well done.

Why not make the notations for 61 and 68 the same, and eliminate their size-reversals ?

Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

George Secor wrote:
Thu Nov 14, 2019 8:15 am
I am submitting the following counter-proposal [to the existing 45-edo notation of ]

45-edo is 7-limit consistent (prime 3 has -8.622 cents or -32.33% error) and has a valid apotome of 2 degrees. I recommend that although (as 1deg45) does not give the best 11/8 (due to 11-limit inconsistency), it does give the best 11/6, 11/7, and 11/9 and results in good 11-limit (and 13-limit) chords:
I'm afraid I don't see this as sufficient reason to change the existing notation for 45-edo, as given in Figure 8 of the XH18 paper. The existing notation has the advantage of a valid assignment of as 35C, and of giving a consistent apotome fraction notation for EDOs with fifth errors between -9.8c and -7.5c. Namely 26, 45, 52 and 64 edos.

Symbol  Apotome fractions represented
-------------------------------------
1/3, 1/2
2/3

Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

For Joe Monzo. A representative sample of the suggested notation for 37-edo, from D to F:

37edo:
D	D	D	D	E	E	E	E	F


The symbols can be found in the Spartan Sagittal single-shaft area of the free SMuFL font called Bravura. [Edit: And the 1 degree symbol is in the Promethean single-shaft section.]

Possible spellings of notes from D to F in 37-edo:

37edo: DE = 7, EF = 1,  = +6,  = -6,  = +3,  = -3;  = +2,  = -2;  = +1,  = -1
D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F


On the staff, the order of the symbols is reversed relative to the above. e.g.
o

o

o