Great approach by John M for bounding n-limit temperaments by generator size

Props to John Moriarty for coming up with this - this is a great idea
and I've never heard this suggested before. I've added a little bit to
it but the core idea is his, so props to him. He gave me permission to
post it because I fear that someone's about to post some Unfortunate
Tuning List Error Flamebait soon, and I think his idea provides a very
elegant solution to the problem of how to bound a temperament's valid
range.

The basic idea: harmony can get pretty out there. We've been exploring
exotemperaments a lot lately, which are recognizable if you try. We've
also been exploring things like mavila a lot over at the XA Facebook
group, as well as some unconventional meantones like 17 and 26-equal.
But all of the generator sizes for EVERY temperament that we've been
exploring - as well as any that has ever been used, in existence, ever
- will always still obey one important rule: they always ensure that,
in the mapping, 4/3 >= 5/4 and 5/4 >= 6/5. This is to say that, in
Bosanquet's terminology, 16/15 and 25/24 are always positive and never
reversed. Always, or else you've mangled the raw structure of 5-limit
harmony so much that the utonal inverse and inversions of the major
triad will now get all out of whack relative to themselves, which is a
simple fact regardless of any personal tolerance for tuning error.

Think about it - if 25/24 is ever negative, then that would mean that
your mapping for 10:12:15 actually has what you're calling 6/5 larger
than what you're calling 5/4. Therefore, 10:12:15 will now have the
major third on the bottom and 4:5:6 has the minor third on the bottom.
So if 25/24 ever flips around, you can bet that the inverse will sound
more like an actual 4:5:6 than the mapping for 4:5:6 itself, no matter
how poorly things are tuned. If you can't at least make major thirds
bigger than minor thirds, then even if you pretend there are no 7 or
11-limit identities you've still completely ruined bare 5-limit
harmony relative to itself. Flipping around 16/15 is even worse - it
means that what you're calling 4/3 is now smaller than what you're
calling 5/4. So if both 25/24 and 16/15 are reversed, then what you're
calling 4:5:6 is actually going to be 3:4:5, which is ridiculous.

If you do this, the border for each temperament will be the EDO at one
extreme where 16/15 vanishes, and the EDO at the other extreme where
25/24 vanishes.

This leads to the following results, copied from John's post

> Syntonic=81/80=7-edo to 5-edo
> Pelogic=135/128 = 2-edo to 7-edo
> Porcupine=250/243= 8-edo to 7-edo
> Magic=3125/3072= 10-edo to 3-edo
> Hanson=15625/15552= 4-edo to 11-edo

For those new to the list and unfamiliar, the terms "Syntonic" and
"Pelogic" are alternate names for meantone and mavila, respectively.

Thoughts?

-Mike

Also, to add - this assumes that, in some sense, 25/24 and 16/15 are
fundamental 5-limit intervals that must always remain positive (or
perhaps vanish), but never disappear, or else the fundamental
structure of 5-limit harmony gets all messed up. This makes intuitive
sense, but to formalize it more rigorously, I thought a good way to
approach it would be to preserve the ordering of the entire n-limit
tonality diamond, which gives you a way to extend this to higher
limits too (and subgroups, if you're clever). So the algorithm would
be something like

1) Work out the n-odd-limit tonality diamond for the n-limit
temperament you're working with
2) Find the difference between every element in this set and every
other (would this be the "Perfect Difference Set?")
3) Find the range of generator tuning where every element in this
resulting set remains positive; it will be bounded by two EDOs where
two of the members vanish

Being as it's more strict than the original, this may produce tighter
bounds than what John listed, but I doubt it - unless you know of a
way to reverse 9/8 without first reversing 16/15 and 25/24. Maybe.

As an unexpected bonus, if you're clever you've noticed that these ET
bounds also implicitly assign each temperament a "mandatory MOS" -
since Meantone runs from 5 EDO-7 EDO, it must produce a 5L2s MOS, if
Mavila runs from 2-EDO to 7-EDO, it must produce a 2L5s MOS, if Magic
runs from 3-EDO to 10-EDO it must produce a 3L7s MOS, etc. This is a
bonus because the goal of this isn't to rehash the discussion on how
to tie together MOS's and temperaments, but this approach gives you a
starting point for that anyway.

Thoughts?

-Mike

PS - let's for the moment ignore the issue of tempered octaves now to
keep things simple; I'm not sure how to define the target criteria if
octaves are tempered just yet.

On Sun, Jun 12, 2011 at 2:54 AM, Mike Battaglia <battaglia01@...> wrote:
> Props to John Moriarty for coming up with this - this is a great idea
> and I've never heard this suggested before. I've added a little bit to
> it but the core idea is his, so props to him. He gave me permission to
> post it because I fear that someone's about to post some Unfortunate
> Tuning List Error Flamebait soon, and I think his idea provides a very
> elegant solution to the problem of how to bound a temperament's valid
> range.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Thoughts?

This sounds a lot like an idea which has been proposed before, that the tuning of a p-limit temperament should always conserve the ordering of the p-limit tonality diamond.

On Sun, Jun 12, 2011 at 3:26 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Thoughts?
>
> This sounds a lot like an idea which has been proposed before, that the tuning of a p-limit temperament should always conserve the ordering of the p-limit tonality diamond.

Oh. Well then, I like it. That's how it should be. Also, being as this
assigns every temperament an MOS, perhaps this could be a point of
departure for naming MOS's.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Jun 12, 2011 at 3:26 AM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > Thoughts?
> >
> > This sounds a lot like an idea which has been proposed before, that the tuning of a p-limit temperament should always conserve the ordering of the p-limit tonality diamond.
>
> Oh. Well then, I like it. That's how it should be. Also, being as this
> assigns every temperament an MOS, perhaps this could be a point of
> departure for naming MOS's.
>
> -Mike
>

I guess I'm missing something. A given EDO can lead to many different MOS through choice of generator. I suppose this means that a given EDO can represent different temperaments via those different MOS.

But saying that 5L3s is between 5 and 8 EDO is just to say 0<s<L.

On Sun, Jun 12, 2011 at 4:14 AM, bobvalentine1 <bob.valentine@...> wrote:
>
> I guess I'm missing something. A given EDO can lead to many different MOS through choice of generator. I suppose this means that a given EDO can represent different temperaments via those different MOS.
>
> But saying that 5L3s is between 5 and 8 EDO is just to say 0<s<L.

It's not just that. We're saying that Father temperament itself runs
between 5 and 8 EDO, because that's the only way to preserve the order
of the tonality diamond. Hence by those bounds, Father must produce a
5L3s MOS. Except with Father I don't think it'll work out that way, I
think it'll be between 5 EDO and 3 EDO instead, but don't quote me on
that.

-Mike

This is precisely the approach we formally defined in Section 3 of our Tuning Continua paper (see Table 1):

http://open.academia.edu/AndrewMilne/Papers/178988/Tuning_continua_and_keyboard_layouts

Of course, it is not the only approach to defining temperaments' tuning ranges, but it perhaps represents a reasonable way to define its outer limits.

Andy

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Props to John Moriarty for coming up with this - this is a great idea
> and I've never heard this suggested before. I've added a little bit to
> it but the core idea is his, so props to him. He gave me permission to
> post it because I fear that someone's about to post some Unfortunate
> Tuning List Error Flamebait soon, and I think his idea provides a very
> elegant solution to the problem of how to bound a temperament's valid
> range.
>
> The basic idea: harmony can get pretty out there. We've been exploring
> exotemperaments a lot lately, which are recognizable if you try. We've
> also been exploring things like mavila a lot over at the XA Facebook
> group, as well as some unconventional meantones like 17 and 26-equal.
> But all of the generator sizes for EVERY temperament that we've been
> exploring - as well as any that has ever been used, in existence, ever
> - will always still obey one important rule: they always ensure that,
> in the mapping, 4/3 >= 5/4 and 5/4 >= 6/5. This is to say that, in
> Bosanquet's terminology, 16/15 and 25/24 are always positive and never
> reversed. Always, or else you've mangled the raw structure of 5-limit
> harmony so much that the utonal inverse and inversions of the major
> triad will now get all out of whack relative to themselves, which is a
> simple fact regardless of any personal tolerance for tuning error.
>
> Think about it - if 25/24 is ever negative, then that would mean that
> your mapping for 10:12:15 actually has what you're calling 6/5 larger
> than what you're calling 5/4. Therefore, 10:12:15 will now have the
> major third on the bottom and 4:5:6 has the minor third on the bottom.
> So if 25/24 ever flips around, you can bet that the inverse will sound
> more like an actual 4:5:6 than the mapping for 4:5:6 itself, no matter
> how poorly things are tuned. If you can't at least make major thirds
> bigger than minor thirds, then even if you pretend there are no 7 or
> 11-limit identities you've still completely ruined bare 5-limit
> harmony relative to itself. Flipping around 16/15 is even worse - it
> means that what you're calling 4/3 is now smaller than what you're
> calling 5/4. So if both 25/24 and 16/15 are reversed, then what you're
> calling 4:5:6 is actually going to be 3:4:5, which is ridiculous.
>
> If you do this, the border for each temperament will be the EDO at one
> extreme where 16/15 vanishes, and the EDO at the other extreme where
> 25/24 vanishes.
>
> This leads to the following results, copied from John's post
>
> > Syntonic=81/80=7-edo to 5-edo
> > Pelogic=135/128 = 2-edo to 7-edo
> > Porcupine=250/243= 8-edo to 7-edo
> > Magic=3125/3072= 10-edo to 3-edo
> > Hanson=15625/15552= 4-edo to 11-edo
>
> For those new to the list and unfamiliar, the terms "Syntonic" and
> "Pelogic" are alternate names for meantone and mavila, respectively.
>
> Thoughts?
>
> -Mike
>

--- In tuning@yahoogroups.com, "andymilneuk" <ANDYMILNE@...> wrote:
>
> This is precisely the approach we formally defined in Section 3 of our Tuning Continua paper

Oops! Sorry about that. The fact that all the DT stuff already had those limits to each temperament is what lead to trying to justify it harmonically in the first place, I definitely don't mean to steal any credit. I saw that paper a long time ago and couldn't make heads or tails of the math back then, but it probably had something to do with why I came up with what I did.

I'll try reading it again now, thanks for setting this straight.

John

On 6/12/2011 3:18 AM, Mike Battaglia wrote:
> Also, to add - this assumes that, in some sense, 25/24 and 16/15 are
> fundamental 5-limit intervals that must always remain positive (or
> perhaps vanish), but never disappear, or else the fundamental
> structure of 5-limit harmony gets all messed up. This makes intuitive
> sense, but to formalize it more rigorously, I thought a good way to
> approach it would be to preserve the ordering of the entire n-limit
> tonality diamond, which gives you a way to extend this to higher
> limits too (and subgroups, if you're clever). So the algorithm would
> be something like
>
> 1) Work out the n-odd-limit tonality diamond for the n-limit
> temperament you're working with
> 2) Find the difference between every element in this set and every
> other (would this be the "Perfect Difference Set?")
> 3) Find the range of generator tuning where every element in this
> resulting set remains positive; it will be bounded by two EDOs where
> two of the members vanish

That's a good start, but if you take an extreme example, the range of generators for father temperament works out to be 400-600 cents. At 600 cents you have 5/4, 4/3, 3/2, and 8/5 all mapped to 600 cents.

I think you'll also want to compare the tempered ratios to the adjacent just ratios. E.g., with a generator around 498 cents, the tempered 5/4 starts to approach the just 4/3. So father would have a range from 407 to 498 cents for its generator. If you want to be even stricter you could set a limit half way between the just ratios, in which case the only valid tuning for father would be 442.179 cents.

Andy,

> This is precisely the approach we formally defined in Section 3 of our Tuning Continua paper

It seems like that paper only takes a set of arbitrarily determined "privileged intervals" and describes keeping them melodically in sequence. Am I missing a more objective motive for the use of these intervals besides their use as the common practice consonances?

We're trying to see how one might go directly from a comma to a VTR. Obviously, given a mapping, some just interval intervals must be chosen to remain intact in relation to each other, but *why should we choose 4/3, 5/4, and 6/5*?

Is it simply because we use 4:5:6 as the fundamental harmonic unit, and it's identity will flip to 1/4:1/5:1/6 when 25/24 or 16/15 is reversed? If so, the use of another triad as "fundamental" would call for a whole new VTR, as you say would be the case for selecting new privileged intervals in your paper. But might it have something to do with retaining an intelligible harmonic series derived from the mapping and tuning, or some other method that would not call for a variation in VTR regardless of your harmonic intent?

Though I've already found it very useful to draw the lines for these temperaments where picking those privileged intervals has, I'd like to figure out why that might be the case.

John M

On Sun, Jun 12, 2011 at 8:50 PM, Herman Miller <hmiller@...> wrote:
>
> I think you'll also want to compare the tempered ratios to the adjacent
> just ratios. E.g., with a generator around 498 cents, the tempered 5/4
> starts to approach the just 4/3. So father would have a range from 407
> to 498 cents for its generator. If you want to be even stricter you
> could set a limit half way between the just ratios, in which case the
> only valid tuning for father would be 442.179 cents.

I'm not sure, because exotemperaments can really screw with your
perception sometimes. Take hedgehog, for example - try the LLsLLLsL
mode in 22-equal. I'll be damned if that LLsL doesn't sound like the
same LLsL in the beginning of the major scale, just really flat. If
you actually manage to hear it that way, then inasfar as the outer
dyad is supposed to represent 3/2, it sounds like you've tempered out
9/8. Try it, you'll see. It's weird.

-Mike

--- In tuning@yahoogroups.com, "jlmoriart" <JlMoriart@...> wrote:
>
> Andy,
>
> > This is precisely the approach we formally defined in Section 3 of our Tuning Continua paper
>
> It seems like that paper only takes a set of arbitrarily determined "privileged intervals" and describes keeping them melodically in sequence. Am I missing a more objective motive for the use of these intervals besides their use as the common practice consonances?
>

No. In the end the choice of privileged intervals is somewhat arbitrary.

> We're trying to see how one might go directly from a comma to a VTR. Obviously, given a mapping, some just interval intervals must be chosen to remain intact in relation to each other, but *why should we choose 4/3, 5/4, and 6/5*?

1/1 and the inversions should also be included. As before, this choice somewhat arbitrary. But these intervals seem a reasonable choice - they are the only intervals designated consonances in common practice and they are only intervals found in the only triads designated consonant in common practice (major and minor triads).

For higher prime limits, this generalises, in the most obvious way, to the intervals found in 4:5:6:7, or 8:10:11:12:14, etc., or building things up in odd-limits - which is Partch's tonality diamond.

>
> Is it simply because we use 4:5:6 as the fundamental harmonic unit, and it's identity will flip to 1/4:1/5:1/6 when 25/24 or 16/15 is reversed?

Not too sure what you mean here

> If so, the use of another triad as "fundamental" would call for a whole new VTR, as you say would be the case for selecting new privileged intervals in your paper.

Yes, the Bohlen-Pierce system, for example, privileges the intervals in 3:5:7 and so implies a different VTR. As would a system that privileges any other subset of primes, or any other set of intervals.

Similarly, the VTR of an MOS scale can be defined by privileging the large step and the small step - so its boundary tunings occur when the small step hits zero size, and the small step and large step become equal size.

> But might it have something to do with retaining an intelligible harmonic series derived from the mapping and tuning, or some other method that would not call for a variation in VTR regardless of your harmonic intent?

Is it any more possible to define a non-arbitrary set of privileged partials than it is musical ratios? Psychoacoustics would indicate that partials above about 10 are not resolvable, so should all ratios of integers up to about 10 be included - that would suggest we need to ensure the tempered 9/8>10/9, hence all meantone tunings are boundary tunings, which seems a bit extreme?

>
> Though I've already found it very useful to draw the lines for these temperaments where picking those privileged intervals has, I'd like to figure out why that might be the case.
>

If a piece of music is made up of consonant harmonies, and the melody is made up voices that connect together these harmonies, the change in melodic direction caused by going beyond the VTR represents an easily noticeable change that will probably break its perceived identity.

Andy

>
> John M
>

just wanted to note that I think that the answer to your youtube tuning mystery is "16"

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Jun 12, 2011 at 8:50 PM, Herman Miller <hmiller@...> wrote:
> >
> > I think you'll also want to compare the tempered ratios to the adjacent
> > just ratios. E.g., with a generator around 498 cents, the tempered 5/4
> > starts to approach the just 4/3. So father would have a range from 407
> > to 498 cents for its generator. If you want to be even stricter you
> > could set a limit half way between the just ratios, in which case the
> > only valid tuning for father would be 442.179 cents.
>
> I'm not sure, because exotemperaments can really screw with your
> perception sometimes. Take hedgehog, for example - try the LLsLLLsL
> mode in 22-equal. I'll be damned if that LLsL doesn't sound like the
> same LLsL in the beginning of the major scale, just really flat. If
> you actually manage to hear it that way, then inasfar as the outer
> dyad is supposed to represent 3/2, it sounds like you've tempered out
> 9/8. Try it, you'll see. It's weird.
>
> -Mike
>

--- In tuning@yahoogroups.com, "jlmoriart" <JlMoriart@...> wrote:

> We're trying to see how one might go directly from a comma to a VTR.

Possibly. What's a VTR?

--- In tuning@yahoogroups.com, "andymilneuk" <ANDYMILNE@...> wrote:
>
> --- In tuning@yahoogroups.com, "jlmoriart" <JlMoriart@> wrote:
> > But might it have something to do with retaining an intelligible harmonic series derived from the mapping and tuning, or some other method that would not call for a variation in VTR regardless of your harmonic intent?
>
> Is it any more possible to define a non-arbitrary set of privileged partials than it is musical ratios? Psychoacoustics would indicate that partials above about 10 are not resolvable, so should all ratios of integers up to about 10 be included - that would suggest we need to ensure the tempered 9/8>10/9, hence all meantone tunings are boundary tunings, which seems a bit extreme?

Yes, I think you're right there. I've been talking about the whole "retaining an intelligible harmonic series' thing with a vague idea of why I thought it would work, but only recently have I gotten into the math and some things didn't work out like I thought they would. Looks like it would be pretty limiting, and it probably wouldn't even be a good indicator for anything like badness or HE are.

I think I might follow Igs's example and just stop trying to think so much. It's fun to read what's going on in these circles, but I'm no mathematician or an expert on cognition or perception. I think I could do microtonality better justice by just making music, and I already know how I want to go about doing that. Whatever it takes, I'm going to start making some music.

John M