Since error weighting is a somewhat contentious topic, I'm just going to list the subgroups on which these small ETs are identical with their larger (and more popular/"generally considered decently accurate") superset-ETs. If you find those larger ETs to be sufficiently accurate on the stated subgroup, you will automatically find the smaller ones equally accurate.

8-EDO: 2.5/3.11/3.13/3.17/3 = 24-EDO

9-EDO: 2.7/3.11/3.13/3.15/3 = 27-EDO

10-EDO: 2.7.13.15 = 60-EDO

11-EDO: 2.7.9.11.15.17 = 22-EDO

13-EDO: 2.5.9.11.13 = 26-EDO

14-EDO: 2.7/5.9/5.11/5 = 28-EDO

15-EDO: 2.13/9.15/9 or 2.5.11 = 45-EDO (optimal) or 2.7.11 = 45-EDO (patent val) or 2.5.7.11 = 45-EDO (45c val; 4th-best mapping by TE weighting)

16-EDO: 2.5.7 = 48-EDO (patent val); 2.5.13 = 48-EDO (optimal val); see also 8-EDO

18-EDO: 2.5.9.21 = 36-EDO; see also 9-EDO with 17/3 added to the basis

20-EDO: 2.7.11.13.15.19 = 60-EDO (optimal mapping on that subgroup)

21-EDO: 2.11/5.13/5.17/5 = 63-EDO

23-EDO: 2.5/3.7/3.11/3 or 2.9.13.15.17.21 = 46-EDO

25-EDO: 2.5.7.9.17 = 50-EDO

## OT: Subgroups & Small EDOs

- Dave Keenan
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### Re: OT: Subgroups & Small EDOs

Can you tell me what the maximum absolute error is for each of those larger ETs on the stated subgroup, assuming untempered octaves, so I can decide whether I find them sufficiently accurate? It would be interesting to see them as decimal fractions of the step of the larger ET, as well as in cents.cryptic.ruse wrote:If you find those larger ETs to be sufficiently accurate on the stated subgroup, you will automatically find the smaller ones equally accurate.

- cryptic.ruse
**Posts:**23**Joined:**Tue Sep 22, 2015 4:38 am

### Re: OT: Subgroups & Small EDOs

That would be quite a bit of calculation on my part, since there are no ready-made tools for calculating unweighted error (unfortunately). Suffice to say, the damage will not be any higher than the larger ET's complete prime limit at the highest prime factor in the subgroup, and in many cases will almost certainly be lower, since the complete prime limit may add primes not present in the subgroup, which are also not well tuned in the larger ET.Can you tell me what the maximum absolute error is for each of those larger ETs on the stated subgroup, assuming untempered octaves, so I can decide whether I find them sufficiently accurate? It would be interesting to see them as decimal fractions of the step of the larger ET, as well as in cents.

I'll see if I can cobble together a spreadsheet to calculate it, though. It would be a worthwhile exercise, since my feelings about weighted error match yours.

- cryptic.ruse
**Posts:**23**Joined:**Tue Sep 22, 2015 4:38 am

### Re: OT: Subgroups & Small EDOs

Okay, here is my unweighted error calculation. What I did was to calculate the errors of each basis interval, then take the minimum and maximum. If the minimum is negative and the max is positive, adding abs(min) to maximum gives the largest error on the coprime grid (aka tonality diamond) formed by the basis. If the minimum and maximum are both positive, then the max error on the basis intervals is also the max error on the coprime grid. If the max error is zero (because all the errors are negative), then abs(min) is the max error on the coprime grid.

Because we're dealing with coprime grids and not prime limits or odd limits, I'm evaluating 12edo (the "benchmark") on the 2.3.5.9.15 basis, rather than 2.3.5, since presumably people using 12edo care about the damage to tetrads and pentads of simple compounds of 2, 3, and 5.

One thing I note is that this approach does explicitly require all intervals of interest be included, unlike the weighted prime-limit approach, because compound intervals (like 9, 15, 21, etc.) multiply the errors of the their factors. So if 3 is the highest-damage basis interval, adding or omitting 9 will change the result.

12edo: 2.3.5.9.15 = 17.596¢ / 0.176 steps, or 35.084¢ / 0.351 steps if 7 is included

For additional comparison:

17edo 2.3.7.9.11.13 = 19.409¢ / 0.275 steps;

19edo 2.3.5.7.9 = 21.457¢ / 0.340 steps, adding 11 to the basis raises to 38.561¢ / 0.611 steps;

31edo 2.3.5.7.9.11 = 11.145¢ / 0.288 steps, adding 13 to the basis raises to 21.447¢ / 0.553 steps;

Keeping these comparisons in mind should help give you some context for the following:

8-EDO: 2.5/3.11/3.13/3.17/3 = 18.642¢ / 0.124 steps (drops to 15.641¢ / 0.104 steps if 17/3 is omitted)

9-EDO: 2.7/3.11/3.13/3.15/3 = 22.543¢ / 0.169 steps (drops to 17.508¢ / 0.131 steps if 13/3 is omitted)

10-EDO: 2.7.13.15 = 8.826¢ / 0.074 steps

11-EDO: 2.7.9.11.15.17 = 20.135¢ / 0.185 steps (drops to 14.272¢ / 0.131 steps if 11 is omitted; either way, identical to 22edo)

13-EDO: 2.5.9.11.13 = 21.823¢ / 0.236 steps (drops to 15.176 / 0.164 steps if 5 & 9 omitted; either way, identical to 26edo)

14-EDO: 2.7/5.9/5.11/5 = 17.488¢ / 0.204 steps (drops to 11.063¢ / 0.129 steps on 2.9/7.11/7)

15-EDO: 2.13/9.15/9 = 7.741¢ / 0.097 steps, or 2.5.7.11 = 22.512¢ / 0.281 steps

16-EDO: 2.11/3.15/3.17/3.21/3 = 17.488¢ / 0.233 steps (drops to 9.175¢ / 0.122 steps if 15/3 is omitted)

18-EDO: 2.5.9.21 = 17.801¢ / 0.267 steps, or 2.7/3.13/3.15/3.17/3 =18.926¢ / 0.284¢

20-EDO: 2.7.11.13.15.19 = 13.805¢ / 0.23 steps

21-EDO: 2.11/5.13/5.17/5 = 10.780¢ / 0.189 steps

23-EDO: 2.5/3.7/3.11/3 = 8.599¢ / 0.165 steps, or 2.9.13.15.17.21 = 13.129¢ / 0.252 steps

25-EDO: 2.5.7.9.17 = 11.910¢ / 0.248 steps

So, while my initial claim that these ETs are "as good as 12edo in the 5-limit" may only hold if we use TE weighted prime-limit error, it's obvious here that they are nearly all as good as more popular ETs like 17, 19, 22, and 31 when compared against those ETs' popularly-exploited resources, and none are as bad as 19edo is in the 11-odd-limit.

Because we're dealing with coprime grids and not prime limits or odd limits, I'm evaluating 12edo (the "benchmark") on the 2.3.5.9.15 basis, rather than 2.3.5, since presumably people using 12edo care about the damage to tetrads and pentads of simple compounds of 2, 3, and 5.

One thing I note is that this approach does explicitly require all intervals of interest be included, unlike the weighted prime-limit approach, because compound intervals (like 9, 15, 21, etc.) multiply the errors of the their factors. So if 3 is the highest-damage basis interval, adding or omitting 9 will change the result.

12edo: 2.3.5.9.15 = 17.596¢ / 0.176 steps, or 35.084¢ / 0.351 steps if 7 is included

For additional comparison:

17edo 2.3.7.9.11.13 = 19.409¢ / 0.275 steps;

19edo 2.3.5.7.9 = 21.457¢ / 0.340 steps, adding 11 to the basis raises to 38.561¢ / 0.611 steps;

31edo 2.3.5.7.9.11 = 11.145¢ / 0.288 steps, adding 13 to the basis raises to 21.447¢ / 0.553 steps;

Keeping these comparisons in mind should help give you some context for the following:

8-EDO: 2.5/3.11/3.13/3.17/3 = 18.642¢ / 0.124 steps (drops to 15.641¢ / 0.104 steps if 17/3 is omitted)

9-EDO: 2.7/3.11/3.13/3.15/3 = 22.543¢ / 0.169 steps (drops to 17.508¢ / 0.131 steps if 13/3 is omitted)

10-EDO: 2.7.13.15 = 8.826¢ / 0.074 steps

11-EDO: 2.7.9.11.15.17 = 20.135¢ / 0.185 steps (drops to 14.272¢ / 0.131 steps if 11 is omitted; either way, identical to 22edo)

13-EDO: 2.5.9.11.13 = 21.823¢ / 0.236 steps (drops to 15.176 / 0.164 steps if 5 & 9 omitted; either way, identical to 26edo)

14-EDO: 2.7/5.9/5.11/5 = 17.488¢ / 0.204 steps (drops to 11.063¢ / 0.129 steps on 2.9/7.11/7)

15-EDO: 2.13/9.15/9 = 7.741¢ / 0.097 steps, or 2.5.7.11 = 22.512¢ / 0.281 steps

16-EDO: 2.11/3.15/3.17/3.21/3 = 17.488¢ / 0.233 steps (drops to 9.175¢ / 0.122 steps if 15/3 is omitted)

18-EDO: 2.5.9.21 = 17.801¢ / 0.267 steps, or 2.7/3.13/3.15/3.17/3 =18.926¢ / 0.284¢

20-EDO: 2.7.11.13.15.19 = 13.805¢ / 0.23 steps

21-EDO: 2.11/5.13/5.17/5 = 10.780¢ / 0.189 steps

23-EDO: 2.5/3.7/3.11/3 = 8.599¢ / 0.165 steps, or 2.9.13.15.17.21 = 13.129¢ / 0.252 steps

25-EDO: 2.5.7.9.17 = 11.910¢ / 0.248 steps

So, while my initial claim that these ETs are "as good as 12edo in the 5-limit" may only hold if we use TE weighted prime-limit error, it's obvious here that they are nearly all as good as more popular ETs like 17, 19, 22, and 31 when compared against those ETs' popularly-exploited resources, and none are as bad as 19edo is in the 11-odd-limit.

- Dave Keenan
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**Posts:**805**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
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### Re: OT: Subgroups & Small EDOs

Thanks for that. Your method of calculation is fine. I must admit I'm surprised by how low some of these are.

However, I don't consider prime 7 to be usable in 12-edo, 11 in 19-edo or 13 in 31-edo, so those errors hold no relevance for me.

Your original claim was:

I replied:

What I'm looking for there is an error greater than 0.5 steps of the superset EDO, In which case I will say that I do not find the larger EDO sufficiently accurate on the stated subgroup.

But if this has any relevance at all to our discussion of possible fractional-3-limit notations for these EDOs, surely it argues

And I have already accepted your argument that these bad-3 EDOs should be notated with apotome-fraction and limma-fraction accidentals. So let's get back to the details of that. I'm sorry to have caused you to digress.

However, I don't consider prime 7 to be usable in 12-edo, 11 in 19-edo or 13 in 31-edo, so those errors hold no relevance for me.

Your original claim was:

Italics added.cryptic.ruse wrote:Since error weighting is a somewhat contentious topic, I'm just going to list the subgroups on which these small ETs are identical with their larger (and more popular/"generally considered decently accurate") superset-ETs.If you find those larger ETs to be sufficiently accurate on the stated subgroup, you will automatically find the smaller ones equally accurate.

I replied:

By "larger ET" here, I was referring to your "generally considered decently accurate superset-ETs". So for each of the EDOs you have listed, you need to tell me what superset EDO you are referring to -- I would expect it to be something like: the smallest multiple whose best fifth has an error of 10 cents or less. e.g.for 8-edo it would be 24-edo, and then express that subgroup error as a fraction of the step size of 24-edo, i.e. triple what you have for 8-edo and obtain 0.37 steps, which I do consider sufficiently accurate, at least for notational purposes.Can you tell me what the maximum absolute error is for each of those larger ETs on the stated subgroup, assuming untempered octaves, so I can decide whether I find them sufficiently accurate? It would be interesting to see them as decimal fractions of the step of the larger ET, as well as in cents.

What I'm looking for there is an error greater than 0.5 steps of the superset EDO, In which case I will say that I do not find the larger EDO sufficiently accurate on the stated subgroup.

But if this has any relevance at all to our discussion of possible fractional-3-limit notations for these EDOs, surely it argues

*against*them. It argues for some kind of 3-free JI-based notation for them, based specifically on that subgroup. I just don't know how to do a 3-free notation on a standard staff, with nominals that are assumed to be in a chain of fifths, no matter how bad those fifths may be.And I have already accepted your argument that these bad-3 EDOs should be notated with apotome-fraction and limma-fraction accidentals. So let's get back to the details of that. I'm sorry to have caused you to digress.

- cryptic.ruse
**Posts:**23**Joined:**Tue Sep 22, 2015 4:38 am

### Re: OT: Subgroups & Small EDOs

So, the only reason I listed the superset ETs in the first place was to try to laze my way out of calculating the errors, since I presumed you had explored the larger ETs enough to have come to your own determination of whether they were harmonically-accurate enough for your tastes. Now that I have gone ahead and calculated the errors, you may disregard the superset ETs entirely.By "larger ET" here, I was referring to your "generally considered decently accurate superset-ETs". So for each of the EDOs you have listed, you need to tell me what superset EDO you are referring to -- I would expect it to be something like: the smallest multiple whose best fifth has an error of 10 cents or less. e.g.for 8-edo it would be 24-edo, and then express that subgroup error as a fraction of the step size of 24-edo, i.e. triple what you have for 8-edo and obtain 0.37 steps, which I do consider sufficiently accurate, at least for notational purposes.

Not so! I brought them up primarily to illustrate the point that "bad" EDOs are not really "bad", just misunderstood; therefore those of us interested in the "good" EDOs are not necessarily going to ignore the "bad" ones. Much of your argument in favor of maintaining the existing Standard Sagittal centers around the idea that, because the "bad" EDOs are "bad", they're not very popular, and as such, it's not a big deal that the "bad" EDOs require unusual/non-standard approaches. When taken together with the "good" EDOs, the "bad" EDO notations increase the total symbol count/system complexity of Standard Sagittal. If I can persuade you that these EDOs offer enough useful JI approximations to get them out of the "bad" pile, I would hope you agree that we should take seriously the amount of complexity they add to Standard Sagittal. While they are often bad in the 3-limit, it's still the case that a 3-limit notation functions for them perfectly well and still reduces the total symbol count over the range of EDOs compared to Standard Sagittal.But if this has any relevance at all to our discussion of possible fractional-3-limit notations for these EDOs, surely it argues against them.

- Dave Keenan
- Site Admin
**Posts:**805**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
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### Re: OT: Subgroups & Small EDOs

Ah. OK. But that was never going to work because I am only familiar with thecryptic.ruse wrote:So, the only reason I listed the superset ETs in the first place was to try to laze my way out of calculating the errors, since I presumed you had explored the larger ETs enough to have come to your own determination of whether they were harmonically-accurate enough for your tastes. Now that I have gone ahead and calculated the errors, you may disregard the superset ETs entirely.

*best*mappings in those superset ETs, and many of those subset mappings do not correspond to the best mapping in the superset. And I'm certainly not familiar with their errors in such bizarre beasts as "2.5/3.11/3.13/3.17/3".

OK. You have successfully made that point. But we agree they are all bad in their accuracy of prime 3 (which I abbreviated to "bad-3" above) since none of your mappings included it, and we agree we want notations whose nominals and sharps and flats are based on prime 3.cryptic.ruse wrote:Not so! I brought them up primarily to illustrate the point that "bad" EDOs are not really "bad", just misunderstood; ...Dave Keenan wrote:But if this has any relevance at all to our discussion of possible fractional-3-limit notations for these EDOs, surely it argues against them.

You have convinced me that I was wrong in that regard....therefore those of us interested in the "good" EDOs are not necessarily going to ignore the "bad" ones. Much of your argument in favor of maintaining the existing Standard Sagittal centers around the idea that, because the "bad" EDOs are "bad", they're not very popular, and as such, it's not a big deal that the "bad" EDOs require unusual/non-standard approaches.

I see. But I only needed to be convinced (a) that they were likely to become more popular (not any particular reason why), and (b) that their existing notations were way too complex, and most importantly of all (c) that there was an alternative way of notating them that could be made compatible with the rest of Sagittal.When taken together with the "good" EDOs, the "bad" EDO notations increase the total symbol count/system complexity of Standard Sagittal. If I can persuade you that these EDOs offer enough useful JI approximations to get them out of the "bad" pile, I would hope you agree that we should take seriously the amount of complexity they add to Standard Sagittal.

We are in violent agreement. :D I really hope George Secor can find time to pitch in here soon. I have no idea what he thinks of all this. I've prompted him by email but he's very busy at the moment.While they are often bad in the 3-limit, it's still the case that a 3-limit notation functions for them perfectly well and still reduces the total symbol count over the range of EDOs compared to Standard Sagittal.