## consistent Sagittal 37-Limit

- Dave Keenan
- Site Admin
**Posts:**1814**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: consistent Sagittal 37-Limit

Re: Comma lists

From George Secor 19/04/2007

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> Hi George,

Hi Dave. I've been very busy lately, so have had to work on this off

and on -- but I *have* been diligently working on it.

> At 05:56 AM 13/04/2007, you wrote:

>> --- Dave Keenan <d.keenan@bigpond.net.au> wrote:

>>

>>> .~|('

>>> ')~|..

>>> seem to be in the wrong order.

>>

>> Yep, it would seem so. However, I again determined both a SoCA

(using

>> 455n or 65:77n) and an alternate SoCA (using 4375n or 13:125n) for

the

>> 3 possible symbols in the 27th mina, which gives the following

figures:

>>

>> )~|'' as 12.897c or 12.883c

>> .~|(' as 13.199c or 13.172c

>> ')~|.. as 13.186c or 13.200c

>>

>> Averaging out the figures for each symbol, .~|(' comes out lower

than

>> ')~|.., but I don't remember exactly why I split this mina into 3

>> parts. Since ')~|.. is more complicated than the other two symbols,

>> and since it's so close in size to .~|(', I now see that it would be

>> better not to use it. (Thanks for spotting that.)

>

> Trying the alternative values of the right accents is something I

> haven't been doing, but I agree it is a good thing to do and I plan

> to modify my spreadsheet to accommodate it.

Okay.

>> So here's how I would now assign the symbols.

>>

>> The five least complex commas for this mina are (in order of size):

>>

>> Cents Name Complex Pop. Rank

>> ------ ------- ------- ---------

>> 12.943 19:121C 49.462 962

>> 13.066 11:133C 44.456 817

>> 13.074 1715C 46.916 110

>> 13.189 35:247C 44.117 924

>> 13.269 11:23C 34.857 129

>

> I think we can ignore those with popularity rank above about 500.

Remember that you said this, because I'll bring it to your attention

below.

>> Taking both complexity and popularity rank into account, 11:23C gets

>> priority for a symbol assignment, with 1715C in 2nd place. They

both

>> compete for .|(', but 11:23C gets it, since it's closer. The choice

>> for )~|'' is clearly 19:121C. With these two symbol definitions,

the

>> symbol boundary would be at 13.106c, which puts 1715C in the )~|''

>> range, giving the two most important commas in the 27th mina

separate

>> symbols.

>>

>> Now I'll look at your spreadsheet selections and comment.

>>

>> You assigned ')~|.. to 1715C, so if we eliminate that symbol, we can

>> delete that assignment.

>

> If we eliminate that symbol then I'd end up assigning it as a

> secondary comma for .~|(' 11:23C. I don't think that considering

> alternate right flags would change that.

That's not what I had in mind. If at all possible, I would want

different symbols for 1715C and 11:23C, the two commas most worthy of

notating. I don't think this is an unreasonable expectation,

considering that they're ~0.195c apart.

> That's because I haven't accepted your practice of assigning commas

> to symbols with the same mina irrespective of whether they are closer

> to that symbol's SoCA.

Irrespective of whether they are closer than what? -- than another,

less popular / more complex comma? If so, then all symbols would have

to be SoCA, otherwise there would always be another less popular / more

complex comma that's closer to SoCA.

> I don't feel that minas are relevant to superolympian at all.

They're relevant if members of the tuning list think that they are.

There's been a lot of discussion on the main list recently in looking

for octave divisions that are both high-limit consistent and divisible

by 12, for the purpose of defining useful logarithmic units of pitch

measurement. See messages 71110, 71115, 71160, 71166, 71249, and

71256. (No, I didn't start that thread!)

I'm working out both olympian (with about 12 minas up to 1/2-apotome

split in two parts) and superolympian (with most minas split in two,

and many in three parts) notations, both of which observe strict

233-EDA boundaries in order that: 1) they be compatible with one

another, 2) the 27-limit consistency offered by 2460-ET be guaranteed,

and 3) symbols may be converted directly to minas, which provide a

quick and convenient unit of measure for integer calculations.

> Or at

> least not _that_ relevant that they could allow a symbol to shift so

> far from its SoCA.

I agree that these shifts should not be large.

What is it in this example that you think gets too far from its SocA?

Here's what I'm proposing, using two symbols for the 27th mina:

Pop.

Cents Name Complex Rank Symbol Dev.1 Dev.2 Comments

------ ------- ------- ---- ------ ------ ------ ---------------

12.897 5:847C 47.654 814 .~|( +0.120 SoCA for )~|''

12.930 27-mina lower boundary )~|'' +0.033 +0.047 ===============

12.943 19:121C 49.462 962 )~|'' +0.046 +0.060 Primary role

13.066 11:133C 44.456 817 )~|'' +0.169 +0.183

13.074 1715C 46.916 110 )~|'' +0.177 +0.191 7-limit default

13.106 symbol boundary ====== )~|'' +0.209 +0.223 ===============

13.106 symbol boundary ====== .~|(' -0.094 -0.067 ===============

13.166 19:23C 56.739 448 .~|(' -0.033 -0.006

13.189 35:247C 44.117 924 .~|(' -0.010 +0.016

13.269 11:23C 34.857 129 .~|(' +0.069 +0.096 Primary role

13.399 71825C 91.989 284 .~|(' +0.200 +0.227 5-limit default

13.418 27-mina upper boundary .~|(' +0.218 +0.245 ===============

The primary roles for the two 27-mina symbols are <0.07c from the

principal SoCA, and no symbol boundary is farther than 1/2 mina

(0.244c) from a symbol's principal SoCA. I've disallowed 5:847C for

)~|'', since it's not atomically correct, substituting 19:121C, which

is of comparable popularity and complexity, and only 0.046c from SoCA.

>> You assigned .~|(' to 78125C -- whoa, that's 5^7, which is much more

>> complicated and less popular than 11:23C. (However, if you cut the

>> prime limit below 19, then this one would get it.)

>

> 78125C was on Gene's list so I thought it should at least be listed

> as a secondary (or tertiary) comma (we need to list these to reduce

> possibilities for Gene to complain). You would have noticed I did not

> put it in the column where all the primary commas are.

Yes, I see now.

>> You assigned )~|'' to 5:847C (SoCA); however, this is 26 minas,

which

>> is not atomically correct. Since there are at least 3 or 4 commas

less

>> complex and/or more popular than 5:847C in the 26th mina, we

shouldn't

>> be obligated to give it a symbol, which would then allow )~|'' to be

>> used in the 27th mina.

>

> 5:847C is another Geneism.

That one has enough badness and low enough popularity that I'd be

content to make it a secondary role for .~|( (primary role 5:17C),

particularly since it gets that symbol when you cut the prime limit to

11.

> Why not use all available 27 mina symbols even though some are very

> close to each other, since we have a lot of commas in this region.

There are only 3 possible symbols, including )~|'':

)~|'' as 12.897c or 12.883c

.~|(' as 13.199c or 13.172c

')~|.. as 13.186c or 13.200c

The overlapping SoCA's for .~|(' and ')~|.. make it very messy (or at

the very least ambiguous as to which is larger), in addition to the

fact that ')~|.. is ugly, and I'd prefer to omit it, providing that no

deserving comma is slighted. Remember what you said above, about

ignoring those with popularity rank above about 500 -- I would say the

same thing about rank approaching 500 if the complexity is also above

45 or 50. Here are the four 23-limit 27-mina commas that rank in the

top 500, in order of size:

Cents Name Complex Pop. Rank

------ ------- ------- ---------

13.074 1715C 46.916 110

13.166 19:23C 56.739 448

13.269 11:23C 34.857 129

13.399 71825C 91.989 284

11:23C and 71825C are both larger than the SoCA of all 3 possible

symbols, so it's impossible not to assign them the same symbol, either

.|(' or ')~|.. (preferably the former, since it's simpler) -- 11:23C

would take precedence over 71825C for a symbol assignment (on the basis

of all 3 considerations: complexity, pop. rank, and size).

19:23C could get ')~|.., but its low popularity rank and high

complexity are IMO not enough to insist that we provide a separate

symbol.

That leaves only 1715C, which shouldn't get the same symbol as 11:23C

and 71825C. If we assign .~|(' to 11:23C and )~|'' to 19:121C, the

boundary would be set at 13.106c, allowing )~|'' to default to 1715C.

>> I'll be looking at the rest of your spreadsheet over the next

several

>> days to see what else needs attention.

>

> Probably a lot. You could just flag what we agree on so far, e.g.

> with a green background, and send it back, and I'll do the alternate

> right accent thing.

Okay, but first I want to continue reviewing whatever differences I

see. If I agree with something you have, then I'm changing what I

have, which will save us some time.

--George

From George Secor 19/04/2007

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> Hi George,

Hi Dave. I've been very busy lately, so have had to work on this off

and on -- but I *have* been diligently working on it.

> At 05:56 AM 13/04/2007, you wrote:

>> --- Dave Keenan <d.keenan@bigpond.net.au> wrote:

>>

>>> .~|('

>>> ')~|..

>>> seem to be in the wrong order.

>>

>> Yep, it would seem so. However, I again determined both a SoCA

(using

>> 455n or 65:77n) and an alternate SoCA (using 4375n or 13:125n) for

the

>> 3 possible symbols in the 27th mina, which gives the following

figures:

>>

>> )~|'' as 12.897c or 12.883c

>> .~|(' as 13.199c or 13.172c

>> ')~|.. as 13.186c or 13.200c

>>

>> Averaging out the figures for each symbol, .~|(' comes out lower

than

>> ')~|.., but I don't remember exactly why I split this mina into 3

>> parts. Since ')~|.. is more complicated than the other two symbols,

>> and since it's so close in size to .~|(', I now see that it would be

>> better not to use it. (Thanks for spotting that.)

>

> Trying the alternative values of the right accents is something I

> haven't been doing, but I agree it is a good thing to do and I plan

> to modify my spreadsheet to accommodate it.

Okay.

>> So here's how I would now assign the symbols.

>>

>> The five least complex commas for this mina are (in order of size):

>>

>> Cents Name Complex Pop. Rank

>> ------ ------- ------- ---------

>> 12.943 19:121C 49.462 962

>> 13.066 11:133C 44.456 817

>> 13.074 1715C 46.916 110

>> 13.189 35:247C 44.117 924

>> 13.269 11:23C 34.857 129

>

> I think we can ignore those with popularity rank above about 500.

Remember that you said this, because I'll bring it to your attention

below.

>> Taking both complexity and popularity rank into account, 11:23C gets

>> priority for a symbol assignment, with 1715C in 2nd place. They

both

>> compete for .|(', but 11:23C gets it, since it's closer. The choice

>> for )~|'' is clearly 19:121C. With these two symbol definitions,

the

>> symbol boundary would be at 13.106c, which puts 1715C in the )~|''

>> range, giving the two most important commas in the 27th mina

separate

>> symbols.

>>

>> Now I'll look at your spreadsheet selections and comment.

>>

>> You assigned ')~|.. to 1715C, so if we eliminate that symbol, we can

>> delete that assignment.

>

> If we eliminate that symbol then I'd end up assigning it as a

> secondary comma for .~|(' 11:23C. I don't think that considering

> alternate right flags would change that.

That's not what I had in mind. If at all possible, I would want

different symbols for 1715C and 11:23C, the two commas most worthy of

notating. I don't think this is an unreasonable expectation,

considering that they're ~0.195c apart.

> That's because I haven't accepted your practice of assigning commas

> to symbols with the same mina irrespective of whether they are closer

> to that symbol's SoCA.

Irrespective of whether they are closer than what? -- than another,

less popular / more complex comma? If so, then all symbols would have

to be SoCA, otherwise there would always be another less popular / more

complex comma that's closer to SoCA.

> I don't feel that minas are relevant to superolympian at all.

They're relevant if members of the tuning list think that they are.

There's been a lot of discussion on the main list recently in looking

for octave divisions that are both high-limit consistent and divisible

by 12, for the purpose of defining useful logarithmic units of pitch

measurement. See messages 71110, 71115, 71160, 71166, 71249, and

71256. (No, I didn't start that thread!)

I'm working out both olympian (with about 12 minas up to 1/2-apotome

split in two parts) and superolympian (with most minas split in two,

and many in three parts) notations, both of which observe strict

233-EDA boundaries in order that: 1) they be compatible with one

another, 2) the 27-limit consistency offered by 2460-ET be guaranteed,

and 3) symbols may be converted directly to minas, which provide a

quick and convenient unit of measure for integer calculations.

> Or at

> least not _that_ relevant that they could allow a symbol to shift so

> far from its SoCA.

I agree that these shifts should not be large.

What is it in this example that you think gets too far from its SocA?

Here's what I'm proposing, using two symbols for the 27th mina:

Pop.

Cents Name Complex Rank Symbol Dev.1 Dev.2 Comments

------ ------- ------- ---- ------ ------ ------ ---------------

12.897 5:847C 47.654 814 .~|( +0.120 SoCA for )~|''

12.930 27-mina lower boundary )~|'' +0.033 +0.047 ===============

12.943 19:121C 49.462 962 )~|'' +0.046 +0.060 Primary role

13.066 11:133C 44.456 817 )~|'' +0.169 +0.183

13.074 1715C 46.916 110 )~|'' +0.177 +0.191 7-limit default

13.106 symbol boundary ====== )~|'' +0.209 +0.223 ===============

13.106 symbol boundary ====== .~|(' -0.094 -0.067 ===============

13.166 19:23C 56.739 448 .~|(' -0.033 -0.006

13.189 35:247C 44.117 924 .~|(' -0.010 +0.016

13.269 11:23C 34.857 129 .~|(' +0.069 +0.096 Primary role

13.399 71825C 91.989 284 .~|(' +0.200 +0.227 5-limit default

13.418 27-mina upper boundary .~|(' +0.218 +0.245 ===============

The primary roles for the two 27-mina symbols are <0.07c from the

principal SoCA, and no symbol boundary is farther than 1/2 mina

(0.244c) from a symbol's principal SoCA. I've disallowed 5:847C for

)~|'', since it's not atomically correct, substituting 19:121C, which

is of comparable popularity and complexity, and only 0.046c from SoCA.

>> You assigned .~|(' to 78125C -- whoa, that's 5^7, which is much more

>> complicated and less popular than 11:23C. (However, if you cut the

>> prime limit below 19, then this one would get it.)

>

> 78125C was on Gene's list so I thought it should at least be listed

> as a secondary (or tertiary) comma (we need to list these to reduce

> possibilities for Gene to complain). You would have noticed I did not

> put it in the column where all the primary commas are.

Yes, I see now.

>> You assigned )~|'' to 5:847C (SoCA); however, this is 26 minas,

which

>> is not atomically correct. Since there are at least 3 or 4 commas

less

>> complex and/or more popular than 5:847C in the 26th mina, we

shouldn't

>> be obligated to give it a symbol, which would then allow )~|'' to be

>> used in the 27th mina.

>

> 5:847C is another Geneism.

That one has enough badness and low enough popularity that I'd be

content to make it a secondary role for .~|( (primary role 5:17C),

particularly since it gets that symbol when you cut the prime limit to

11.

> Why not use all available 27 mina symbols even though some are very

> close to each other, since we have a lot of commas in this region.

There are only 3 possible symbols, including )~|'':

)~|'' as 12.897c or 12.883c

.~|(' as 13.199c or 13.172c

')~|.. as 13.186c or 13.200c

The overlapping SoCA's for .~|(' and ')~|.. make it very messy (or at

the very least ambiguous as to which is larger), in addition to the

fact that ')~|.. is ugly, and I'd prefer to omit it, providing that no

deserving comma is slighted. Remember what you said above, about

ignoring those with popularity rank above about 500 -- I would say the

same thing about rank approaching 500 if the complexity is also above

45 or 50. Here are the four 23-limit 27-mina commas that rank in the

top 500, in order of size:

Cents Name Complex Pop. Rank

------ ------- ------- ---------

13.074 1715C 46.916 110

13.166 19:23C 56.739 448

13.269 11:23C 34.857 129

13.399 71825C 91.989 284

11:23C and 71825C are both larger than the SoCA of all 3 possible

symbols, so it's impossible not to assign them the same symbol, either

.|(' or ')~|.. (preferably the former, since it's simpler) -- 11:23C

would take precedence over 71825C for a symbol assignment (on the basis

of all 3 considerations: complexity, pop. rank, and size).

19:23C could get ')~|.., but its low popularity rank and high

complexity are IMO not enough to insist that we provide a separate

symbol.

That leaves only 1715C, which shouldn't get the same symbol as 11:23C

and 71825C. If we assign .~|(' to 11:23C and )~|'' to 19:121C, the

boundary would be set at 13.106c, allowing )~|'' to default to 1715C.

>> I'll be looking at the rest of your spreadsheet over the next

several

>> days to see what else needs attention.

>

> Probably a lot. You could just flag what we agree on so far, e.g.

> with a green background, and send it back, and I'll do the alternate

> right accent thing.

Okay, but first I want to continue reviewing whatever differences I

see. If I agree with something you have, then I'm changing what I

have, which will save us some time.

--George

- Dave Keenan
- Site Admin
**Posts:**1814**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: consistent Sagittal 37-Limit

Re: Comma lists

From George Secor 21/04/2007

Hi Dave,

I'm answering 2 out of 3 messages today.

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> At 07:20 AM 19/04/2007, you wrote:

>> --- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> ...

>>> Trying the alternative values of the right accents is something I

>>> haven't been doing, but I agree it is a good thing to do and I

plan

>>> to modify my spreadsheet to accommodate it.

>>

>> Okay.

>

> Still not done. Sorry.

No problem. Take your time.

>>> I think we can ignore those with popularity rank above about 500.

>>

>> Remember that you said this, because I'll bring it to your attention

>> below.

>

> A foolish consistency is the hobgoblin of little minds.

Here's something to which you can pay little mind:

Q: What is mind?

A: No matter.

Q: What is matter?

A: Never mind.

> ...

>>> That's because I haven't accepted your practice of assigning

commas

>>> to symbols with the same mina irrespective of whether they are

closer

>>> to that symbol's SoCA.

>>

>> Irrespective of whether they are closer than what? -- than another,

>> less popular / more complex comma?

>

> No. Closer than the other adjacent symbol's SoCA.

But suppose that we've agreed to use only one of two symbols in a

particular instance for superolympian, and the symbol with SoCA closer

to the ratio to be notated is more complicated than the farther one?

What then?

1) Must we use the more complicated symbol and discard the simpler one?

2) Or must we define the simpler symbol with a less popular / more

complex (LPMC) ratio and then have the symbol default to the more

popular / less complex (MPLC) ratio at the current level of precision

(even if that's superolympian)?

3) Or should we disregard the more complicated symbol and simply assign

the simpler symbol to the MPLC ratio. If someone later decides to take

Sagittal up to a higher level of precision that requires both symbols,

then the superolympian symbol assignment becomes the superolympian

default, and the two symbols may then be redefined according to commas

that are closer to their SoCA's.

> ...

>>> I don't feel that minas are relevant to superolympian at all.

>>

>> They're relevant if members of the tuning list think that they are.

>> There's been a lot of discussion on the main list recently in

looking

>> for octave divisions that are both high-limit consistent and

divisible

>> by 12, for the purpose of defining useful logarithmic units of pitch

>> measurement. See messages 71110, 71115, 71160, 71166, 71249, and

>> 71256. (No, I didn't start that thread!)

>

> I don't see how that makes minas relevant to a level of Sagittal

> specifically intended to go beyond them. Olympian is the domain of

> minas.

We have a few issues to deal with in this situation, where there aren't

anywhere near enough symbols to notate all of the degrees in whatever

finer unit of measure you intend to replace the mina.

From what you're saying, I'm concluding that if a comma doesn't come

within a unit or so (tina, or whatever) of the SoCA of any symbol, then

it can't be assigned to a symbol, and we must look for something LPMC.

What symbol does the MPLC symbol then default to, or (in other words),

will *all* of the boundaries between symbols be set halfway between

their defining commas (or between their SoCA's, as I see you're doing

it), or will the tina (or whatever unit) boundaries have any relevance?

If EDA-unit boundaries have no relevance, or if you're allowing that

symbols may be assigned to adjacent tinas, then in superolympian we can

no longer guarantee the 27-limit consistency we already have with

minas.

> A lot of other numbers were tossed around there besides 2460. In any

> case there is already a slight discrepancy between the 2460_EDO

> degree and the mina we are using, that is bound to cause problems for

> commas near the boundaries anyway.

The consistency of 2460-EDO guarantees that these problems will be

encountered only with very complicated commas. The whole point of the

discussion was to find divisions of the octave where this would be the

case.

> So I don't think we should allow

> ourselves to be constrained by what may well be a momentary fad.

This subject has come up before on the list (on several occasions), so

I don't thinks it's a passing interest.

> Did you mention on the list that the pronunciation is "meena"?

No, I was more concerned with explaining its excellent properties.

> If, after defining athenian as 21-EDA, we had insisted that higher

> precision notations must have boundaries that correspond exatly to

> those, we would not now be using 233-EDA for olympian. We must allow

> the possibility that someone in future may define a new accent or

> something that will take Sagittal to new extremes of ridiculousness

> of resolution and should not do anything that would make that

> difficult.

Yes, that's a worthy goal.

> One way to make sure of that would be to invent it ourselves (but no

> need to tell anyone).

Tee hee!

> 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 (see below).

> and probably distinguishes the standard and alternative

> values for the right accents (although I haven't checked).

>

> Why not consider superolympian as an incomplete notation for 2151-EDA

> and set boundaries accordingly?

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?

>> I agree that these shifts should not be large.

>>

>> What is it in this example that you think gets too far from its

SocA?

>

> 1715C is significantly closer to the SoCA of .~|( than it is to the

> SoCA of )~|''

1715C is 13.074c. I'm repeating this from a previous message:

There are only 3 possible symbols, including )~|'':

)~|'' as 12.897c or 12.883c

.~|(' as 13.199c or 13.172c

')~|.. as 13.186c or 13.200c

I think you mean closer to SoCA of .~|(' than of )~|'' -- okay!

However, to get that, we would have to use all 3 symbols. (You've

convinced me.)

If we assign the two MPLC commas to .~|(' and ')~|.., then we get

option 1:

Pop.

Cents Name Complex Rank Symbol Dev.1 Dev.2 Comments

------ ------- ------- ---- ------ ------ ------ ---------------

12.897 5:847C 47.654 814 .~|( +0.120 SoCA for )~|''

12.930 27-mina lower boundary= )~|'' +0.033 +0.047 ===============

12.943 19:121C 49.462 962 )~|'' +0.046 +0.060 Primary role

13.008 symbol boundary ======= )~|'' +0.111 +0.125 ===============

13.008 symbol boundary ======= .~|(' -0.191 -0.164 ===============

13.066 11:133C 44.456 817 .~|(' -0.133 -0.106

13.074 1715C 46.916 110 .~|(' -0.126 -0.099 Primary role

13.166 19:23C 56.739 448 .~|(' -0.033 -0.006

13.171 symbol boundary ======= .~|(' -0.028 -0.001 ===============

13.171 symbol boundary ======= ')~|.. 0.015 -0.029 ===============

13.189 35:247C 44.117 924 ')~|.. +0.003 -0.011

13.269 11:23C 34.857 129 ')~|.. +0.083 +0.069 Primary role

13.399 71825C 91.989 284 ')~|.. +0.213 +0.199 5-limit default

13.418 27-mina upper boundary= ')~|.. +0.232 +0.218 ===============

Only 2 commas on our list would get )~|'': 19:121C and 47C, since it

would get only about 1/6 mina of territory. The olympian-level symbol

for the 27th mina would be the simplest one, .~|(', which gets 1/3

mina. The ugly symbol ')~|.. not only gets slightly more than 1/2

mina, but also the least complex comma, 11:23C -- not as good as I had

hoped for.

If we use both principal values for the SoCA's, then .~|(' > ')~|..,

giving option 2:

Pop.

Cents Name Complex Rank Symbol Dev.1 Dev.2 Comments

------ ------- ------- ---- ------ ------ ------ ---------------

12.897 5:847C 47.654 814 .~|( +0.120 SoCA for )~|''

12.930 27-mina lower boundary= )~|'' +0.033 +0.047 ===============

12.943 19:121C 49.462 962 )~|'' +0.046 +0.060 Primary role

13.008 symbol boundary ======= )~|'' +0.111 +0.125 ===============

13.008 symbol boundary ======= ')~|.. 0.177 -0.191 ===============

13.066 11:133C 44.456 817 ')~|.. -0.120 -0.134

13.074 1715C 46.916 110 ')~|.. -0.112 -0.126 Primary role

13.166 19:23C 56.739 448 ')~|.. -0.019 -0.033

13.171 symbol boundary ======= ')~|.. 0.015 -0.029 ===============

13.171 symbol boundary ======= .~|(' -0.028 -0.001 ===============

13.189 35:247C 44.117 924 .~|(' +0.010 -0.016

13.269 11:23C 34.857 129 .~|(' +0.069 +0.096 Primary role

13.399 71825C 91.989 284 .~|(' +0.200 +0.227 5-limit default

13.418 27-mina upper boundary= .~|(' +0.218 +0.245 ===============

Again, the same 2 commas on our list get )~|'' in the lowest 1/6 mina.

The ugly symbol ')~|.. now gets 1/3 mina. The olympian-level symbol

would still be .~|(', but now it gets slightly more than 1/2mina and

the least complex comma, 11:23C, in addition to making 71825C (from

Gene's list) easier to notate.

> ...

> Why does 19:121C deserve a symbol primary role?

It doesn't. It's just that )~|'' deserves to be used. In promethean,

the most natural place for a boundary is between )~| and ~|( is between

27 and 28 minas (same place as an athenian boundary), so DAFLR is

promoted by devoting as much 27-mina territory as possible to )~|-cored

symbols.

> I don't think it

> does. So do we need to use )~|'' at all? If we use it, it seems to me

> that 5:847C deserves it more than 19:121C does, as 5:847C is SoCA and

> ever so slightly more popular and lower prime limit, though more

> complex according to your rating.

>

> You may remember that my only real criticism of your complexity

> rating was that it seemed to introduce higher primes too soon.

>

> So maybe we only need one 27 mina symbol.

Yes, maybe. Using only .~|(' (defined as 1715C) would minimize the

territory given to ')~|.. by completely eliminating it.

I'll have to give this some thought.

> But if we have two, why

> must the boundary be at 13.106. My approach ignores minas and puts

> the boundary midway between symbol SoCAs (assuming only the standard

> definitions of the right accents at this stage). How is your boundary

> calculated?

I put the boundaries halfway between the commas that define the

symbols.

--George

From George Secor 21/04/2007

Hi Dave,

I'm answering 2 out of 3 messages today.

--- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> At 07:20 AM 19/04/2007, you wrote:

>> --- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> ...

>>> Trying the alternative values of the right accents is something I

>>> haven't been doing, but I agree it is a good thing to do and I

plan

>>> to modify my spreadsheet to accommodate it.

>>

>> Okay.

>

> Still not done. Sorry.

No problem. Take your time.

>>> I think we can ignore those with popularity rank above about 500.

>>

>> Remember that you said this, because I'll bring it to your attention

>> below.

>

> A foolish consistency is the hobgoblin of little minds.

Here's something to which you can pay little mind:

Q: What is mind?

A: No matter.

Q: What is matter?

A: Never mind.

> ...

>>> That's because I haven't accepted your practice of assigning

commas

>>> to symbols with the same mina irrespective of whether they are

closer

>>> to that symbol's SoCA.

>>

>> Irrespective of whether they are closer than what? -- than another,

>> less popular / more complex comma?

>

> No. Closer than the other adjacent symbol's SoCA.

But suppose that we've agreed to use only one of two symbols in a

particular instance for superolympian, and the symbol with SoCA closer

to the ratio to be notated is more complicated than the farther one?

What then?

1) Must we use the more complicated symbol and discard the simpler one?

2) Or must we define the simpler symbol with a less popular / more

complex (LPMC) ratio and then have the symbol default to the more

popular / less complex (MPLC) ratio at the current level of precision

(even if that's superolympian)?

3) Or should we disregard the more complicated symbol and simply assign

the simpler symbol to the MPLC ratio. If someone later decides to take

Sagittal up to a higher level of precision that requires both symbols,

then the superolympian symbol assignment becomes the superolympian

default, and the two symbols may then be redefined according to commas

that are closer to their SoCA's.

> ...

>>> I don't feel that minas are relevant to superolympian at all.

>>

>> They're relevant if members of the tuning list think that they are.

>> There's been a lot of discussion on the main list recently in

looking

>> for octave divisions that are both high-limit consistent and

divisible

>> by 12, for the purpose of defining useful logarithmic units of pitch

>> measurement. See messages 71110, 71115, 71160, 71166, 71249, and

>> 71256. (No, I didn't start that thread!)

>

> I don't see how that makes minas relevant to a level of Sagittal

> specifically intended to go beyond them. Olympian is the domain of

> minas.

We have a few issues to deal with in this situation, where there aren't

anywhere near enough symbols to notate all of the degrees in whatever

finer unit of measure you intend to replace the mina.

From what you're saying, I'm concluding that if a comma doesn't come

within a unit or so (tina, or whatever) of the SoCA of any symbol, then

it can't be assigned to a symbol, and we must look for something LPMC.

What symbol does the MPLC symbol then default to, or (in other words),

will *all* of the boundaries between symbols be set halfway between

their defining commas (or between their SoCA's, as I see you're doing

it), or will the tina (or whatever unit) boundaries have any relevance?

If EDA-unit boundaries have no relevance, or if you're allowing that

symbols may be assigned to adjacent tinas, then in superolympian we can

no longer guarantee the 27-limit consistency we already have with

minas.

> A lot of other numbers were tossed around there besides 2460. In any

> case there is already a slight discrepancy between the 2460_EDO

> degree and the mina we are using, that is bound to cause problems for

> commas near the boundaries anyway.

The consistency of 2460-EDO guarantees that these problems will be

encountered only with very complicated commas. The whole point of the

discussion was to find divisions of the octave where this would be the

case.

> So I don't think we should allow

> ourselves to be constrained by what may well be a momentary fad.

This subject has come up before on the list (on several occasions), so

I don't thinks it's a passing interest.

> Did you mention on the list that the pronunciation is "meena"?

No, I was more concerned with explaining its excellent properties.

> If, after defining athenian as 21-EDA, we had insisted that higher

> precision notations must have boundaries that correspond exatly to

> those, we would not now be using 233-EDA for olympian. We must allow

> the possibility that someone in future may define a new accent or

> something that will take Sagittal to new extremes of ridiculousness

> of resolution and should not do anything that would make that

> difficult.

Yes, that's a worthy goal.

> One way to make sure of that would be to invent it ourselves (but no

> need to tell anyone).

Tee hee!

> 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 (see below).

> and probably distinguishes the standard and alternative

> values for the right accents (although I haven't checked).

>

> Why not consider superolympian as an incomplete notation for 2151-EDA

> and set boundaries accordingly?

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?

>> I agree that these shifts should not be large.

>>

>> What is it in this example that you think gets too far from its

SocA?

>

> 1715C is significantly closer to the SoCA of .~|( than it is to the

> SoCA of )~|''

1715C is 13.074c. I'm repeating this from a previous message:

There are only 3 possible symbols, including )~|'':

)~|'' as 12.897c or 12.883c

.~|(' as 13.199c or 13.172c

')~|.. as 13.186c or 13.200c

I think you mean closer to SoCA of .~|(' than of )~|'' -- okay!

However, to get that, we would have to use all 3 symbols. (You've

convinced me.)

If we assign the two MPLC commas to .~|(' and ')~|.., then we get

option 1:

Pop.

Cents Name Complex Rank Symbol Dev.1 Dev.2 Comments

------ ------- ------- ---- ------ ------ ------ ---------------

12.897 5:847C 47.654 814 .~|( +0.120 SoCA for )~|''

12.930 27-mina lower boundary= )~|'' +0.033 +0.047 ===============

12.943 19:121C 49.462 962 )~|'' +0.046 +0.060 Primary role

13.008 symbol boundary ======= )~|'' +0.111 +0.125 ===============

13.008 symbol boundary ======= .~|(' -0.191 -0.164 ===============

13.066 11:133C 44.456 817 .~|(' -0.133 -0.106

13.074 1715C 46.916 110 .~|(' -0.126 -0.099 Primary role

13.166 19:23C 56.739 448 .~|(' -0.033 -0.006

13.171 symbol boundary ======= .~|(' -0.028 -0.001 ===============

13.171 symbol boundary ======= ')~|.. 0.015 -0.029 ===============

13.189 35:247C 44.117 924 ')~|.. +0.003 -0.011

13.269 11:23C 34.857 129 ')~|.. +0.083 +0.069 Primary role

13.399 71825C 91.989 284 ')~|.. +0.213 +0.199 5-limit default

13.418 27-mina upper boundary= ')~|.. +0.232 +0.218 ===============

Only 2 commas on our list would get )~|'': 19:121C and 47C, since it

would get only about 1/6 mina of territory. The olympian-level symbol

for the 27th mina would be the simplest one, .~|(', which gets 1/3

mina. The ugly symbol ')~|.. not only gets slightly more than 1/2

mina, but also the least complex comma, 11:23C -- not as good as I had

hoped for.

If we use both principal values for the SoCA's, then .~|(' > ')~|..,

giving option 2:

Pop.

Cents Name Complex Rank Symbol Dev.1 Dev.2 Comments

------ ------- ------- ---- ------ ------ ------ ---------------

12.897 5:847C 47.654 814 .~|( +0.120 SoCA for )~|''

12.930 27-mina lower boundary= )~|'' +0.033 +0.047 ===============

12.943 19:121C 49.462 962 )~|'' +0.046 +0.060 Primary role

13.008 symbol boundary ======= )~|'' +0.111 +0.125 ===============

13.008 symbol boundary ======= ')~|.. 0.177 -0.191 ===============

13.066 11:133C 44.456 817 ')~|.. -0.120 -0.134

13.074 1715C 46.916 110 ')~|.. -0.112 -0.126 Primary role

13.166 19:23C 56.739 448 ')~|.. -0.019 -0.033

13.171 symbol boundary ======= ')~|.. 0.015 -0.029 ===============

13.171 symbol boundary ======= .~|(' -0.028 -0.001 ===============

13.189 35:247C 44.117 924 .~|(' +0.010 -0.016

13.269 11:23C 34.857 129 .~|(' +0.069 +0.096 Primary role

13.399 71825C 91.989 284 .~|(' +0.200 +0.227 5-limit default

13.418 27-mina upper boundary= .~|(' +0.218 +0.245 ===============

Again, the same 2 commas on our list get )~|'' in the lowest 1/6 mina.

The ugly symbol ')~|.. now gets 1/3 mina. The olympian-level symbol

would still be .~|(', but now it gets slightly more than 1/2mina and

the least complex comma, 11:23C, in addition to making 71825C (from

Gene's list) easier to notate.

> ...

> Why does 19:121C deserve a symbol primary role?

It doesn't. It's just that )~|'' deserves to be used. In promethean,

the most natural place for a boundary is between )~| and ~|( is between

27 and 28 minas (same place as an athenian boundary), so DAFLR is

promoted by devoting as much 27-mina territory as possible to )~|-cored

symbols.

> I don't think it

> does. So do we need to use )~|'' at all? If we use it, it seems to me

> that 5:847C deserves it more than 19:121C does, as 5:847C is SoCA and

> ever so slightly more popular and lower prime limit, though more

> complex according to your rating.

>

> You may remember that my only real criticism of your complexity

> rating was that it seemed to introduce higher primes too soon.

>

> So maybe we only need one 27 mina symbol.

Yes, maybe. Using only .~|(' (defined as 1715C) would minimize the

territory given to ')~|.. by completely eliminating it.

I'll have to give this some thought.

> But if we have two, why

> must the boundary be at 13.106. My approach ignores minas and puts

> the boundary midway between symbol SoCAs (assuming only the standard

> definitions of the right accents at this stage). How is your boundary

> calculated?

I put the boundaries halfway between the commas that define the

symbols.

--George

- Dave Keenan
- Site Admin
**Posts:**1814**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: consistent Sagittal 37-Limit

Re: Comma lists

From George Secor 25/04/2007

Hi Dave,

I'm going to reply to the off-comma-topic stuff in a separate message,

as a new subject, so the comma lists material continues here:

> ...

>> 3) Or should we disregard the more complicated symbol and simply

assign

>> the simpler symbol to the MPLC ratio. If someone later decides to

take

>> Sagittal up to a higher level of precision that requires both

symbols,

>> then the superolympian symbol assignment becomes the superolympian

>> default, and the two symbols may then be redefined according to

commas

>> that are closer to their SoCA's.

>

> That certainly seems consistent with the way we've been going from

> athenian to olympian.

Okay!

>> We have a few issues to deal with in this situation, where there

aren't

>> anywhere near enough symbols to notate all of the degrees in

whatever

>> finer unit of measure you intend to replace the mina.

>>

>> From what you're saying, I'm concluding that if a comma doesn't

come

>> within a unit or so (tina, or whatever) of the SoCA of any symbol,

then

>> it can't be assigned to a symbol, and we must look for something

LPMC.

>>

>> What symbol does the MPLC symbol then default to, or (in other

words),

>> will *all* of the boundaries between symbols be set halfway between

>> their defining commas (or between their SoCA's, as I see you're

doing

>> it), or will the tina (or whatever unit) boundaries have any

relevance?

>>

>> If EDA-unit boundaries have no relevance, or if you're allowing that

>> symbols may be assigned to adjacent tinas, then in superolympian we

can

>> no longer guarantee the 27-limit consistency we already have with

>> minas.

>

> That's correct. I'm assuming that some symbols will cover multiple

> tinas. Superolympian must be accepted as incomplete in a sense. But

> then again if we don't tell anyone anything about the relationship

> between superolympian and tinas they wouldn't _expect_ any kind of

> consistency of adding up tina numbers. Remember that the top level of

> Sagittal is _not_ supposed to be based on any equal division. It is

> strictly JI. We use the equal divisions as a ladder to climb up, but

> then we must throw away the ladder.

Must we, now? That doesn't seem appropriate to me, if we're

entertaining the thought of developing the semantics (if not the

symbols) of a still higher level of resolution.

> I see that there are two different sets of boundaries needed here

> during the process. Initially we need boundaries to help us decide

> what commas to assign to what symbols. These could be tina

> boundaries. Then once we have our set of symbols and their primary

> commas we can throw away those boundaries and place them midway

> between primary commas.

Hmmm, that's not what I had in mind, because it throws 27-limit

consistency out the door.

> Currently _I_ am using boundaries midway beween SoCAs for the initial

> boundaries and _you_ are apparently somehow trying to use boundaries

> midway between primary commas as the initial boundaries, before you

> have primary commas! Although I understand you are using mina

> boundaries to bootstrap this process.

Yes, and no. Inasmuch there aren't at present enough symbols to notate

all of the steps of a finer-than-mina division, I'm keeping *all* of

the mina boundaries, both before and after assigning primary commas,

for the simple reason that they're meaningful. It's only between two

commas *within the same mina* that I assign a boundary based on those

commas. This preserves the guarantee of 27-limit consistency based on

mina numbers, while still enabling one to make distinctions between

commas falling within the same mina boundaries. All minas are split

into 2 parts, excepting 1) those at 0, 44, 55, and 56 minas (where it's

not feasible), and 2) those that, for various reasons, are split into 3

parts.

As I've described it, this really shouldn't be called superolympian,

but rather atomic. It retains the boundaries (and therefore the

27-limit consistency) of olympian and, at the same time, allows greater

precision *within* most minas. In olympian, I also split 12 minas (in

2 parts) under 1/2-apotome, in those instances where two fairly popular

commas fall within the same mina boundaries.

> Or alternatively we could say that the superolympian level is not a

> level designed for approximation at all.

Symbol boundaries would then be meaningless.

> If you have a comma that

> doesn't have a superolympian symbol then you should go to the

> olympian level and use the symbol for the nearest mina.

As I've outlined it above, split-mina approximation can do better than

that.

> In theory, if someone extended the notation so every tina could be

> notated, then the superolympian level would become a level that could

> be used for approximation. But we would then undoubtedly have

> multiple symbols for many tinas and therefore there would also be a

> super-superolympian level which was not for approximation.

Could we now call levels by the following names?

Olympian: Minas of 233-EDA, with most unsplit, some split in 2;

Atomic: Minas of 233-EDA, with most split in 2, a few unsplit, some

split in 3;

Superolympian: Tinas of ?-EDA, with most unsplit, some split in 2;

Ultraolympian: Tinas of ?-EDA, with most split.

The mina boundaries could probably be adjusted by small amounts to

coincide with tina boundaries without compromising 27-limit

olympian/atomic consistency.

>> The consistency of 2460-EDO guarantees that these problems will be

>> encountered only with very complicated commas. The whole point of

the

>> discussion was to find divisions of the octave where this would be

the

>> case.

>

> I hadn't appreciated that before. You're quite right. Thanks.

>

> But the errors will change slightly and so the maximum consistent

> powers of low primes may change.

What do you mean? Between the EDO and the EDA? I pointed out on

tuning-math that the differences are very slight (as long as the

3-error is very low), and I think you agreed with that.

>> This subject has come up before on the list (on several occasions),

so

>> I don't thinks it's a passing interest.

>

> OK. But I don't think we should allow EDO divisibility by 12 to

> influence us too much.

I think we can have it, as well. >27-limit consistency is the greater

challenge.

>> 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.)

>

> 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:

http://tech.groups.yahoo.com/group/tuni ... nstncy.xls

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.

>

>> 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?

>

> So which of the above properties _don't_ 576 and 809 have. An actual

> columnar comparison of the properties you mention would be good.

One shortcoming of 576 and 809-EDA is non-divisibility of the EDO by

12.

One undesirable thing about 576-EDA that I didn't spot in the 6079-EDO

consistency figures is that some commas are dangerously close to

boundaries. For example, 5C is only 0.0070c below the 109th degree

upper boundary, and 4375n barely makes it above the 2nd degree |' lower

boundary (with only 0.0010c to spare). This seems too close for

comfort.

While assigning commas to symbols, I used a spreadsheet that allows me

to compare the boundaries for 1105, 809, 576, 233, 58, 47, and 21-EDA,

and this is what alerted me to the dangers of 576-EDA.

Both 809 and 1105-EDA look a lot more "secure", particularly the

latter. Since 809-EDA splits minas into 1/3's, a single mini-accent

up/down pair is all that's needed to notate it. I'd be a little wary

about that 8.9% 7-error, however. If we're going into more precision,

then we're going to see higher 7-exponents: 7^5 takes you close to 45%

error, so only a miniscule additional amount contributed by some other

prime (e.g., 3^4 would add another 5.2%) would send it over the 50%

mark.

For 1105 you'd need both single and double mini-accents, since it

divides the mina into 1/4's, just as the mina divides the 5-schisma.

Its ET (11664) really delivers low error: -0.3% for 3, 3.1% for 5, 1.2%

for 7, and 19.0% for 11 (each of these figures is better than for

2460-ET). Plus it's 12-divisible. Unless we can find something

suitable that has a higher consistency limit, I'd say we go with this

one.

The rest of this message is about the 27th mina:

>> 1715C is 13.074c. I'm repeating this from a previous message:

>>

>> There are only 3 possible symbols, including )~|'':

>> )~|'' as 12.897c or 12.883c

>> .~|(' as 13.199c or 13.172c

>> ')~|.. as 13.186c or 13.200c

>>

>> I think you mean closer to SoCA of .~|(' than of )~|'' -- okay!

>

> Yeah. Sorry. Well spotted.

>

>> However, to get that, we would have to use all 3 symbols. (You've

>> convinced me.)

>

> But have I convinced me?

(Groan!) See below.

>> Again, the same 2 commas on our list get )~|'' in the lowest 1/6

mina.

>> The ugly symbol ')~|.. now gets 1/3 mina. The olympian-level symbol

>> would still be .~|(', but now it gets slightly more than 1/2mina and

>> the least complex comma, 11:23C, in addition to making 71825C (from

>> Gene's list) easier to notate.

>

> I'm afraid I just can't muster the concentration to follow this at

> the moment. But it sounds OK.

>

>>> Why does 19:121C deserve a symbol primary role?

>>

>> It doesn't. It's just that )~|'' deserves to be used. In

promethean,

>> the most natural place for a boundary is between )~| and ~|( is

between

>> 27 and 28 minas (same place as an athenian boundary), so DAFLR is

>> promoted by devoting as much 27-mina territory as possible to

)~|-cored

>> symbols.

>

> I see the problem.

Do you mean: 1) you see a problem with this, or 2) you now see the

problem (concern) that I've been attempting to address, or 3) both?

>>> I don't think it

>>> does. So do we need to use )~|'' at all? If we use it, it seems to

me

>>> that 5:847C deserves it more than 19:121C does, as 5:847C is SoCA

and

>>> ever so slightly more popular and lower prime limit, though more

>>> complex according to your rating.

>>>

>>> You may remember that my only real criticism of your complexity

>>> rating was that it seemed to introduce higher primes too soon.

>>>

>>> So maybe we only need one 27 mina symbol.

>>

>> Yes, maybe. Using only .~|(' (defined as 1715C) would minimize the

>> territory given to ')~|.. by completely eliminating it.

>>

>> I'll have to give this some thought.

>

> OK

I've given it some thought, and I've concluded that if it's a matter of

choosing between splitting into 3 or not splitting at all, then I'd go

for 3, because this is supposed to be higher resolution than olympian.

I'd avoid splitting minas only where it's not feasible.

I hope we're getting somewhere.

--George

From George Secor 25/04/2007

Hi Dave,

I'm going to reply to the off-comma-topic stuff in a separate message,

as a new subject, so the comma lists material continues here:

> ...

>> 3) Or should we disregard the more complicated symbol and simply

assign

>> the simpler symbol to the MPLC ratio. If someone later decides to

take

>> Sagittal up to a higher level of precision that requires both

symbols,

>> then the superolympian symbol assignment becomes the superolympian

>> default, and the two symbols may then be redefined according to

commas

>> that are closer to their SoCA's.

>

> That certainly seems consistent with the way we've been going from

> athenian to olympian.

Okay!

>> We have a few issues to deal with in this situation, where there

aren't

>> anywhere near enough symbols to notate all of the degrees in

whatever

>> finer unit of measure you intend to replace the mina.

>>

>> From what you're saying, I'm concluding that if a comma doesn't

come

>> within a unit or so (tina, or whatever) of the SoCA of any symbol,

then

>> it can't be assigned to a symbol, and we must look for something

LPMC.

>>

>> What symbol does the MPLC symbol then default to, or (in other

words),

>> will *all* of the boundaries between symbols be set halfway between

>> their defining commas (or between their SoCA's, as I see you're

doing

>> it), or will the tina (or whatever unit) boundaries have any

relevance?

>>

>> If EDA-unit boundaries have no relevance, or if you're allowing that

>> symbols may be assigned to adjacent tinas, then in superolympian we

can

>> no longer guarantee the 27-limit consistency we already have with

>> minas.

>

> That's correct. I'm assuming that some symbols will cover multiple

> tinas. Superolympian must be accepted as incomplete in a sense. But

> then again if we don't tell anyone anything about the relationship

> between superolympian and tinas they wouldn't _expect_ any kind of

> consistency of adding up tina numbers. Remember that the top level of

> Sagittal is _not_ supposed to be based on any equal division. It is

> strictly JI. We use the equal divisions as a ladder to climb up, but

> then we must throw away the ladder.

Must we, now? That doesn't seem appropriate to me, if we're

entertaining the thought of developing the semantics (if not the

symbols) of a still higher level of resolution.

> I see that there are two different sets of boundaries needed here

> during the process. Initially we need boundaries to help us decide

> what commas to assign to what symbols. These could be tina

> boundaries. Then once we have our set of symbols and their primary

> commas we can throw away those boundaries and place them midway

> between primary commas.

Hmmm, that's not what I had in mind, because it throws 27-limit

consistency out the door.

> Currently _I_ am using boundaries midway beween SoCAs for the initial

> boundaries and _you_ are apparently somehow trying to use boundaries

> midway between primary commas as the initial boundaries, before you

> have primary commas! Although I understand you are using mina

> boundaries to bootstrap this process.

Yes, and no. Inasmuch there aren't at present enough symbols to notate

all of the steps of a finer-than-mina division, I'm keeping *all* of

the mina boundaries, both before and after assigning primary commas,

for the simple reason that they're meaningful. It's only between two

commas *within the same mina* that I assign a boundary based on those

commas. This preserves the guarantee of 27-limit consistency based on

mina numbers, while still enabling one to make distinctions between

commas falling within the same mina boundaries. All minas are split

into 2 parts, excepting 1) those at 0, 44, 55, and 56 minas (where it's

not feasible), and 2) those that, for various reasons, are split into 3

parts.

As I've described it, this really shouldn't be called superolympian,

but rather atomic. It retains the boundaries (and therefore the

27-limit consistency) of olympian and, at the same time, allows greater

precision *within* most minas. In olympian, I also split 12 minas (in

2 parts) under 1/2-apotome, in those instances where two fairly popular

commas fall within the same mina boundaries.

> Or alternatively we could say that the superolympian level is not a

> level designed for approximation at all.

Symbol boundaries would then be meaningless.

> If you have a comma that

> doesn't have a superolympian symbol then you should go to the

> olympian level and use the symbol for the nearest mina.

As I've outlined it above, split-mina approximation can do better than

that.

> In theory, if someone extended the notation so every tina could be

> notated, then the superolympian level would become a level that could

> be used for approximation. But we would then undoubtedly have

> multiple symbols for many tinas and therefore there would also be a

> super-superolympian level which was not for approximation.

Could we now call levels by the following names?

Olympian: Minas of 233-EDA, with most unsplit, some split in 2;

Atomic: Minas of 233-EDA, with most split in 2, a few unsplit, some

split in 3;

Superolympian: Tinas of ?-EDA, with most unsplit, some split in 2;

Ultraolympian: Tinas of ?-EDA, with most split.

The mina boundaries could probably be adjusted by small amounts to

coincide with tina boundaries without compromising 27-limit

olympian/atomic consistency.

>> The consistency of 2460-EDO guarantees that these problems will be

>> encountered only with very complicated commas. The whole point of

the

>> discussion was to find divisions of the octave where this would be

the

>> case.

>

> I hadn't appreciated that before. You're quite right. Thanks.

>

> But the errors will change slightly and so the maximum consistent

> powers of low primes may change.

What do you mean? Between the EDO and the EDA? I pointed out on

tuning-math that the differences are very slight (as long as the

3-error is very low), and I think you agreed with that.

>> This subject has come up before on the list (on several occasions),

so

>> I don't thinks it's a passing interest.

>

> OK. But I don't think we should allow EDO divisibility by 12 to

> influence us too much.

I think we can have it, as well. >27-limit consistency is the greater

challenge.

>> 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.)

>

> 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:

http://tech.groups.yahoo.com/group/tuni ... nstncy.xls

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.

>

>> 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?

>

> So which of the above properties _don't_ 576 and 809 have. An actual

> columnar comparison of the properties you mention would be good.

One shortcoming of 576 and 809-EDA is non-divisibility of the EDO by

12.

One undesirable thing about 576-EDA that I didn't spot in the 6079-EDO

consistency figures is that some commas are dangerously close to

boundaries. For example, 5C is only 0.0070c below the 109th degree

upper boundary, and 4375n barely makes it above the 2nd degree |' lower

boundary (with only 0.0010c to spare). This seems too close for

comfort.

While assigning commas to symbols, I used a spreadsheet that allows me

to compare the boundaries for 1105, 809, 576, 233, 58, 47, and 21-EDA,

and this is what alerted me to the dangers of 576-EDA.

Both 809 and 1105-EDA look a lot more "secure", particularly the

latter. Since 809-EDA splits minas into 1/3's, a single mini-accent

up/down pair is all that's needed to notate it. I'd be a little wary

about that 8.9% 7-error, however. If we're going into more precision,

then we're going to see higher 7-exponents: 7^5 takes you close to 45%

error, so only a miniscule additional amount contributed by some other

prime (e.g., 3^4 would add another 5.2%) would send it over the 50%

mark.

For 1105 you'd need both single and double mini-accents, since it

divides the mina into 1/4's, just as the mina divides the 5-schisma.

Its ET (11664) really delivers low error: -0.3% for 3, 3.1% for 5, 1.2%

for 7, and 19.0% for 11 (each of these figures is better than for

2460-ET). Plus it's 12-divisible. Unless we can find something

suitable that has a higher consistency limit, I'd say we go with this

one.

The rest of this message is about the 27th mina:

>> 1715C is 13.074c. I'm repeating this from a previous message:

>>

>> There are only 3 possible symbols, including )~|'':

>> )~|'' as 12.897c or 12.883c

>> .~|(' as 13.199c or 13.172c

>> ')~|.. as 13.186c or 13.200c

>>

>> I think you mean closer to SoCA of .~|(' than of )~|'' -- okay!

>

> Yeah. Sorry. Well spotted.

>

>> However, to get that, we would have to use all 3 symbols. (You've

>> convinced me.)

>

> But have I convinced me?

(Groan!) See below.

>> Again, the same 2 commas on our list get )~|'' in the lowest 1/6

mina.

>> The ugly symbol ')~|.. now gets 1/3 mina. The olympian-level symbol

>> would still be .~|(', but now it gets slightly more than 1/2mina and

>> the least complex comma, 11:23C, in addition to making 71825C (from

>> Gene's list) easier to notate.

>

> I'm afraid I just can't muster the concentration to follow this at

> the moment. But it sounds OK.

>

>>> Why does 19:121C deserve a symbol primary role?

>>

>> It doesn't. It's just that )~|'' deserves to be used. In

promethean,

>> the most natural place for a boundary is between )~| and ~|( is

between

>> 27 and 28 minas (same place as an athenian boundary), so DAFLR is

>> promoted by devoting as much 27-mina territory as possible to

)~|-cored

>> symbols.

>

> I see the problem.

Do you mean: 1) you see a problem with this, or 2) you now see the

problem (concern) that I've been attempting to address, or 3) both?

>>> I don't think it

>>> does. So do we need to use )~|'' at all? If we use it, it seems to

me

>>> that 5:847C deserves it more than 19:121C does, as 5:847C is SoCA

and

>>> ever so slightly more popular and lower prime limit, though more

>>> complex according to your rating.

>>>

>>> You may remember that my only real criticism of your complexity

>>> rating was that it seemed to introduce higher primes too soon.

>>>

>>> So maybe we only need one 27 mina symbol.

>>

>> Yes, maybe. Using only .~|(' (defined as 1715C) would minimize the

>> territory given to ')~|.. by completely eliminating it.

>>

>> I'll have to give this some thought.

>

> OK

I've given it some thought, and I've concluded that if it's a matter of

choosing between splitting into 3 or not splitting at all, then I'd go

for 3, because this is supposed to be higher resolution than olympian.

I'd avoid splitting minas only where it's not feasible.

I hope we're getting somewhere.

--George

- Dave Keenan
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### Re: consistent Sagittal 37-Limit

And you already have the next and last message from George that contains "11:23", here.

- cmloegcmluin
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### Re: consistent Sagittal 37-Limit

Thanks for sharing all this @Dave Keenan !

The 2nd page of it is all the 11:23C, so we don't have to worry about that stuff.

Here is evidence of a historical difference between

I understand the concept of default values per precision level which potentially deviate from the "once-and-for-all" primary comma... but I really don't like it. And it would seem it wasn't retained in the end.

If, at least at some point, 47S was to be merely the default value in Olympian, while 11:23 was the "once-and-for-all" primary comma, I feel like our confidence in dismissing the splitting of the 75th mina to include the 47S has grown that much stronger.

Perhaps we (well, you know, Dave) could now find evidence of where/when/how exactly the concept of precision level default values was eliminated. Then we'd be that much closer to simplifying this bit of the Extreme precision level notation.

The 2nd page of it is all the 11:23C, so we don't have to worry about that stuff.

This seems to confirm your theory that:75 .(|(. 11:23S instead of ~|)''

But the real paydirt you hit, I think, is this bit:But which of the two valid (by mina arithmetic) symbols was assigned to the whole 75th mina before it was split? I can only assume it must have been because if it had been there would not have been any DAFLL violation, which George clearly states as the reason for the split.

Then we must ask, why was George so resistant to assigning to 47S? Surely the pre-existing boundary at the VH precision level (one level down), between and would have either been at 18.5/58 of an apotome, or midway between their commas 49S and 11S, which you have shown amount to practically the same thing, which is very near 74.5 minas, which should have predisposed George to using for the 75th mina.

>>> 2. Since there's such a thing as a Herculean default (58-EDA) for a symbol,

>>> which may be different from its once-and-for-all primary role, then might

>>> there not also be Olympian defaults (233-EDA) that are different from

>>> primary roles derived from considering all possible accented symbols.

>>

>> I hope not. But I do think that there should be promethean defaults

>> for symbols that are different from their primary roles.

>

> I can accept that there might be a few promethean defaults that differ from

> their primary roles, but there had better be a simple rule explaining why,

> not a bunch of random exceptions.

I already replied:

<< There will be -- with no exceptions! >>

but that was without thinking that we had tentatively agreed that all

primary ratios would be 23-limit. If the most popular ratio for a

particular number of minas is >23-limit, then that would be the

olympian default, but the primary ratio would be the most popular

23-limit ratio.

Here are the instances in which I think that this principle can be

applied without controversy:

# of default primary minas ratio ratio ----- ------- ------- 1 31:49n 455n 2 13:37n 65:77n 11 11:31k 605k 22 5:53k 5:161k 31 5:47C 7:143C 42 19:73C 19:169C 59 13:47C 605C 66 53C 19:49C 69 29S 13:17S 75 47S 11:23S 91 499S 17:49S 98 83M 5:187M 108 7:29M 2375M 110 47M 11:85M

Here is evidence of a historical difference between

*default*and*primary*values(/commas/ratios). I thought we had discussed this somewhere and decided there was no meaningful difference between default and primary. But it would seem at some point there was!I understand the concept of default values per precision level which potentially deviate from the "once-and-for-all" primary comma... but I really don't like it. And it would seem it wasn't retained in the end.

If, at least at some point, 47S was to be merely the default value in Olympian, while 11:23 was the "once-and-for-all" primary comma, I feel like our confidence in dismissing the splitting of the 75th mina to include the 47S has grown that much stronger.

Perhaps we (well, you know, Dave) could now find evidence of where/when/how exactly the concept of precision level default values was eliminated. Then we'd be that much closer to simplifying this bit of the Extreme precision level notation.

Last edited by Dave Keenan on Fri May 29, 2020 8:36 am, edited 1 time in total.

**Reason:***Restored the formatting of George's table*- cmloegcmluin
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### Re: consistent Sagittal 37-Limit

My methodology was not airtight. A lot of these schisma-sized steps are split, since there are 54 symbols in the space of just 35 schisma steps. So I only checked the lower bounds of whole-numbered steps. Of these 35, 23 of them were closer to 58-EDA and 12 were closer to 59-EDA. So I wouldn't say there's a clear winner; we don't even quite have a 2/3rds majority.I'm not quite sure what I mean by "58/59 eda". It's possible I'm simply mistaken about there being any 59-EDA aspect to the VHP Herculean level at all. I may have been seduced by the fact that both 612edo and 624edo have highly-accurate JI approximations. I'm pretty sure 624edo corresponds to 59-EDA. You might check the VHP boundaries to see if any of them are closer to odd half steps of 59 than odd half steps of 58.

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### Re: consistent Sagittal 37-Limit

Sigh. Too hard. When you just said, "Search on <x> in your [George] email", I could cope with that.Cml wrote: Perhaps we (well, you know, Dave) could now find evidence of where/when/how exactly the concept of precision level default values was eliminated. Then we'd be that much closer to simplifying this bit of the Extreme precision level notation.

I don't understand why you excluded the "splits". It seems like that would be

*more*work. not less. Were they split 58-EDA buckets or split 59-EDA buckets (rhetorical)? There's a bias right there. Which is, I guess, what you meant by "not airtight".

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### Re: consistent Sagittal 37-Limit

It wasn't more work. Maybe I'm not being clear, or maybe I don't understand something about the step names I inherited from George's work. What I meant was that on the Boundaries tab of the JI calculator spreadsheet that some of the 58-EDA step names are whole and some are not. I considered the ones with decimal places to be "split" 58-/59- EDA and thus those boundaries would not even be trying to fall on a 58- / 59- EDA midpoint, thus wouldn't help make this call.Dave Keenan wrote: ↑Thu May 28, 2020 6:05 pm I don't understand why you excluded the "splits". It seems like that would bemorework. not less. Were they split 58-EDA buckets or split 59-EDA buckets (rhetorical)? There's a bias right there. Which is, I guess, what you meant by "not airtight".

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### Re: consistent Sagittal 37-Limit

Maybe there's some relevant stuff on the Yahoo Tuning Group backup. I'll hunt soon.Dave Keenan wrote: ↑Thu May 28, 2020 6:05 pmSigh. Too hard. When you just said, "Search on <x> in your [George] email", I could cope with that.Cml wrote: Perhaps we (well, you know, Dave) could now find evidence of where/when/how exactly the concept of precision level default values was eliminated. Then we'd be that much closer to simplifying this bit of the Extreme precision level notation.

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### Re: consistent Sagittal 37-Limit

The ones with decimal places are split 58-EDA buckets, and thus not trying to fall on a 58-EDA midpoint, but some of them might fall near a 59-EDA midpoint due to invisible forces of mathematics. I suspect this because I think of both 612edo and 624edo as "good" divisions.cmloegcmluin wrote: ↑Fri May 29, 2020 6:03 am What I meant was that on the Boundaries tab of the JI calculator spreadsheet that some of the 58-EDA step names are whole and some are not. I considered the ones with decimal places to be "split" 58-/59- EDA and thus those boundaries would not even be trying to fall on a 58- / 59- EDA midpoint, thus wouldn't help make this call.