## Magrathean diacritics

cmloegcmluin
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### Re: Magrathean diacritics

Ash9903b4 wrote:
Thu May 28, 2020 3:22 am
The idea to make dots map to the unison sounds interesting, but I don't think it would be that much of a problem if we skipped JI entirely and just made them stand for 1 degree of either 20203 or 36269 by default.
I had that thought too, yes.

I want to push a little harder for a JI value, though. I'm not ready to throw in the towel yet.

Here are mappings for the three 0.5 commas we're considering under the three EDA's @Ash9903b4 suggests:

comma		1914eda	3436eda	5587eda
49:9765625n	 0	 7	-2
13:4675n	 3	-6	15
17:9295n	 3	 5	 0


cmloegcmluin
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### Re: Magrathean diacritics

George Secor, 21/04/2007 wrote: Dave wrote:
> I've mentioned 2151-EDA in the past. That has a
> little over 9 times the resolution of 233-EDA (call them ninas or
> tinas?)

For that, I may prefer ninas.
...
I did a quick consistency check for 2151-EDA. This is the virtual equivalent of 22704-ET, which is 27-limit consistent. So far, so good.

To see how 2151-EDA compares with other EDO's of interest, I made some consistency checks and came up with the following figures:

Consis error of odd harmonic (% of 1 deg of EDO)
EDA	EDO	Limit	3	5	7	min / max at 27-limit
----	-----	------	----	----	----	--------------------
233	2460	27	-0.8	+5.7	-9.3	-20.2 / +11.4
275	2901	17	+2.4	+8.7	13.7	-23.8 / +27.0
576	6079	29	+1.3	-0.1	+8.9	+31.3 / -15.1
809	8539	27	+0.5	+5.6	-0.4	+30.4 / -8.7
1105	11664	27	-0.3	+3.1	+1.2	+19.0 / -16.7
2151	22704	27	+1.1	-5.5	18.6	+24.4 / -18.6


The lower the 3-error, the more closely the EDO will agree with the
EDA. The lower the 5- and 7-errors, the higher the allowable
prime-exponent will be in a comma without encountering an
inconsistency. (Notice that 2901-EDO is not very good.)

One problem I have with 2151-EDA/22704-EDO is that 7 deviates by -18.6%
of 1deg22704. This indicates that a comma containing 7^3 will have a
cumulative error of -55.8% of a degree and probably won't be a
consistent number of tinas with respect to one containing a lower power
of 7.

1105-EDA/11664-EDO looks like a better choice across the board. I just
noticed that the maximum error of the 15-limit consonances (expressed
in actual cents!) in 11664-EDO (0.0200c, for 11/9) is less than that of
22704-EDO (0.0213c, for 13/7). It not only has 455n and 4375n *easily*
falling within the same (4-unit) |' boundaries, but also 65:77n and
13:125n *easily* within the same (8-unit) |'' boundaries, so it's a
fourfold division of the mina. (11664 is also divisible by 12, which
was one of the requirements on the tuning list for a measuring unit.)
I already suggested the name "quartina", so this is my nomination for
the "tina".

Will the real "Tina" please stand up?
Last edited by Dave Keenan on Thu May 28, 2020 3:24 pm, edited 4 times in total.
Reason: Extended the quote and attributed it to George

Dave Keenan
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### Re: Magrathean diacritics

Thanks, but I just I had to cram your columns some more.

                                    		occur's	3exp    limit   SoPF>3  error   any
prime           tina	as	ALAGA   ALAGA   ALAGA   ALAGA   ALAGA
tina	comma	abs3exp	limit	SoPF>3	error	mina	George	George	George  George  George  monzo						ratio	                whose

0.5 tinas
0.50	49:9765625n 13	7	64	0.00		yes	yes	yes	yes	yes	[   3 -13  10  -2                             ⟩	78125000/78121827	George
0.73	13:4675n    6	17	51	0.23		yes		yes		yes	[ -18   6   2   0   1  -1   1                 ⟩	3408075/3407872		Cmloegcmluin
0.33	17:9295n    7	17	59	0.17		yes		yes		yes	[   2  -7   1   0   1   2  -1                 ⟩	37180/37179		Ash9903b4

1 tina
1.26	121:1225n   4	11	46	0.26		yes	yes	yes	yes	yes	[  -3   4  -2  -2   2                         ⟩	9801/9800		George
0.90	65:833n     8	17	49	0.10					yes	yes	[   9  -8  -1   2   0  -1   1                 ⟩	426496/426465		Ash9903b4
0.83	7:221n      12	17	37	0.17				yes	yes	yes	[ -24  12   0  -1   0   1   1                 ⟩	117448461/117440512	Cmloegcmluin
1.07	196625n     1	13	50	0.07		yes			yes	yes	[ -16  -1   3   0   2   1                     ⟩	196625/196608		Ash9903b4
1.20	5:10241n    0	19	49	0.20		yes			yes	yes	[ -11   0  -1   2   1   0   0   1             ⟩	10241/10240		Ash9903b4

2 tinas
1.85	13:6655n    0	13	51	0.15		yes	yes	yes	yes	yes	[   9   0  -1   0  -3   1                     ⟩	6656/6655		George
1.75	35:1573n    6	13	47	0.25			yes	yes		yes	[  15  -6   1   1  -2  -1                     ⟩	1146880/1146717		Cmloegcmluin
1.82	49:145n     11	29	48	0.18				yes		yes	[  19 -11  -1   2   0   0   0   0   0  -1     ⟩	25690112/25686315	Cmloegcmluin
1.87	1705n       9	31	47	0.13				yes	yes	yes	[ -25   9   1   0   1   0   0   0   0   0   1 ⟩	33559515/33554432	Cmloegcmluin
2.11	5831n       6	17	38	0.11	2			yes	yes	yes	[   3   6   0  -3   0   0  -1                 ⟩	5832/5831		Ash9903b4

3 tinas
3.01	455n        0	13	25	0.01	21	yes	yes	yes	yes	yes	[  12  -2  -1  -1   0  -1                     ⟩	4096/4095		Sagittal

4 tinas
4.07	7:3025n     3	11	39	0.07	3	yes	yes	yes	yes	yes	[  -4  -3   2  -1   2                         ⟩	3025/3024		George
3.84	17:245n     10	17	36	0.16	2			yes		yes	[ -12  10  -1  -2   0   0   1                 ⟩	1003833/1003520		Cmloegcmluin
4.24	85:637n     7	17	49	0.24							[ -14   7  -1   2   0   1  -1                 ⟩	1393119/1392640		Cmloegcmluin
3.93	49:1045n    1	19	49	0.07	1	yes			yes	yes	[   6  -1  -1   2  -1   0   0  -1             ⟩	3136/3135		Ash9903b4
4.00	17:6875n    4	17	48	0.00					yes	yes	[  15  -4  -4   0  -1   0   1                 ⟩	557056/556875		Ash9903b4
4.17	49:1859n    3	13	51	0.17		yes				yes	[ -10   3   0  -2   1   2                     ⟩	50193/50176		Ash9903b4

                                    		occur's	3exp    limit   SoPF>3  error   any
prime           tina	as	ALAGA   ALAGA   ALAGA   ALAGA   ALAGA
tina	comma	abs3exp	limit	SoPF>3	error	mina	George	George	George  George  George  monzo						ratio	                whose

5 tinas
5.13	25:2401n    1	7	38	0.13	2	yes	yes	yes	yes	yes	[  -5  -1  -2   4                             ⟩	2401/2400		George
4.76	20449n      8	13	48	0.24							[  27  -8   0   0  -2  -2                     ⟩	134217728/134165889	Cmloegcmluin
4.86	232925n     2	11	50	0.14							[  21  -2  -2  -1  -3                         ⟩	2097152/2096325		Cmloegcmluin
4.93	625:833n    1	17	51	0.07		yes			yes	yes	[   2  -1   4  -2   0   0  -1                 ⟩	2500/2499		Ash9903b4
5.07	5:2431n     5	17	46	0.07					yes	yes	[  -1  -5  -1   0   1   1   1                 ⟩	2431/2430		Ash9903b4
5.19	11:2375n    3	19	45	0.19							[   3   3  -3   0   1   0   0  -1             ⟩	2376/2375		Ash9903b4

6 tinas
5.92	65:77n      0	13	36	0.08		yes	yes	yes	yes	yes	[   5  -3   1  -1  -1   1                     ⟩	2080/2079		Sagittal

7 tinas
7.18	143:1715n   1	13	50	0.18		yes	yes	yes	yes	yes	[   2   1  -1  -3   1   1                     ⟩	1716/1715		George
6.78	5:1729n     11	19	44	0.22				yes		yes	[  -9  11   1  -1   0  -1   0  -1             ⟩	885735/885248		Cmloegcmluin
6.82	11:119n     11	17	35	0.18	1			yes	yes	yes	[  14 -11   0   1  -1   0   1                 ⟩	1949696/1948617		Cmloegcmluin
7.10	5:2527n     12	19	50	0.10			yes	yes	yes	yes	[ -28  12  -1   1   0   0   0   2             ⟩	1342951407/1342177280	Cmloegcmluin
7.13	1729n       3	19	39	0.13				yes	yes	yes	[  -6  -3   0   1   0   1   0   1             ⟩	1729/1728		Ash9903b4
7.24	7:425n      5	17	34	0.24	2			yes		yes	[  -2   5  -2   1   0   0  -1                 ⟩	1701/1700		Ash9903b4

8 tinas
7.95	420175n     8	7	45	0.05		yes	yes	yes	yes	yes	[  -6  -8   2   5                             ⟩	420175/419904		George
7.82	5:253n      4	23	39	0.18	3	yes	yes	yes		yes	[ -12   4  -1   0   1   0   0   0   1         ⟩	20493/20480		Cmloegcmluin
7.98	13:77n      11	13	31	0.02	2		yes	yes	yes	yes	[ -20  11   0   1   1  -1                     ⟩	13640319/13631488	Cmloegcmluin
8.00	19:385n     4	19	42	0.00	1	yes		yes	yes	yes	[   2  -4   1   1   1   0   0  -1             ⟩	1540/1539		Ash9903b4
8.08	175:187n    7	17	45	0.08	1	yes		yes		yes	[  11  -7  -2  -1   1   0   1                 ⟩	382976/382725		Ash9903b4
8.10	95:169n     2	19	50	0.10		yes				yes	[  -4   2  -1   0   0   2   0  -1             ⟩	1521/1520		Ash9903b4
8.14	343:1625n   3	13	49	0.14		yes				yes	[   7  -3  -3   3   0  -1                     ⟩	43904/43875		Ash9903b4

9 tinas
8.93	539n        5	11	25	0.07		yes	yes	yes	yes	yes	[  17  -5   0  -2  -1                         ⟩	131072/130977		George
8.77	125:2401n   9	7	43	0.23			yes			yes	[ -10   9   3  -4                             ⟩	2460375/2458624		Cmloegcmluin
8.83	25:1183n    11	13	43	0.17							[  23 -11   2  -1   0  -2                     ⟩	209715200/209564901	Cmloegcmluin
8.90	19:2401n    12	19	47	0.10							[ -26  12   0   4   0   0   0  -1             ⟩	1275989841/1275068416	Cmloegcmluin
9.21	41503n      4	11	43	0.21		yes	yes			yes	[  -9  -4   0   3   2                         ⟩	41503/41472		Ash9903b4
8.97	125:833n    9	17	46	0.03					yes	yes	[ -17   9  -3   2   0   0   1                 ⟩	16395939/16384000	Ash9903b4

Last edited by cmloegcmluin on Fri May 29, 2020 2:46 am, edited 1 time in total.

Dave Keenan
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### Re: Magrathean diacritics

I figure the choice for 7 tinas is between these:
                                    		occur's	3exp    limit   SoPF>3  error   any
prime           tina	as	ALAGA   ALAGA   ALAGA   ALAGA   ALAGA
tina	comma	abs3exp	limit	SoPF>3	error	mina	George	George	George  George  George

6.82	11:119n     11	17	35	0.18	1			yes	yes	yes
7.24	7:425n      5	17	34	0.24	2			yes		yes

When I consider that the dot might end up representing intervals smaller than a half-tina, I think the 0.24 tina error disqualifies the second one, despite its lower 3-exponent and additional occurrence as two minas in Olympian. They are essential tied for both prime limit and SoPF>3.

What do you guys think?

Dave Keenan
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### Re: Magrathean diacritics

Can anyone explain why most of the commas for 1 tina and 0.5 tina, mentioned by Ash in his first two posts, are no longer on the list?
Ash9903b4 wrote:
Fri Mar 27, 2020 8:52 am
I'd like to reopen this thread to suggest a different, somewhat simpler interpretation for the single-tina diacritic, and one that actually came up when studying temperaments in 494: the 55:343-schismina (14765025303/14763950080). It's the difference between the 7:11-comma () and the 5:49 comma, or about 0.126 cents.

An even simpler but higher-limit mapping would be the 13:119-schismina (0.117 cents, 117448461/117440512), or the difference between the 17-comma () and the 7:13-comma.
Here are some relatively simple 19-limit mappings for each Magrathean diacritic:

tempered out: 1990625n ≈ 169:1925n ≈ 104975n

0.5 tina: 17:9295n

1 tina: 65:833n ≈ 5:10241n ≈ 232925n ≈ 25:14399n

Dave Keenan
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### Re: Magrathean diacritics

Here's an except from a later email from George, with quotes of my responses re EDAs, to George's email above.
George wrote: Dave wrote:
> I think 3 error needs to be below 2%, 5 error below 7%, 7 error below
> 10% and 11 error below 25%.
>
> My method of finding such things is clearly wrong.
>
> Can you do the calculations for the actual EDA's, or at least show
> the 11 error too? 275 and 2151-EDA's are definietely out.

I thought you knew about my consistency spreadsheet, but if not, then
enter the EDO# in the cyan cell to see all of the odds up to 51:

I haven't taken the trouble to adjust any of this for EDA's, but the
differences in the boundaries are very small. The apotome will change
by 7 times the 3-error, for 2460-ET <-> 233-EDA a shift of 0.02647c.
The half-apotome will shift by half that amount, or 0.01323c, and the
smaller minas by even less; e.g., in the 5C region the shift is ~0.005c
(about 1% of the width of a mina).

For EDO's that have a relative 3-error greater than that of 2460
(-0.8%), the relative shift will be somewhat more. But I don't think
that will be the case in whatever EDA we decide on for the tina.

> 576, 809 and 1105 are all very good. Is there no 31-limit consistent
> ET in this region? I keep thinking of the fact that Ben Johnston once
> composed in 31-limit.

That's a tough one. 20203-EDO (8539+11664, 1914-EDA) is 45-limit
consistent, with 0.3% 3-error, 0.9% 7-error, and 10.3% 11-error, but
its 5-error is 8.7% -- a little more than we would like. (Also, the
23-error, at 47.8%, though consistent, is rather excessive.)

I don't have any systematic way of looking for very many of these, so I
hope Gene will come up with some more suggestions.

>> One problem I have with 2151-EDA/22704-EDO is that 7 deviates by -18.6%
>> of 1deg22704. This indicates that a comma containing 7^3 will have a
>> cumulative error of -55.8% of a degree and probably won't be a
>> consistent number of tinas with respect to one containing a lower power
>> of 7.
>
> Not probably, certainly. I agree 2151-EDA is out.
>

cmloegcmluin
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### Re: Magrathean diacritics

Dave Keenan wrote:
Thu May 28, 2020 3:15 pm
Can anyone explain why most of the commas for 1 tina and 0.5 tina, mentioned by Ash in his first two posts, are no longer on the list?
Of the single 0.5 mina and the four 1 mina candidates Ash (sorry @Ash9903b4, you're just "Ash" now, hehe) shared in his first post, only one of them is truly gone. I tracked all five of them, and four of them have appeared in all five tables (two on page 2, two on page 4, one on page 6).

However, the 25:14399n turned out not to be a 1-tina but rather a 5-tina; Ash corrected it by his 2nd post.

The 25:14399n met its demise on page 4; it did not survive the journey to my table, because it did not meet Dave's stated criteria (specifically, SoPF ≤ 51).

As for the rest of the 1-tina commas Ash brought to light in his 2nd post, they were also lost due to not meeting the SoPF ≤ 51 criteria.

Perhaps we should relax that criteria for the 1-tina?

cmloegcmluin
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### Re: Magrathean diacritics

Re: the 7th, I kinda just like George's. But I feel terribly biased about it.

Dave Keenan
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### Re: Magrathean diacritics

Thanks for explaining re Ash's earlier commas.
cmloegcmluin wrote:
Thu May 28, 2020 3:54 pm
Re: the 7th, I kinda just like George's. But I feel terribly biased about it.
You were a bad boy putting all that stuff in the table about who first suggested each comma, and all that ALAGA George stuff.

It should have been double-blind placebo-controlled. That means I poke both your eyes out and say "Here, take this".

Did you hear about the woman who came home from the doctor and proudly told her husband:

"The doctor said I have hypochondria, and he gave me a broad-spectrum placebo".

Dave Keenan
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### Re: Magrathean diacritics

Do we get any useful 1 tina candidates by subtracting yellow commas for successive tinas?

I agree that any comma we might give for the dot, should map to zero tinas, using the 8539.00834edo mapping.

Thanks for checking all those mappings, Cml. (Hey, I get four letters because I'm the owner of this forum. OK? )

Shame the mappings didn't help with the decisions. Yet.