Yahoo Tuning Groups Ultimate Backup tuning 55 EDO and 2.3.5.11 243/242 and 81/80 temperament

I don't really understand how things are named around here, so I'm
going to talk in numbers. :)

I like the intervals 11/9, 11/8, and 11/6. (11/7? Not so much. Not
sure why. Doesn't matter for now.)

I also like having two neutral thirds stacking up to be a 3/2 perfect
fifth, which requires that 243/242 be tempered out.

Tempering out 243/242, of course, also means tempering out its square,
59049/58654. This happens to be the difference between five stacked
11/9s and 11/4. (I wouldn't have expected that relationship; squaring
the comma would make me expect to see a stack of four 11/9s or two
3/2s. I was looking around for 59049/58654 on the wiki for a while
before I realized that it's just 243/242 squared.)

So far, I'm in the 2.3.11 subgroup. Sometimes, when generating scales
for this subgroup, close approximations to other intervals also
appear. When the approximations are close to 5/4 or 6/5, I like to
have 81/80 tempered out.

Several EDOs temper out these commas using their patent vals: 7, 24,
31, 38, and 55.

7 does a lot of damage. 31 has 10-cent flat 11/8s, and 38 is worse. 24
is very good if you're happy with 12 EDO's sharp major thirds. 55
gives very good approximations to pretty much everything: The 6/5
minor thirds get the most damage, but at 305 cents they're still
better than 12 EDO.

I get MOSs at 7, 10, and 17 notes. There are some improprieties, but
they're not outrageous. At 17 notes, some approximations to 7 start
creeping in, so we could define the temperament out to the full
11-limit if we wanted to -- I just haven't gotten there yet, in part
because I find 17 tones to be unwieldy.

Here's the 10-tone MOS in 55 EDO:

! C:\Program Files (x86)\Scala22\neutrality.scl
!
2.3.5.11 with ~11/9 generator and 243/242 and 81/80 tempered out.
Tempered to 55 EDO.
10
!
43.63637
196.36364
349.09091
392.72728
545.45455
698.18182
741.81819
894.54546
1047.27273
2/1

It sounds nice, so I thought I'd share it.

Regards,
Jake

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> I don't really understand how things are named around here, so I'm
> going to talk in numbers. :)

Perfectly fine. =)

> I like the intervals 11/9, 11/8, and 11/6. (11/7? Not so much. Not
> sure why. Doesn't matter for now.)

Me too!

> I also like having two neutral thirds stacking up to be a 3/2 perfect
> fifth, which requires that 243/242 be tempered out.
>
> Tempering out 243/242, of course, also means tempering out its square,
> 59049/58654. This happens to be the difference between five stacked
> 11/9s and 11/4. (I wouldn't have expected that relationship; squaring
> the comma would make me expect to see a stack of four 11/9s or two
> 3/2s. I was looking around for 59049/58654 on the wiki for a while
> before I realized that it's just 243/242 squared.)
>
> So far, I'm in the 2.3.11 subgroup. Sometimes, when generating scales
> for this subgroup, close approximations to other intervals also
> appear. When the approximations are close to 5/4 or 6/5, I like to
> have 81/80 tempered out.

Ding ding! That's known as "mohaha" temperament:
http://xenharmonic.wikispaces.com/Chromatic+pairs#Mohaha
http://xenharmonic.wikispaces.com/Neutral+third+scales

It's definitely one of the all-stars. Everybody ought to know mohaha. Many (if not most) maqamat fit quite well into mohaha temperament (http://maqamworld.com/) which means there's already a significant corpus of music out there that's compatible with this temperament. (I won't say it's *in* mohaha temperament because it doesn't use quite enough harmony to be able to say "this music sounds like it's approximating these JI chords".)

The page http://maqamworld.com/maqamat.html contains this:

"The main reason why harmony is rarely used is that chords dont sound very pleasant when they include quarter tones or microtonal variations. Harmony sounds best when notes have a natural harmonic relationship (3rd, 4th, 5th and 6th harmonic, etc). In Arabic music, this is true for a tonic and its fifth (3rd harmonic), but most of the time not true for any other note combination."

Personally I think that if you kept everything else about maqam music the same, but added harmony with the specific goal of proving the above wrong, something magical would happen. =)

[snip]
> It sounds nice, so I thought I'd share it.

I wholeheartedly agree.

Keenan

🔗Mike Battaglia <battaglia01@...>

🔗Jake Freivald <jdfreivald@...>

Thanks for the replies, Keenan and Mike.

> Ding ding! That's known as "mohaha" temperament:
> http://xenharmonic.wikispaces.com/Chromatic+pairs#Mohaha
> http://xenharmonic.wikispaces.com/Neutral+third+scales

Oy! Okay, and thanks for providing me a name.

I had heard of Mohaha before -- even played with it a little bit,
maybe a year or so ago, without really grokking it -- but when I was
looking around for this temperament on the wiki (and believe me, I
always assume that whatever I'm looking at has been identified,
analyzed, categorized, described, catalogued, scraped, sanded, and
polyurethaned already) I looked for 243/242 and 81/80.

Little did I realize that (81/80) / (243/242) = 121/120 (just
calculated that now, realizing there must be a relationship) and
*that's* what I should be looking for instead of 243/242!

Moreover, the wiki doesn't list 55 EDO as a possible Mohaha tuning --
should we add that, or am I missing something? Without doing the math,
it looks like the best of the bunch if you want to spread the damage
gently and evenly across 11/9, 5/4, 3/2, and 11/8.

The whole "comma basis" thing drives me a bit crazy, I have to say:
81/80 (four 3/2s = 5/4) and 243/242 (two 11/9s = 3/2) is more
important to me than 121/120 (12/11 = 11/10) in this context, because
I'm considering generators and harmony, but I can see how 121/120
might be more important for someone else melodically, say. I've been
stubbornly reading for two years now, and I still frequently don't
grok why the various commas are valuable musically -- until I want to
do something specific musically, and stumble across the comma (or
comma combination) as I did here. That's why I keep experimenting and,
occasionally, posting the results. I always see something new. (And
thank you to those who help by pointing the way.)

But anyway, now that we're here, I'm happy to know where I am. Half
the battle is getting a map, and the other half is knowing how to read
it, and if I'm a half-wit with half a mind to try then I'm already a
quarter of a way there.

> It's definitely one of the all-stars. Everybody ought to know mohaha.
> Many (if not most) maqamat fit quite well into mohaha temperament
> (http://maqamworld.com/) which means there's already a significant
> corpus of music out there that's compatible with this temperament.

I'm totally checking this out. Thanks.

> The page http://maqamworld.com/maqamat.html contains this:
>
> "The main reason why harmony is rarely used is that chords dont
> sound very pleasant when they include quarter tones or microtonal
> variations. [snip]
>
> Personally I think that if you kept everything else about maqam
> music the same, but added harmony with the specific goal of proving
> the above wrong, something magical would happen. =)

Are you volunteering? :)

Regards,
Jake

🔗genewardsmith <genewardsmith@...>

🔗jdfreivald@...

> > Moreover, the wiki doesn't list 55 EDO as a possible Mohaha tuning --
> > should we add that, or am I missing something? Without doing the math,
> > it looks like the best of the bunch if you want to spread the damage
> > gently and evenly across 11/9, 5/4, 3/2, and 11/8.
>
> It's OK, but the 131 tuning used in the example scales does a better job.

(1) The fact that X may do a better job doesn't mean that Y shouldn't be listed. I see that you agree, since the chromatic pairs page is updated. :)

(2) How do you figure that 131 "does a better job"? Its error on 11/8 is double that of 55, and its error on other important intervals doesn't seem to be that much better. I'm assuming that you're following a formula here, perhaps one that places less value on higher-limit intervals, but if I value 11/8 that may not be the formula I want in this case.

Regards,
Jake

Sent from my Verizon Wireless BlackBerry

🔗genewardsmith <genewardsmith@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, jdfreivald@ wrote:
>
> > (2) How do you figure that 131 "does a better job"? Its error on 11/8 is double that of 55, and its error on other important intervals doesn't seem to be that much better.
>
> 131 is all-flat, and 55 isn't. If you look at the errors of eg 9/5 or 11/5, that makes a difference. Anyway, 55 is fine.

Yes, that's the reason - Gene is considering all intervals, not just primes. If the primes are mostly off in the same direction, the intervals *between* the primes are improved.

Note that 55 and 131 are on opposite sides of 31edo - that is, those tunings disagree on whether the mohaha[31] MOS is 24L7s or 7L24s. So, if both 55 and 131 are acceptable, then 31edo must also be acceptable. Here's the layout of octave fractions, for reference:

2\7 11\38 20\69 29\100 38\131 9\31 16\55 7\24 5\17 3\10

Note that 131 is quite close to 31, but on the opposite side of 31 from 55. For me, 31edo is totally fine and dandy.

Keenan

🔗Jake Freivald <jdfreivald@...>

Gene said:

> 131 is all-flat, and 55 isn't. If you look at the errors of eg 9/5 or 11/5, that makes a difference. Anyway, 55 is fine.

Super, Gene, thanks for the explanation.

> Yes, that's the reason - Gene is considering all intervals, not just primes.
> If the primes are mostly off in the same direction, the intervals *between*
> the primes are improved.

I get that intervals other than primes are important. It's why, for
instance, 29 EDO can be good for 14/11 and 13/11 even though the
errors in 7, 11, and 13 are 17.10, 13.39, and 12.94 cents
respectively. The errors cancel each other out in certain
combinations.

I admit that I wasn't looking at 9/5 or 11/5, which are indeed pretty
bad -- I was eyeballing an optimization for 3/2, 11/9, 11/8, and 11/6
only -- and that I wasn't looking at whether all of the errors were
sharp or flat.

> Note that 55 and 131 are on opposite sides of 31edo - that is, those
> tunings disagree on whether the mohaha[31] MOS is 24L7s or 7L24s.

I hadn't noticed that. I like the way you phrase what you're saying
here, because I don't remember people talking like this before (though
I'm sure they have).

I was trying to work out something similar in other temperaments, but
I can't, or at least not quickly. A porcupine[15] scale can be
generated in (or perhaps "as") 15 EDO by choosing 2\15 as a generator,
but every other EDO that tempers out 250/243 (e.g., 15, 22, 29, 21c,
30) and is sufficiently large (i.e., 7 EDO doesn't count because it
can't generate porcupine[15]) has a generator larger than 2\15; that
is, they're all "on the same side" of 15 EDO. A meantone[12] scale can
be generated in/as 12 EDO by choosing 7\12 as a generator, but every
other sufficiently large EDO that tempers out 81/80 has a generator
smaller than 7\12. Maybe I'm not using a large scale, and should look
at meantone[19] or [31] instead....

Anyway, it actually looks like a possibly interesting property of
mohaha[31] that it can flip from XL+YS to YL+XS without using crazily
distorted mappings. Same temperament, similar mappings, flipped MOS
structure for specific tunings.

Is this related to the (e.g.) "7&12" notation for meantone that I've
never properly understood? (I understand that it means that we use the
vals of 7 EDO and 12 EDO somehow, but I don't know how those vals
relate to the generators used.)

> So, if both 55 and 131 are acceptable, then 31edo must also be acceptable.

Right, I wasn't arguing that either 131 or 31 are "unacceptable". I
was eyeballing the scales with particular value set on particular
intervals, and I can see how the things I was valuing at that time
could be different from what other people value at other times and in
other contexts.

Regards,
Jake

🔗genewardsmith <genewardsmith@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> > Note that 55 and 131 are on opposite sides of 31edo - that is, those
> > tunings disagree on whether the mohaha[31] MOS is 24L7s or 7L24s.
>
> I hadn't noticed that. I like the way you phrase what you're saying
> here, because I don't remember people talking like this before (though
> I'm sure they have).
>
> I was trying to work out something similar in other temperaments, but
> I can't, or at least not quickly. A porcupine[15] scale can be
> generated in (or perhaps "as") 15 EDO by choosing 2\15 as a generator,
> but every other EDO that tempers out 250/243 (e.g., 15, 22, 29, 21c,
> 30) and is sufficiently large (i.e., 7 EDO doesn't count because it
> can't generate porcupine[15]) has a generator larger than 2\15; that
> is, they're all "on the same side" of 15 EDO. A meantone[12] scale can
> be generated in/as 12 EDO by choosing 7\12 as a generator, but every
> other sufficiently large EDO that tempers out 81/80 has a generator
> smaller than 7\12. Maybe I'm not using a large scale, and should look
> at meantone[19] or [31] instead....

It seems like you're making the mistake (which, BTW, lots of smart people were making for years and years) of assuming patent vals. You're probably thinking something like "17edo doesn't temper out 81/80", but that statement is nonsensical because 17edo is simply a collection of intervals. with no specific relationship to JI. What you should say is that the 17edo *patent val* (which maps both 6/5 and 5/4 to 5 steps) doesn't temper out 81/80. On the other hand, the 17c val, which maps 6/5 to 4 steps and 5/4 to 6 steps, is almost as good as, if not better than, the patent val (however you care to measure goodness), and that equal temperament *does* temper out 81/80.

So, for porcupine, it looks like you're correct that no patent vals do the trick, but the 23bc val (which Mike Battaglia has suggested before as a tool to learn porcupine categories) tempers out 250/243 and is on the "wrong side" of 15. Similarly, for meantone, 17c is one of the equal temperaments you seek; it's on the opposite side of 12 from the familiar meantone temperaments like 19 and 31.

Of course, these specific equal temperaments (23bc and 17c) are much higher in error than the rank-2 temperaments porcupine and meantone. We can all feel comfortable saying that meantone[12] is 7L5s rather than 5L7s, even though there are technically meantone tunings (such as 17c) that make it 5L7s. Why? Because the optimal meantone tuning is *unambiguously* on the 7L5s side; there's no doubt about it.

But, for any rank-2 temperament, there's always some number of notes that's large enough that you might as well call it equal at that point because it's ambiguous which way the MOS sequence goes.

For porcupine, 15 should definitely be 7L8s, and 22 should *almost* certainly be 7L15s (although some people may prefer things like 29edo), but by the time you get to porcupine[37], it's totally unclear whether it should be 22L15s or 15L22s, so you might as well stop there and call it 37edo.

For meantone, 12 should definitely be 7L5s, and 19 should probably be 12L7s for the usual meanings of "meantone" (although "flattone land" on the other side of 19 is very interesting), but by the time you get to 31edo you might as well call it equal because both 19L12s and 12L19s are perfectly reasonable.

> Anyway, it actually looks like a possibly interesting property of
> mohaha[31] that it can flip from XL+YS to YL+XS without using crazily
> distorted mappings. Same temperament, similar mappings, flipped MOS
> structure for specific tunings.

This property is in no way particular to mohaha, as I explained above. It's bound to happen at some point with every conceivable rank-2 temperament.

> Is this related to the (e.g.) "7&12" notation for meantone that I've
> never properly understood? (I understand that it means that we use the
> vals of 7 EDO and 12 EDO somehow, but I don't know how those vals
> relate to the generators used.)

"7&12" indeed refers to the patent vals of 7edo and 12edo. I could explain the rest if you really want, but we might want to take it over to tuning-math because it could get nitty gritty really quick. =) A key concept is the "extended Euclidean algorithm".

Keenan

🔗Jake Freivald <jdfreivald@...>

> It seems like you're making the mistake...of assuming patent vals.

No! I win! :)

Seriously, though, I do understand the difference between patent vals
and others; I'm afraid my language got sloppy. When I was looking at
porcupine and meantone, I went to Graham's temperament finder, went to
the unison vector search, and entered my comma. Entering 250/243 gives
7, 15, 8, 14c, 22, 1c, 29, 21c, 6bc, and 30 -- no 23bc. The only
non-patent val of those that has more than 15 notes is 21c, which I
checked out, and it's on the "right side" of 15 EDO. Entering 81/80
gives 12, 7, 19, 5, 24, 31, 14c, 26, 17c, and 2c. And oops, there's
17c, but apparently I didn't try it. [Hangs head...]

My penchant for patent vals comes mostly from a pragmatic issue: I
have a spreadsheet that I do a lot of calculations in (e.g., on the
way to work, when I don't have an Internet connection), and in Excel
it's easy to generate the patent vals.

In one sheet, I plug in up to four commas and see every EDO that
tempers out each of them: and by "every EDO", I mean that I
automatically see whether the patent val tempers them out. If I want
to look at an alternate val I have to manually enter it.

In another sheet, I plug in the EDO that I want, and it shows me the
values for all of the intervals contained in Scala (plus a few other
cats and dogs) using the patent val: If I want to use an alternate
val, I have to manually change it. If I have access to Graham's
temperament finder then I can get the alternate val easily enough --
but only if it's on the list, like 17c was for 81/80. If it's not on
the list, like 23bc for 250/243, I'd have to manually mess around
until I can find it. That's why I said this:

>> I was trying to work out something similar in other temperaments, but
>> I can't, or at least not quickly.

You said:

> Of course, these specific equal temperaments (23bc and 17c) are much
> higher in error than the rank-2 temperaments porcupine and meantone.

No doubt. A 730-cent "perfect fifth" (23bc-tuned porcupine) no longer
seems so perfect.

> Why? Because the optimal meantone tuning is *unambiguously* on the
> 7L5s side; there's no doubt about it.

Works for me.

> But, for any rank-2 temperament, there's always some number of notes that's
> large enough that you might as well call it equal at that point because it's
> ambiguous which way the MOS sequence goes.

That makes some sense. I was starting to intuit something like that,
which is why I said this:

>> Maybe I'm not using a large scale, and should look
>> at meantone[19] or [31] instead....

...and your explanation makes it clear why this is a property of every
rank-2 temperament rather than of mohaha.

> "7&12" indeed refers to the patent vals of 7edo and 12edo. I could explain
> the rest if you really want, but we might want to take it over to tuning-math
> because it could get nitty gritty really quick. =) A key concept is the
> "extended Euclidean algorithm".

I don't know if I want the full explanation or not. :) I just want to
know why it's important. When someone says "that's the 7&12
temperament", I know that means meantone -- but that's just
memorization. When someone says, "that's the 12f&29 temperament", I
have no idea how this identifies the temperament as Cassandra. And
maybe it's easier not to care, if Graham's temperament finder will
tell me what commas it tempers out, etc., anyway.

I'll look up "extended Euclidean algorithm" and see if I can
understand that, and then, if I don't go into vapor lock, I may join
the tuning-math list and ask you about it there.

Thanks as always for your patience,
Jake