A proposal to simplify the notation of EDOs with bad fifths

[Dave Keenan]

= 35:36 has the tempered value of a tempered whole-tone minus 32:35, which, in 59b, is 9\59 minus 32:35 (7.73\59), which is 1.37\59, so yeah, it is valid as 1\59b.

Thanks for that. I've nudged the purple/rose boundary in this post to include 59b in purple. In fact, I thought, why not define the boundary as the noble number that is the limit of this fibonacci-like sequence
4\7, 15\26, 19\33, 34\59b, 53\92b, 87\151bb, 140\243bbbb, 227\394bbbbbb, ...
i.e. 691.3671 ¢.

Maybe all colour boundaries can be noble fractions of the octave.

Of course none of the above ETOs beyond 59b will actually have a notation. Too many steps per apotome or limma. But I don't think anyone will care.

It turns out that, if ||( was an actual symbol in Revo, it would complete the matching shaft sequences for the Rose notation: , with being ||(.

Ha! That's a great observation. There are plenty of spare codepoints at the end of the spartan multi-shaft table,
https://w3c.github.io/smufl/gitbook/tab ... ntals.html

I don't remember why George wanted to leave that gap at U+E31A, U+E31B, but it doesn't seem likely that it was for ||( !!( . If it was for those, it would have made more sense to have gaps at U+E310, U+E311 and U+E32C, U+E32D. And it seems that, if we're keeping the gap at U+E31A, U+E31B then we should keep a corresponding gap an apotome higher, at U+E336, U+E337.

so ||( !!( X( and Y( could be added there. If we do, I suggest U+E338 thru U+E33B.

[Edit: Turns out it's pointless to leave gaps that are not at the end of a named section, since SMuFL insists on naming them, like they did U+E31A (accSagittalUnused1) and U+E31B (accSagittalUnused2), after which those names can never be changed. So best to start assigning new symbols from U+E336.]

[volleo6144]

Thanks for that. I've nudged the purple/rose boundary in this post to include 59b in purple. In fact, I thought, why not define the boundary as the noble number that is the limit of this fibonacci-like sequence
4\7, 15\26, 19\33, 34\59b, 53\92b, 87\151bb, 140\243bbbb, 227\394bbbbbb, ...
i.e. 691.3671 ¢.

Maybe all colour boundaries can be noble fractions of the octave.

This specific noble number is (515-√5)\890, or 691.36710¢. It, as a fraction of an octave, satisfies the equation 445x² - 515x + 149 = 0, and, as a number of cents, satisfies 89x² - 123600x + 42912000 = 0.
(All numbers of the form (A+√B)/C with A,B,C integers satisfy some equation of the form Dx² - Ex + F = 0.)

The last element in the sequence you described whose EDO is less than 10,000 is 4071\7066b¹²⁴.

Ha! That's a great observation. There are plenty of spare codepoints at the end of the spartan multi-shaft table,
https://w3c.github.io/smufl/gitbook/tab ... ntals.html
so ||( !!( X( and Y( could be added there. If we do, I suggest U+E338  thru U+E33B .

I don't remember why George wanted to leave that gap at U+E31A , U+E31B , but it doesn't seem likely that it was for ||( !!( . If it was for those, it would have made more sense to have gaps at U+E310 , U+E311  and U+E32C , U+E32D . And it seems that, if we're keeping the gap at U+E31A , U+E31B  then we should keep a corresponding gap an apotome higher, at U+E336 , U+E337 .

I don't really know either. There's also )|| = and ~|| = and their X-shaft counterparts )X = and , and things like )/||\ for which I'm not sure what their canonical equivalents would be ( in this case?), and also down-accidental versions, and there probably isn't room to fit all of these in.



[Dave Keenan]

There's a simpler way to define that noble number (and all noble numbers), using smaller integer coefficients, namely as a phi-weighted mediant of the first two terms in the series. i.e. As a fraction of an octave it is:

4 + 15ϕ
-------
7 + 26ϕ

where ϕ = (√5 + 1)/2

However

a + cϕ
------
b + dϕ

is a noble number only when a, b, c, d, are whole numbers and abs(ad - bc) = 1

[Dave Keenan]

Here's a proposal for precise colour boundaries at fifth-sizes which are noble fractions of an octave. I haven't always given the noble ratios in lowest terms. Instead I've used the simplest form where the denominator refers to EDOs of the two colours immediately on either side of the boundary.

Boundary
(fifth	Colour	Description, EDOs
error)		Fifth size	Boundary fifth as a noble fraction of an octave
------	------	-----------------------------------------------------------------------------------
	Black
+166.3732¢	868.3282¢	(1+2ϕ)\(1+3ϕ)
	Gold	Bad fifths apotome fraction notation, 5n, 32, 37, 42 (6,8,13,18 pref ⊂ 12,24,26,36)
+10.0690¢	712.0240¢	(13+3ϕ)\(22+5ϕ)
	Green	Super pythagorean notation, 22n, 27n, 49, 59, 61, 71 [note 59 changed to green]
+6.0989¢	708.0539¢	(10+13ϕ)\(17+22ϕ)
	Blue	17n, 39, 46, 56, 63
+2.1406¢	704.0956¢	(10+17ϕ)\(17+29ϕ)
	Magenta	29n, 70
+0.7910¢	702.7460¢	(17+24ϕ)\(29+41ϕ)
	Grey	Pythagorean notation (JI), 41, 53, 65
-1.1998¢	700.7552¢	(66+7ϕ)\(113+12ϕ)	[Edit: Simplified from (7+73ϕ)\(12+125ϕ) thanks to volleo6144]
	Orange	12-relative notation (Trojan notation), 12n
-2.7719¢	699.1831¢	(60+7ϕ)\(103+12ϕ)	[Edit: Simplified from (7+67ϕ)\(12+115ϕ) thanks to volleo6144]
	Pink	(Possible future colour) 91, 103
-3.1122¢	698.8428¢	(46+53ϕ)\(79+91ϕ)
	Yellow	43, 55, 67
-4.5581¢	697.3969¢	(18+25ϕ)\(31+43ϕ)	[Edit: Corrected from (18+25ϕ)\(31+45ϕ) thanks to volleo6144]
	Cyan	Meantone notation, 19n, 31n, 50, 69
-8.0520¢	693.9030¢	(26+11ϕ)\(45+19ϕ)
	Purple	Sub meantone notation, 26n, 45, 64
-10.5879¢	691.3671¢	(4+15ϕ)\(7+26ϕ)		[See viewtopic.php?p=1976#p1976]
	Rose	Bad fifths limma fraction notation, 7n, 9, 16, 23, 33, 40, 47 (11 pref subset of 22)
-52.9196¢	649.0354¢	(5+1ϕ)\(9+2ϕ)		[Edit: Simplified from (1+6ϕ)\(2+11ϕ) thanks to volleo6144]
	White

What's a reasonable criterion for deciding which "b-grade" ETs should be included on the periodic table? Some maximum value for EDO-number multiplied by absolute error in cents? I'm thinking we should only include 23b, 30b, 35b, 37b, 42b, 47b and 59b.

[volleo6144]

What's a reasonable criterion for deciding which "b-grade" ETs should be included on the periodic table? Some maximum value for EDO-number multiplied by absolute error in cents? I'm thinking we should only include 23b, 30b, 35b, 37b, 42b, 47b and 59b.

Note: any non-"b" EDO will have EDO × error < 600¢, and any "b" EDO will have EDO × error > 600¢. For example, 6 has 6 × 98¢ = 588¢, while 6b has 6 × 102¢ = 612¢; 47 has 47 × 12.59¢ = 591.9¢, while 47b has 47 × 12.94¢ = 608.1¢. There are infinitely many b-EDOs with EDO × error < any specific number above 600¢.

[Dave Keenan]

Ah. Thanks for that. How about a limit on abs(group-10)+steps_per_CD?

[volleo6144]

Ah. Thanks for that. How about a limit on abs(group-10)+steps_per_CD?

Seems maybe okay; this essentially means drawing two diagonal lines parallel to the two diagonals with 72 on them. If this limit is 12 (as the value for 72), then ten EDOs on the sharp side (42, 49, 56, 63, 70, 54, 61, 68, 59, 66, 71) and two EDOs on the flat side (64, 69) get omitted, and if it's the maximal value of the non-"b"s below 72 (71 has group = (71-13)/10 = 5.8 and CD = 13, so 4.2 + 13 = 17.2), then it has to include the orange stack all the way up to 96.

edit: 96 not 204 what was I thinking

[volleo6144]

Here's a proposal for precise colour boundaries at fifth-sizes which are noble fractions of an octave. I haven't always given the noble ratios in lowest terms. I've left them in a form where the denominator refers to EDOs of the colours on either side of the boundary.

Boundary
(fifth
error)	Colour	Description, EDOs
------	------	-----------------------------------------------------------------------------------
	Black
+166.3732¢	868.3282¢	(1+2ϕ)\(1+3ϕ)

The Fibonacci-like sequences for the boundaries (similar to the purple/rose one seen in a previous post) are, up to the point where you start needing "b"s, and including some terms before the ones of the right color (up to the point at which a higher number precedes a lower number):

Boundary	Sequence
Black/Gold	2, 1, 3, 4b (4 = 2)...
Gold/Green	22, 5, 27, 32, 59, 91b (91)...
Green/Blue	12, 5, 17, 22, 39, 61, 100b (100 = 50)...
Blue/Magenta	7, 5, 12, 17, 29, 46, 75, 121, 186, 317b (317§)...
Magenta/Grey	17, 12, 29, 41, 70, 111, 181, 292, 473, 765b (765 = 255 is the same color as 765b)...
Grey/Orange	113, 12, 125, 137, 262, 399, 661b (661 is the same color as 661b)...
Orange/Pink	103, 12, 115, 127, 242b (242 = 121)...
Pink/Yellow	67‡, 12, 79, 91, 170, 261b (261 = 29)...
Yellow/Cyan	19‡, 12, 31, 43*, 74, 117, 191b (191)...
Cyan/Purple	45, 19, 64, 83b (83)...
Purple/Rose	19, 7, 26, 33, 59b (59)...
Rose/White	9†, 2, 11, 13b (13)...

* Uh... you wrote 45 in your table... (fixed)
† This boundary is listed as the noble-mediant of 2 and 11, not of 9 and 2, in your table. (fixed, as well as the orange/grey and orange/pink ones I didn't notice)
‡ This happens to still be of the right color, but the one after it isn't.
§ 317 is so high it's orange without being part of the 12n stack.

edit: 61, not 51, in green/blue list (only noticed this because I happen to know 3×17 off the top of my head); added the non-"b" versions and subsets of high edos because why not

[Dave Keenan]

Ah. Thanks for that. How about a limit on abs(group-10)+steps_per_CD?

Seems maybe okay; this essentially means drawing two diagonal lines parallel to the two diagonals with 72 on them. If this limit is 12 (as the value for 72), then ten EDOs on the sharp side (42, 49, 56, 63, 70, 54, 61, 68, 59, 66, 71) and two EDOs on the flat side (64, 69) get omitted, and if it's the maximal value of the non-"b"s below 72 (71 has group = (71-13)/10 = 5.8 and CD = 13, so 4.2 + 13 = 17.2), then it has to include the orange stack all the way up to 96.

Thanks for that. If it was used, this would only be a cutoff for b-ETs. I would still include all the non-b's.

But I have a better criterion. I realised that what makes a second-best fifth worth considering for notation, is when it is not much worse than the best fifth. A convenient cutoff is where the error in the second-best fifth is no more than 4/3 of the error in the best fifth.

Putting it another way: For a b-ET to be included, the fractional part of its number of EDO steps in a pure (2:3) fifth, frac(log₂(3/2)×EDO), must be between 3/7 and 4/7.

That includes all those I listed earlier, except 37b, which is OK with me. I realise this gives an infinite number of b-ETs, but that's OK too. As you might guess, it adds on average one b-ET for every 7-EDOs. Up to 72-EDO it includes only 6b, 11b, 18b, 23b, 30b, 35b, 42b, 47b, 59b, 64b, 71b.

I wouldn't show 6b on the table as it would require a new top row and two more groups to the right, and it would be white, and the recommended notation would still be as a subset of 12-EDO anyway. And it's debatable whether it's worth including 11b and 18b, given that they are subset notations. Although I suppose it would indicate that you could, if you wish, give them a native-fifth notation of the type indicated by the colour.

[Dave Keenan]

The Fibonacci-like sequences for the boundaries (similar to the purple/rose one seen in a previous post) are, up to the point where you start needing "b"s, and including some terms before the ones of the right color (up to the point at which a higher number precedes a lower number):
...
* Uh... you wrote 45 in your table...
† This boundary is listed as the phidiant of 2 and 11, not of 9 and 2, in your table.

Thanks for checking those. I have made these (and other implied) corrections to my earlier post.

Good name, "phidiant". I have previously called it the "noble-mediant".