Yahoo Tuning Groups Ultimate Backup tuning Optimal Lumma Stability

Given some MOS, we know that

If L/s > 2, the MOS will be Rothenberg improper, except for Carl's exception
If L/s = 2, the MOS will be Rothenberg proper, but not strictly proper
If L/s < 2, the MOS will be Rothenberg strictly proper

Also, we know that

If L=s, the Lumma stability of the MOS will be 1
If s=0, the Lumma stability of the MOS will be 0

It's been noted that the optimal stability for initial learning of a
scale isn't 1, which it's obviously not - can you imagine learning the
diatonic scale by listening to 7-equal? It's also been noted that a
scale's "learnability" is not necessarily correlated with its tuning
accuracy. So independent of any particular measures of tuning, I pose
this question to everyone: at what value of L/s will an MOS sound
"most unequal?"

Or, in other words, at what value will the specific interval classes
seem to have the most categorical independence?

1/1 is the opposite of what we want by definition, but if you go in
the other direction towards infinity/1, that actually just brings you
to another equal temperament with a step size of 0.

I think that the most possible unequal arrangement should work out to
be L/s = phi, yes? This also, not so coincidentally, has the following
properties
1) your MOS and all of its chromatic, enharmonic, etc descendents will
be strictly proper
2) One implication of this is that "sharps" will be lower than "flats"
for your chromatic scale (as with 19-equal vs 17-equal) (e.g. c < s)
3) Thus chromatic harmony will lay out in extremely "comprehensible"
fashion, which can serve as nice "training wheels" for someone first
learning chromatic harmony for a temperament

After experimenting with mavila in 25-equal vs 23-equal, and
"porcupine" in 23-equal vs 22-equal, I have to say that this is
exactly how it works for me, and furthermore that once I get the hang
of the chromatic structure, it then carries over to other improper
tunings, kind of like how I can hear meantone in everything.

For example, in 25-equal, using Armodue notation, the progression
1-2-7b -> 1-3b-5 is a contracting tritone resolving to a 0-240-673
"subminor" chord. In 23-equal, it resolves to a 0-209-678 chord.
Before spending time in 25-equal, this always sounded like I was
moving by microtonal inflections to a "sus2" chord. After spending
time in 25-equal, it now sounds like I'm moving to a really, really
flat "subminor" chord. I can snap 0-209-678 into a "subminor"
perception now, or perhaps what I'm really snapping it into is my
newly formed 0-3b-6 perception, except I think what's going on is that
strong and independent categories make for ease of association with
chord qualities...

Thoughts?

-Mike

🔗martinsj013 <martinsj@...>

🔗Mike Battaglia <battaglia01@...>

On Mon, Aug 22, 2011 at 5:02 AM, martinsj013 <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > ... at what value of L/s will an MOS sound "most unequal?"
> > ... 1/1 is the opposite of what we want by definition, but if you go in the other direction towards infinity/1, that actually just brings you to another equal temperament with a step size of 0.
> > ... L/s = phi, yes?
>
> Mike, I won't pretend to have thought about this deeply, but the obvious answer seems to be 2/1. You didn't say the MOS has to be strictly proper. In some sense, 2/1 seems like the "mid-point" between your two extremes, 1/1 and infinity/1. Why are values between phi/1 and 2/1 less good than phi/1?

I doubt that there's going to be a radical difference between 2/1 and
phi/1 in terms of categorical perception; I was just trying to figure
out some kind of hypothetical "most possible unequal" ratio. I've
since come to realize that porcupine[7] in 22-equal has always bugged
me mostly because of little things that boil down to it having a L/s =
4/3 ratio - the diminished fifths and perfect fifths sound similar in
size, the 10/9's and 9/8's as well, etc. Porcupine[8] has always
bothered me as well because the ~55 cent small step sounds as though
it might be an intonational inflection and not an actual change in
interval class. After switching to porcupine/opossum/whatever in
23-equal, I realized I liked it -way- better, because although the
intonational accuracy wasn't as good as 22, it didn't sound mostly
like 7-equal anymore.

I then realized what exactly had happened - I was saying that
diminished fifths and perfect fifths sounded too similar, that 10/9
and 9/8 sounded too similar, that the small step in porcupine[8]
sounded like an intonational shift in unison. Or, in other words, I
was saying that from a porcupine[7] perspective, the specific interval
categories in every generic interval class were getting confused,
meaning that in porcupine[7], c=L-s was too small. And, in fact, L/s
is 4/3 in 22-equal, which makes it about as confusing as flattone is
in 26-equal. From a porcupine[8] perspective I was saying that the
small step was too small to sound like a new interval and sounded like
a shift in another interval.

So if porcupine[7] being L/s = 4/3 sounds too much like 7-equal... and
porcupine[8] with L/s = 3/1 sounds too much like 7-equal too... then
what generator would make porcupine[7] and porcupine[8] both sound
maximally far from 7-equal, and hence maximally stable? I assumed phi
would do the trick, but perhaps there's no one size fits all solution
- L/s = 3/1 works wonders for the diatonic scale, I notice, but
doesn't do so well for porcupine[8]...

-Mike

🔗Kalle Aho <kalleaho@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Aug 23, 2011 at 5:05 AM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > this question to everyone: at what value of L/s will an MOS sound
> > "most unequal?"
> >
> > Or, in other words, at what value will the specific interval classes
> > seem to have the most categorical independence?
>
> I've been thinking about very similar things lately!

Good. Let the revolution begin! I've been waiting for a discussion
about this for a while now.

> I don't think unequalness and categorical independence are the same thing. L/s =
> Phi gives the most unequal MOSes but the greatest categorical
> independence must be calculated for each MOS separately.

I agree, I've since come to realize I was a bit hasty in my
generalization. I find that Blackwood[10] in 20-equal is a bit more
"comprehensible" than in 15-equal - the 15-equal version sounds almost
like 10-equal. I've found that I like somewhere between 15- and
20-equal the best, 5 2 5 2 5 2 5 2 5 2 is pretty good.

> What exactly does it mean is not yet really decided but here's a suggestion: every
> specific interval is as far in size from the closest specific
> interval as possible.

I'm not sure I understand, can you clarify? You mean in the same
generic interval class, or not?

>For example, this gives 3 3 1 3 3 3 1 from
> 17-equal as the most categorically independent diatonic scale if we
> don't mind about the impropriety.

I don't understand, how did you derive this? This means that the minor
thirds and the perfect fourths are a lot closer than in 2 2 1 2 2 2 1.
How would you work it out for porcupine[8]?

Does your experience with porcupine and mavila match up to my
experiences from my earlier message?

-Mike

🔗Carl Lumma <carl@...>

This bit of (unposted I think) text from 2000 may also be
of interest:

To that end, "rank standard deviation" and "rank range" are designed to measure the complexity of a scale's interval matrix. The higher these values, the more difficult it will be for the listener to construct the matrix, and the more likely it will be that he uses a matrix he has already learned instead (thus hearing the scale in question as a re- or mis-tuning of a scale he already knows). The measures are easy to calculate. Imagine a log-frequency ruler whose total length is the interval of equivalence ("formal octave") of our periodic scale. Take all of the unique intervals in the scale's interval matrix and mark them off on the ruler. Then, make a list of the distances between all pairs of consecutive values marked on the ruler. The measures are then the standard deviation and the range of the values on this list.

The idea is that since the interval matrix works by ranking intervals by size, the easiest matrices to use will be ones in which the sizes are most evenly distributed. Since for periodic scales the intervals between the intervals (ratios between consecutive marks on the ruler) must sum to the formal octave, the mean ratio occurs between every pair of marks in the ideal case, and the standard deviation will be in direct proportion to the complexity. Range will work in a similar way; I'm not sure which is more intuitive.

-C.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I'm struggling to keep up with things at this point, but
> assuming we're interested in proper MOS, wouldn't the
> point of maximum distinguishably be midway between equal
> temperament and the point of non-strict propriety?
> That is, when L/s is "midway" between 2 and 1?
>
> -Carl
>

🔗Kalle Aho <kalleaho@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Aug 23, 2011 at 5:05 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > I don't think unequalness and categorical independence are the same thing. L/s =
> > Phi gives the most unequal MOSes but the greatest categorical
> > independence must be calculated for each MOS separately.
>
> I agree, I've since come to realize I was a bit hasty in my
> generalization. I find that Blackwood[10] in 20-equal is a bit more
> "comprehensible" than in 15-equal - the 15-equal version sounds almost
> like 10-equal. I've found that I like somewhere between 15- and
> 20-equal the best, 5 2 5 2 5 2 5 2 5 2 is pretty good.
>
> > What exactly does it mean is not yet really decided but here's a suggestion: every
> > specific interval is as far in size from the closest specific
> > interval as possible.
>
> I'm not sure I understand, can you clarify? You mean in the same
> generic interval class, or not?

Not. Remember that you tend to confuse the characteristic dissonance
of TOP Pajara Pentachordal Decatonic with its' ~3:4. They are in
different interval classes.

> >For example, this gives 3 3 1 3 3 3 1 from
> > 17-equal as the most categorically independent diatonic scale if we
> > don't mind about the impropriety.
>
> I don't understand, how did you derive this? This means that the minor
> thirds and the perfect fourths are a lot closer than in 2 2 1 2 2 2 1.

Yes, but in 12-equal the augmented fourth and the diminished fifth are
the same size so this is potentially confusing. Of course we also
know that 12-equal works just fine and the feature gives us tritone
substitutions. So I don't know if it's that important after all.

> How would you work it out for porcupine[8]?

That would be the 15-equal version, I believe.

> Does your experience with porcupine and mavila match up to my
> experiences from my earlier message?

I don't have enough experience of either to say anything about that.

As I said, I'm not so sure if the maximal categorical distinctness is
that important given 12-equal diatonic. But the TOP Pajara
Pentachordal Decatonic problem suggests there's more to optimal
tuning than just making the consonances as consonant as possible. And
just look at this, the TOP Mavila[7]:

1 2 3 4 5 6 7
1/1 : 194.5 358.0 521.5 716.0 879.5 1043.0 1206.5
194.5 : 163.5 327.0 521.5 685.0 848.5 1012.0 1206.5
358.0 : 163.5 358.0 521.5 685.0 848.5 1043.0 1206.5
521.5 : 194.5 358.0 521.5 685.0 879.5 1043.0 1206.5
716.0 : 163.5 327.0 490.5 685.0 848.5 1012.0 1206.5
879.5 : 163.5 327.0 521.5 685.0 848.5 1043.0 1206.5
1043.0: 163.5 358.0 521.5 685.0 879.5 1043.0 1206.5

That's just horrible!

Perhaps I just want the intended dissonances to be as distinct (in
size) from the intended consonances as possible. So in this respect
Mavila[7] would probably work best in 16-equal and the Pentachordal
Decatonics in 22-equal.

Kalle

🔗Mike Battaglia <battaglia01@...>

Sorry for the late replies, folks, I'm working 60 hour weeks here...

On Tue, Aug 23, 2011 at 3:01 PM, Carl Lumma <carl@...> wrote:
>
> I'm struggling to keep up with things at this point, but
> assuming we're interested in proper MOS, wouldn't the
> point of maximum distinguishably be midway between equal
> temperament and the point of non-strict propriety?
> That is, when L/s is "midway" between 2 and 1?

Well, the mediant of those two is L/s = 3/2, and if we keep taking
successive noble mediants, we end up with L/s = phi, which is the same
thing you get if you start with L/s midway between 1 and infinity.

But if anything, all of my experiences over the past few weeks have
led to me feeling that something closer to "stability" is really
what's important, with propriety being secondary. I'm not sure whether
it's Rothenberg stability or Lumma stability or some other stability
metric yet undiscovered that's the key culprit.

Outside of propriety, the L/s point < 2 thing may be important because
it means that we're in a "negative" temperament, using Bosanquet's
terminology, where sharps are lower than flats. I'm not sure how
perceptually important that is, but I note that 25-equal is way more
comprehensible to me than 23-equal for mavila. I think.

Actually, I'd appreciate it if you could humor me with a little
experiment: Can you do, in 23-equal, 0-4-13? Does that sound like a
sus2 chord to you? Then, turn it off, give your brain 10 seconds to
reset, and then do 0-3-14 -> 0-4-13. Does it now sound like a really
flat "subminor?" Or still sus, with the intonation shifting slightly
for the bottom note?

At any rate I'm starting to feel though, and I see Kalle suspects the
same thing as well, that the point of "maximum categorical
independence" isn't going to be the same L/s ratio for all MOS. Andy
Milne has a model involving spreading each interval category out with
a Gaussian curve, presumably rooted in the same sort of thing
contributing to pitch uncertainty which we model with harmonic
entropy. We might call Milne stability. Perhaps that's the way to go,
at least for the initial learning of the scale.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Tue, Aug 23, 2011 at 3:03 PM, Carl Lumma <carl@...> wrote:
>
> This bit of (unposted I think) text from 2000 may also be
> of interest:
>
> To that end, "rank standard deviation" and "rank range" are designed to measure the complexity of a scale's interval matrix. The higher these values, the more difficult it will be for the listener to construct the matrix, and the more likely it will be that he uses a matrix he has already learned instead (thus hearing the scale in question as a re- or mis-tuning of a scale he already knows). The measures are easy to calculate. Imagine a log-frequency ruler whose total length is the interval of equivalence ("formal octave") of our periodic scale. Take all of the unique intervals in the scale's interval matrix and mark them off on the ruler. Then, make a list of the distances between all pairs of consecutive values marked on the ruler. The measures are then the standard deviation and the range of the values on this list.

So the range for any MOS is the same thing as the chroma, right? And
is what you're calling "rank standard deviation" the same thing as
what's now called Lumma stability?

> The idea is that since the interval matrix works by ranking intervals by size, the easiest matrices to use will be ones in which the sizes are most evenly distributed. Since for periodic scales the intervals between the intervals (ratios between consecutive marks on the ruler) must sum to the formal octave, the mean ratio occurs between every pair of marks in the ideal case, and the standard deviation will be in direct proportion to the complexity. Range will work in a similar way; I'm not sure which is more intuitive.

OK, here's another experiment. Load up porcupine[7] in 22-equal, the
Lssssss mode. Play 1 3 5, which should be major, and then transpose it
diatonically up and down the scale, so you go through the following
sequence of chords:

1) major
2) diminished
3) diminished
4) diminished
5) minor
6) minor
7) major

Now, here's the question: do you find the diminished chords
extra-unpleasant, even more so than diminished chords in 12-equal,
because they sort of sound like flat dicot-tempered major chords? And
in general, do you also feel like the 163 cent "minor seconds" and the
209 cent "whole tones" sometimes sound like ambiguous versions of one
another? Lastly, if you switch to porcupine[8], do you feel sometimes
like that motion by the 55-cent small step sounds like it might be
just an intonational shift over a unison, especially with the 3 3 3 3
1 3 3 3 mode and 3 1 3 3 3 3 3 3 modes?

Now try the same things in opossum in 23-tet, which uses 3\23 as a
generator. Do you feel like the intervals are more independent now and
less ambiguous, despite their relative lack of intonational purity?
Also note that the small step in porcupine[8] is now a 95-cent leading
tone, does that make any difference?

I hear it that way and got some consensus on this when I asked in the
XA chat. To me, when you compare it to 23-equal, the 22-equal version
sounds kinda like flattone does in 26-equal when you compare it to
something like 31-equal - uncomfortably compressed and slightly
ambiguous. However, in this case, the ambiguity is related to the
tuning with greater harmonic purity, not the other way around. When
you play the rest of the scale, it sounds like the fifth actually
should be rather sharp - much like I thought that 7/4 was flat the
first time I listened to JI after coming into it from 12-equal. You
can mess with gorgo in 16-equal and note that in the context of that
scale, the 750 cent interval sounds more like "a 3/2," which may
really mean something deeper and more subtly than an actual 3/2, than
the 675 cent interval does.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Wed, Aug 24, 2011 at 5:20 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > I'm not sure I understand, can you clarify? You mean in the same
> > generic interval class, or not?
>
> Not. Remember that you tend to confuse the characteristic dissonance
> of TOP Pajara Pentachordal Decatonic with its' ~3:4. They are in
> different interval classes.

Yeah. That's exactly why I didn't like it. That's a good example.
That's the same thing I'm saying about mavila in 25 and porcupine in
23.

> Yes, but in 12-equal the augmented fourth and the diminished fifth are
> the same size so this is potentially confusing. Of course we also
> know that 12-equal works just fine and the feature gives us tritone
> substitutions. So I don't know if it's that important after all.

How are you deriving this? What equation are you using?

> > How would you work it out for porcupine[8]?
>
> That would be the 15-equal version, I believe.

OK, how about Blackwood[10]? I'm convinced you may be onto something here.

> As I said, I'm not so sure if the maximal categorical distinctness is
> that important given 12-equal diatonic. But the TOP Pajara
> Pentachordal Decatonic problem suggests there's more to optimal
> tuning than just making the consonances as consonant as possible. And
> just look at this, the TOP Mavila[7]:
>
> That's just horrible!

All of your problems are solved in 25-equal... :)

> Perhaps I just want the intended dissonances to be as distinct (in
> size) from the intended consonances as possible. So in this respect
> Mavila[7] would probably work best in 16-equal and the Pentachordal
> Decatonics in 22-equal.

I would agree with this were it not for my experiences with porcupine.
It would be helpful for me if you checked out the example I just gave
to Carl - also the mavila one as well. It would be good to get some
data points on this. It's very confusing to sort out.

-Mike

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

On Fri, Aug 26, 2011 at 1:36 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > So the range for any MOS is the same thing as the chroma, right?
> > And is what you're calling "rank standard deviation" the same
> > thing as what's now called Lumma stability?
>
> It's different than Lumma stability. It considers the
> intervals between intervals. The range is the usual range
> (of the sample) of these meta intervals. For an MOS there's
> only one meta interval inside each interval class (the chroma)
> so their range is zero, but there's still the intervals
> between interval classes, e.g. between the largest 2nd and
> smallest 3rd, etc.

OK, so then the easiest scales to learn should, generally speaking be
MOS's of low-numbered EDOs?

> > OK, here's another experiment.
>
> Those seem like interesting experiments, which I'd like
> to try. Unfortunately: no time.

OK, here's one I've already made, which is vaguely related. Have you
heard this yet?

http://soundcloud.com/mikebattagliamusic/flat-major-scale

This is even flatter than mavila. Does it still sound recognizable as
the first 5 notes of a major scale to you, with the outer interval of
the pentachord being a flat 3/2 and all that? This is for Kalle as
well.

-Mike

🔗Carl Lumma <carl@...>

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

> > It's different than Lumma stability. It considers the
> > intervals between intervals. The range is the usual range
> > (of the sample) of these meta intervals. For an MOS there's
> > only one meta interval inside each interval class (the chroma)
> > so their range is zero, but there's still the intervals
> > between interval classes, e.g. between the largest 2nd and
> > smallest 3rd, etc.
>
> OK, so then the easiest scales to learn should, generally
> speaking be MOS's of low-numbered EDOs?

For proper MOS it would be something like

range(s, chroma, (ss - L), (sss - LL)...)

where "ss" really means "smallest 3rd" (whether that's
really ss will depend on the MOS pattern).

I should add that it's a purely speculative idea that
this is related to scale learnability. I would love to
have funding to test it.

> OK, here's one I've already made, which is vaguely related. Have you
> heard this yet?
> http://soundcloud.com/mikebattagliamusic/flat-major-scale
> This is even flatter than mavila. Does it still sound
> recognizable as the first 5 notes of a major scale to you,
> with the outer interval of the pentachord being a flat 3/2
> and all that? This is for Kalle as well.

A hard question to answer...

-Carl

🔗Mike Battaglia <battaglia01@...>

On Fri, Aug 26, 2011 at 3:19 AM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > It's different than Lumma stability. It considers the
> > > intervals between intervals. The range is the usual range
> > > (of the sample) of these meta intervals. For an MOS there's
> > > only one meta interval inside each interval class (the chroma)
> > > so their range is zero, but there's still the intervals
> > > between interval classes, e.g. between the largest 2nd and
> > > smallest 3rd, etc.
> >
> > OK, so then the easiest scales to learn should, generally
> > speaking be MOS's of low-numbered EDOs?
>
> For proper MOS it would be something like
>
> range(s, chroma, (ss - L), (sss - LL)...)
>
> where "ss" really means "smallest 3rd" (whether that's
> really ss will depend on the MOS pattern).

Right, and all of those expressions will be equal if L/s is 2, meaning
that s = c, and so on, hence minimizing the range for some MOS. And
you claim that minimizing the range makes the scale easiest to learn,
right? In general, I think that every one of those expressions except
for "s" is going to equal c for any MOS, so what you're basically
saying is that scales in which s=c will be easiest to learn, despite
that they're only Rothenberg proper and not strictly proper. Since the
only 2/1 MOS's are embedded in low-numbered EDO's (unless the MOS is
stupidly large), those end up being the magic scales.

> I should add that it's a purely speculative idea that
> this is related to scale learnability. I would love to
> have funding to test it.

I will say that I was fixated on some magic L/s = 2 ratio a while ago,
but after playing around a bit got into the idea that the magic L/s =
phi ratio was more important. It implies not only a proper scale but a
proper chromatic scale involving the meta-intervals between the
intervals, and so on and so on all to infinity, and also that it'd be
"maximally unequal" in some kind of sense, making the categories more
independent, which was supported by my experiments with porcupine and
mavila. It also means that every MOS will be negative in a generalized
sense, with sharps being lower than flats.

At any rate, Kalle's statements about having all intervals be
maximally distinct seems to be an even closer correlate to my
experience, although I'm still not sure how he's deriving those
figures. And I note that Blackwood where L/s = 3/1 seems to be more
coherent than where L/s = 2/1. Lastly, I also note that flattone[7]
sounds just as "compressed" as porcupine in 22-equal, even though I
already know the diatonic scale quite well, so there may be a
difference between scale coherence and scale learnability.

> > OK, here's one I've already made, which is vaguely related. Have you
> > heard this yet?
> > http://soundcloud.com/mikebattagliamusic/flat-major-scale
> > This is even flatter than mavila. Does it still sound
> > recognizable as the first 5 notes of a major scale to you,
> > with the outer interval of the pentachord being a flat 3/2
> > and all that? This is for Kalle as well.
>
> A hard question to answer...

Yes, it's a trick question. But to non-trickify it a bit, does it
sound as though, if you were to play root, third, and fifth in that
scale together, it would sound like an uncomfortably flat,
high-entropy chord that you might assign the "major chord" moniker to?
Keep in mind that I haven't played the notes at the same time - my
goal is to ascertain what your imagination is telling you what would
sound if you did play them simultaneously, even though they're not
being played simultaneously.

-Mike

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

--- Mike Battaglia <battaglia01@...> wrote:

> > For proper MOS it would be something like
> >
> > range(s, chroma, (ss - L), (sss - LL)...)
> >
> > where "ss" really means "smallest 3rd" (whether that's
> > really ss will depend on the MOS pattern).
>
> Right, and all of those expressions will be equal if L/s is 2,
> meaning that s = c, and so on, hence minimizing the range for
> some MOS.

Not sure which expressions you mean. If you mean the
terms inside the range function, yes, s = chroma but
(ss - L) = 0 and (sss - LL) = -1.

And for any fixed L/s value it still depends on the MOS
pattern. The general form of the above expression is

range(s, chroma, (Smallest3rd - Biggest2nd),
(Smallest4th - Biggest3rd), ...)

for proper MOS only.

> And you claim that minimizing the range makes the scale
> easiest to learn, right?

That's the idea.

> In general, I think that every one of those expressions except
> for "s" is going to equal c for any MOS,

Take the 5-tone Pythagorean. c = 90 cents and ss-L = 114.

> > A hard question to answer...
>
> Yes, it's a trick question. But to non-trickify it a bit,
> does it sound as though, if you were to play root, third,
> and fifth in that scale together, it would sound like an
> uncomfortably flat, high-entropy chord that you might
> assign the "major chord" moniker to?

I really don't know what to make of these subjective
listening things. First, how do I know what I'm hearing?
And second, how can we talk about it in a way that
makes sense?

-Carl

🔗Mike Battaglia <battaglia01@...>

On Fri, Aug 26, 2011 at 1:55 PM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
>
> Not sure which expressions you mean. If you mean the
> terms inside the range function, yes, s = chroma but
> (ss - L) = 0 and (sss - LL) = -1.
>
> And for any fixed L/s value it still depends on the MOS
> pattern. The general form of the above expression is
>
> range(s, chroma, (Smallest3rd - Biggest2nd),
> (Smallest4th - Biggest3rd), ...)
>
> for proper MOS only.

OK, I think I see what's happening. First off, you mentioned doing
this between all consecutive intervals, then I assume you want largest
second - smallest second in there too, as well as largest third -
smallest third, all of which would be c. However, let's say for the
diatonic scale, all of the "smallest 3rd - biggest 2nd" and "smallest
4th - largest 3rd" expressions will equate to s, except for the
smallest fifth and the largest fourth, which will equate to (s-c), the
diesis (or second-order chroma using the MODMOS terminology I laid out
before). So I think the range for any MOS, proper or not, will end up
being something like

range(s, c, s-c)

There are three possible orderings for these values, from largest to
smallest (examples given for meantone[7], I'm in a rush)

s > c > s-c - 3/2 < L/s < 2/1 - for meantone between 19-equal and 12-equal
s > s-c > c - 1/1 < L/s < 3/2 - for meantone between 7-equal and 19-equal
c > s > s-c - 2/1 < L/s < Inf - for meantone sharper than 12-equal

The other three are pathologic, as they would imply that s-c > s,
which means that s > L, which is stupid and evil.

If we're to apply

> > Yes, it's a trick question. But to non-trickify it a bit,
> > does it sound as though, if you were to play root, third,
> > and fifth in that scale together, it would sound like an
> > uncomfortably flat, high-entropy chord that you might
> > assign the "major chord" moniker to?
>
> I really don't know what to make of these subjective
> listening things. First, how do I know what I'm hearing?
> And second, how can we talk about it in a way that
> makes sense?

Yes, you're describing the characteristic failure of introspection to
distinguish between categorical perception and f0 estimation. It was
only a minor goal of mine with this example to bring that up, but it
should make people think a little harder when they say something
"sounds like 3/2" or "sounds like 5/4." They could just be saying that
it sounds like <ineffable mental construct that I've given the name
"3/2" to>. Although that wasn't the main point of this example.

Here's the gist of it - when I arpeggiate C-E-G-C-E-G-etc, no F0
estimation takes place unless I play those notes within a few hundred
milliseconds of one another. Nonetheless, the chord still seems to
retain the same "quality" as when I play the notes together. An
arpeggiated C-E-G-C-E-G-C-E-G will seem to be more consonant than an
arpeggiated C-Eb-Gb-A-C-Eb-Gb-A-C. This is good, because it allows us
to hear triadic harmony in something like a Bach 2-part invention. If
this didn't happen, Bach's Invention no 8 would sound no different
than an atonal composition consisting of random 12-equal dyads.

My hypothesis is that this is something that's learned, something
that's tuning-specific, and that will fail when someone listens to a
new tuning or a new scale that they're completely unused to - that
this process will "misfire" in characteristic ways as we reify in the
wrong harmonic data.

So the question is, relative to your categorical perception - if you
believe, with the best of your knowledge, that the root, third, and
fifth of this scale, when played at the same time, ---would--- sound
like what you'd categorically call a "major chord." This is as opposed
to something you'd categorically call a "minor chord" or an "augmented
chord" or a "diminished chord." If you have categories that you've
assigned numerical labels to, such as "4:5:6" or "5:6:7," as I suspect
most people on here do, then you can use those as well. I'm trying to
figure out how you'd harmonically reason through the scale fragment I
played.

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

--- Mike Battaglia <battaglia01@...> wrote:

> > Not sure which expressions you mean. If you mean the
> > terms inside the range function, yes, s = chroma but
> > (ss - L) = 0 and (sss - LL) = -1.
> > And for any fixed L/s value it still depends on the MOS
> > pattern. The general form of the above expression is
> > range(s, chroma, (Smallest3rd - Biggest2nd),
> > (Smallest4th - Biggest3rd), ...)
> > for proper MOS only.
>
> So I think the range for any MOS, proper or not, will end up
> being something like range(s, c, s-c)

That sounds right, but it only works for proper MOS.
An improper MOS could have arbitrarily small overlap,
and that would have to be a term in the range function.

By Chalmers' formula, proper MOS have ss > L unless
there's only one L. And since L = s+c, s > c.

Given this, the minimum range obtains when s = 2c
and therefore, L/s = 3/2. Which is the same as we got
before by taking the mediant. So 3/2 must be right,
unless of course you were to tell me that the other
statistic I mention, the standard deviation, were to
minimize with L/s = phi....

> I'm trying to figure out how you'd harmonically reason
> through the scale fragment I played.

I wish I knew.

-Carl

🔗Mike Battaglia <battaglia01@...>

On Sat, Aug 27, 2011 at 2:38 AM, Carl Lumma <carl@...> wrote:
> >
> > So I think the range for any MOS, proper or not, will end up
> > being something like range(s, c, s-c)
>
> That sounds right, but it only works for proper MOS.
> An improper MOS could have arbitrarily small overlap,
> and that would have to be a term in the range function.

What do you mean arbitrarily small overlap? It should still work out
for proper MOS, except s-c will be negative. Note the possible
orderings I gave for s, c, and s-c, and how some of them only happen
if L/s > 2.

But if we're marking intervals off on a ruler and considering adjacent
ones, should s-c really be negative? If it's s, c, and |s-c|, I think
we'll get a different result.

> By Chalmers' formula, proper MOS have ss > L unless
> there's only one L.

Does ss mean smallest third, or 2s? If the former, then any truly
proper MOS should have ss > L, because propriety is defined that way.
If the latter, then do you mean "only one s" above?

> And since L = s+c, s > c.
> Given this, the minimum range obtains when s = 2c
> and therefore, L/s = 3/2.

How do you derive this?

> Which is the same as we got
> before by taking the mediant. So 3/2 must be right,
> unless of course you were to tell me that the other
> statistic I mention, the standard deviation, were to
> minimize with L/s = phi....

I didn't really understand what you were saying before. I've been more
concerned with what allows for a stable heirarchical perception, both
for the diatonic scale in question and for its chromatic scale and so
on. I think that L/s = phi should minimize the range for any
temperament's MOS's.

> > I'm trying to figure out how you'd harmonically reason
> > through the scale fragment I played.
>
> I wish I knew.

Are you saying that you can't figure out what the quality of the tonic triad is?

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

--- Mike Battaglia <battaglia01@...> wrote:

> > > So I think the range for any MOS, proper or not, will end up
> > > being something like range(s, c, s-c)
> >
> > That sounds right, but it only works for proper MOS.
> > An improper MOS could have arbitrarily small overlap,
> > and that would have to be a term in the range function.
>
> What do you mean arbitrarily small overlap? It should still
> work out for proper MOS, except s-c will be negative. Note the
> possible orderings I gave for s, c, and s-c, and how some
> of them only happen if L/s > 2.
> But if we're marking intervals off on a ruler and
> considering adjacent ones, should s-c really be negative?
> If it's s, c, and |s-c|, I think we'll get a different result.

s-c shouldn't be negative. It'd be |s-c| because it's
just like any other gap on the ruler. But I don't know
if the whole 'smallest N - largest M' thing even holds.
Weird things can happen in wildly improper MOS.

> > By Chalmers' formula, proper MOS have ss > L unless
> > there's only one L.
>
> Does ss mean smallest third, or 2s?

Literally 2s for Chalmers' formula.

> If the former, then any truly proper MOS should have ss > L,
> because propriety is defined that way.

Except for MOS with only one L (as Ryan pointed out).

> > And since L = s+c, s > c.
> > Given this, the minimum range obtains when s = 2c
> > and therefore, L/s = 3/2.
>
> How do you derive this?

If c is bigger, s-c gets smaller and L stays the same.
If c is smaller, it gets smaller and L stays the same. :)

> I didn't really understand what you were saying before.
> I've been more concerned with what allows for a stable
> heirarchical perception, both for the diatonic scale in
> question and for its chromatic scale and so on. I think
> that L/s = phi should minimize the range for any
> temperament's MOS's.

The range is definitely minimal with L/s = 3/2. I don't
know about the std. I posted the excerpt of my old essay
without even thinking about it - actually I was worried
it was unrelated. But it turns out to agree with what I
came up with originally, just thinking generally. So that's
reassuring (to me).

> > > I'm trying to figure out how you'd harmonically reason
> > > through the scale fragment I played.
> >
> > I wish I knew.
>
> Are you saying that you can't figure out what the quality
> of the tonic triad is?

Honestly, I have no idea what you're asking me. :(

-Carl

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

On Sat, Aug 27, 2011 at 3:22 AM, Carl Lumma <carl@...> wrote:
>
> > If the former, then any truly proper MOS should have ss > L,
> > because propriety is defined that way.
>
> Except for MOS with only one L (as Ryan pointed out).

You mean only one s, right? Porcupine[8] in 22-equal is proper despite
being L/s = 3/1. If you have only one L, and L < 2s, that would mean
that you're dealing with something like 3 1 1 1 1 1 1, which is
negri[7] in 9-equal, and that's improper.

> > > And since L = s+c, s > c.
> > > Given this, the minimum range obtains when s = 2c
> > > and therefore, L/s = 3/2.
> >
> > How do you derive this?
>
> If c is bigger, s-c gets smaller and L stays the same.
> If c is smaller, it gets smaller and L stays the same. :)
and then
> Replace "L" with "s" there, sorry. -C.

But if c is getting smaller, that means that L/s is approaching 1/1,
and the s in 7-equal is larger than the s for the diatonic scale in
12-equal...

> > I didn't really understand what you were saying before.
> > I've been more concerned with what allows for a stable
> > heirarchical perception, both for the diatonic scale in
> > question and for its chromatic scale and so on. I think
> > that L/s = phi should minimize the range for any
> > temperament's MOS's.
>
> The range is definitely minimal with L/s = 3/2. I don't
> know about the std. I posted the excerpt of my old essay
> without even thinking about it - actually I was worried
> it was unrelated. But it turns out to agree with what I
> came up with originally, just thinking generally. So that's
> reassuring (to me).

Yes, it is, but my train of thought was -
- The haplotonic scale has the lowest range at L/s = 3/2.
- Once you understand the haplotonic scale, you'll want to get your
head wrapped around the albitonic scale next.
- If the haplotonic scale is set to L/s = 3/2, then the albitonic
scale is 2/1, which isn't optimal. So say you set that to 3/2 instead.
OK.
- Once you understand the albitonic scale, you'll want to get your
head wrapped around the chromatic scale next.
- If the albitonic scale is set to L/s = 3/2, then the chromatic scale
is set to 2/1, which isn't optimal. So say you set that to 3/2
instead. OK

And so on. So I was saying that the L/s value that will minimize the
range for all of the meta-intervals, and the meta-meta intervals, and
the meta-meta-meta-meta intervals all the way down the chain ad
infinitum, was phi.

Obviously you don't need to go down that far, probably no further than
the chromatic or perhaps enharmonic level, but the situation does get
cloudy when you have a scale which has two useful MOS's right next to
each other, like porcupine or mohajira.

For example, the optimal range for porcupine[7] implies 15-equal,
whereas the optimal range for porcupine[8] implies 23-equal. Which one
is better for porcupine temperament in general? How about mohajira[7]
vs mohajira[10]? Probably both, and setting L/s = phi sets, in a
certain sense, a maximally easy to learn structure all the way up and
down the entire MOS chain and temperament at large, at least as far as
range and propriety are both concerned.

OK, how important are range and propriety? I have no idea, but if
you've noted my continued posts about mavila and porcupine recently,
there's something around the vicinity of these ideas that has
immediate and apparent importance to my ear. 25-equal mavila and
23-equal porcupine are so refreshingly "clear" to my mind that in
retrospect it's obvious why I hated porcupine in 22-equal so much, and
why this seems to be such a common reaction - it sounds like it might
as well be 7-equal. It's like getting used to meantone by listening to
26-equal.

But at any rate, Kalle's construct seems to be giving some interesting
results, in that he seems to be doing the same thing you are, but
getting L/s = 3/1 for meantone as optimal. So I'm not sure how he's
tweaked it compared to you, but I do note that I find the diatonic
scale in 17-equal to be more comprehensible than the one in 19-equal.

> > > > I'm trying to figure out how you'd harmonically reason
> > > > through the scale fragment I played.
> > >
> > > I wish I knew.
> >
> > Are you saying that you can't figure out what the quality
> > of the tonic triad is?
>
> Honestly, I have no idea what you're asking me. :(

OK, here's another example (apologies for the poor sound quality)

http://soundcloud.com/mikebattagliamusic/not-flat-major-scale

This one is also LLsL.

Does it sound like the root, third, and fifth, if played together,
would make a different fundamental type of chord than the root, third,
and fifth of the other example, or just a differently intoned version
of the same chord?
And do you feel that in the other example would make a different
fundamental type of chord than the root, third, and fifth of the
diatonic scale in 12-equal, or just a differently intoned version of
the same chord?

I'm just asking for your initial impression. The left-brained
awareness of what "differently intoned version of the same chord"
would actually entail in a model of music cognition is besides the
point. Whatever it means, you have access to this information just by
experiencing it, so I'm asking you what you experience.

-Mike

🔗Graham Breed <gbreed@...>

🔗Carl Lumma <carl@...>

--- Mike Battaglia <battaglia01@...> wrote:

> > > If the former, then any truly proper MOS should have ss > L,
> > > because propriety is defined that way.
> >
> > Except for MOS with only one L (as Ryan pointed out).
>
> You mean only one s, right?

grrr yes.

> > > > And since L = s+c, s > c.
> > > > Given this, the minimum range obtains when s = 2c
> > > > and therefore, L/s = 3/2.
> > >
> > > How do you derive this?
> >
> > If c is bigger, s-c gets smaller and L stays the same.
> > If c is smaller, it gets smaller and L stays the same. :)
> > Replace "L" with "s" there, sorry. -C.
>
> But if c is getting smaller, that means that L/s is approaching
> 1/1, and the s in 7-equal is larger than the s for the diatonic
> scale in 12-equal...

So? c = s minimizes the range and makes L/s = 3/2.

> setting L/s = phi sets, in a certain sense, a maximally easy
> to learn structure all the way up and down the entire MOS chain
> and temperament at large, at least as far as range and propriety
> are both concerned.

Really?

> Does it sound like the root, third, and fifth,
> if played together, would make a different fundamental type of
> chord than the root, third, and fifth of the other example,

It's this kind of question that I can't answer.

-Carl

🔗Mike Battaglia <battaglia01@...>

On Sat, Aug 27, 2011 at 1:33 PM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > > > > And since L = s+c, s > c.
> > > > > Given this, the minimum range obtains when s = 2c
> > > > > and therefore, L/s = 3/2.
> > > >
> > > > How do you derive this?
> > >
> > > If c is bigger, s-c gets smaller and L stays the same.
> > > If c is smaller, it gets smaller and L stays the same. :)
> > > Replace "L" with "s" there, sorry. -C.
> >
> > But if c is getting smaller, that means that L/s is approaching
> > 1/1, and the s in 7-equal is larger than the s for the diatonic
> > scale in 12-equal...
>
> So? c = s minimizes the range and makes L/s = 3/2.

Right, and I asked how you derived that, and you gave the above
analysis, and I pointed out that s in fact does not stay the same, so
here we are.

> > setting L/s = phi sets, in a certain sense, a maximally easy
> > to learn structure all the way up and down the entire MOS chain
> > and temperament at large, at least as far as range and propriety
> > are both concerned.
>
> Really?

It should make it so that every MOS is maximally close to L/s = 3/2, yes?

> > Does it sound like the root, third, and fifth,
> > if played together, would make a different fundamental type of
> > chord than the root, third, and fifth of the other example,
>
> It's this kind of question that I can't answer.
It would be helpful if you could explicitly state what part of my
question you don't understand. I feel like I've given you enough
information to understand which specific percept I'm talking about
now, that I've adequately distinguished between categorical perception
and f0 estimation, and that no lingering doubt should exist about
which one I'm getting at.

-Mike

🔗Carl Lumma <carl@...>

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

> > > But if c is getting smaller, that means that L/s is approaching
> > > 1/1, and the s in 7-equal is larger than the s for the diatonic
> > > scale in 12-equal...
> >
> > So? c = s minimizes the range and makes L/s = 3/2.
>
> Right, and I asked how you derived that, and you gave the above
> analysis, and I pointed out that s in fact does not stay the same,
> so here we are.

Huh??

> > > setting L/s = phi sets, in a certain sense, a maximally easy
> > > to learn structure all the way up and down the entire MOS chain
> > > and temperament at large, at least as far as range and propriety
> > > are both concerned.
> >
> > Really?
>
> It should make it so that every MOS is maximally close
> to L/s = 3/2, yes?

I don't know. It makes it so that the MOS series is slowest
to converge, I think.

> > It's this kind of question that I can't answer.
>
> It would be helpful if you could explicitly state what part
> of my question you don't understand.

All of it. I don't understand your subjective descriptions
of scales/chords/identities at all and seldom if ever have.

> I feel like I've given you enough
> information to understand which specific percept I'm talking
> about now,

Er...

> that I've adequately distinguished between categorical
> perception and f0 estimation, and that no lingering doubt
> should exist about which one I'm getting at.

What makes you think I even know the difference between
"categorical perception" and "f0 estimation"?

-Carl

🔗Mike Battaglia <battaglia01@...>

On Sat, Aug 27, 2011 at 5:53 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > > But if c is getting smaller, that means that L/s is approaching
> > > > 1/1, and the s in 7-equal is larger than the s for the diatonic
> > > > scale in 12-equal...
> > >
> > > So? c = s minimizes the range and makes L/s = 3/2.
> >
> > Right, and I asked how you derived that, and you gave the above
> > analysis, and I pointed out that s in fact does not stay the same,
> > so here we are.
>
> Huh??

It does appear that L/s does minimize the range, which I can see
empirically, but I don't get how you derived that. Your explanation
depends on s staying the same size as c gets smaller, but that doesn't
happen.

> > > > setting L/s = phi sets, in a certain sense, a maximally easy
> > > > to learn structure all the way up and down the entire MOS chain
> > > > and temperament at large, at least as far as range and propriety
> > > > are both concerned.
> > >
> > > Really?
> >
> > It should make it so that every MOS is maximally close
> > to L/s = 3/2, yes?
>
> I don't know. It makes it so that the MOS series is slowest
> to converge, I think.

I think it'll also make it closest to 3/2 but don't have the time to
formally prove it now.

> > It would be helpful if you could explicitly state what part
> > of my question you don't understand.
>
> All of it. I don't understand your subjective descriptions
> of scales/chords/identities at all and seldom if ever have.

How is the term "flat major chord" ambiguous to you? Is it ambiguous
to anyone else?

> > that I've adequately distinguished between categorical
> > perception and f0 estimation, and that no lingering doubt
> > should exist about which one I'm getting at.
>
> What makes you think I even know the difference between
> "categorical perception" and "f0 estimation"?

Consider major thirds in superpyth[7] vs major thirds in meantone[7].
The former are 9/7, whereas the latter are 5/4. The minor thirds in
the former are 7/6, whereas the minor thirds in the latter are 6/5.
Despite this, they're both "minor thirds," in some sense, despite that
they evoke different virtual fundamentals. The former sense reflects
that they evoke a shared categorical perception, and the latter
reflects that they evoke different f0's, where f0 is a term in the
psychoacoustics literature used to refer to the perceived fundamental
tone of a complex pitch.

-Mike

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

On Sat, Aug 27, 2011 at 8:17 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It does appear that L/s does minimize the range, which I can see
> > empirically, but I don't get how you derived that. Your
> > explanation depends on s staying the same size as c gets smaller,
> > but that doesn't happen.
>
> The range expression we came up with doesn't include L,
> so s and c are independent.

c = L-s.

> > How is the term "flat major chord" ambiguous to you?
>
> I know what a flat pitch is. I have no idea what a
> flat chord is.

Let's say a chord in which every dyad is flatter than you'd expect.

> > Despite this, they're both "minor thirds," in some sense,
>
> Yes, they're both the small 3rds of a MOS.

Their gestalt doesn't seem to radically change if you use meantone[5]
vs superpyth[5], nor diminished[8].

> > despite that they evoke different virtual fundamentals
>
> They do?

For me and I wager most people reading this, 7/6 and 6/5 evoke
different VFs for you, as do 9/7 and 5/4. Do they not for you?

-Mike

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I think it'll also make it closest to 3/2 but don't have the time to
> formally prove it now.

Here's a proof. (Specifically, a proof of the statement that L/s = phi minimizes the maximum deviation of L/s from 3/2 for all larger MOSes.)

Clearly if L/s = phi then the deviation for the first MOS is

phi - 3/2

Let the large and small intervals of the next larger MOS be L' and s'. Since phi < 2 we have

L' = s
s' = L - s

L'/s' = s/(L-s) = 1/(L/s-1) = 1/(phi-1) = phi

Therefore, (as I'm sure many of you know), the deviation for the second MOS, and all larger MOSes are the same as for the first one. They're all (phi - 3/2).

Now, assume we have some L/s for which the maximum deviation is smaller than this. That means L/s < phi. But then

L/s - 1 < phi - 1
1/(L/s - 1) > 1/(phi - 1)
1/(L/s - 1) > phi
L'/s' > phi

so the deviation for the next larger MOS is in fact larger than (phi - 3/2). This contradicts our assumption and shows that L/s = phi has the stated optimization property.

All this basically means that there really is something to these "golden section" scales - they're the scales where the "badness" of the step size ratio L/s is smallest in BOTH the "albitonic" and "chromatic" scales (as well as all hyper-chromatic scales too).

L/s = phi scales are also the unique scales for which every larger MOS is strictly proper, but that proof is left as an exercise for the reader. =P

Keenan

🔗Mike Battaglia <battaglia01@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> This does show that if L/s < phi, that the next larger MOS is > phi,
> which I think will end up creating an oscillating pattern (centered
> around phi?) as the structure continues. So in that case it does
> minimize the distance from phi as the MOS structure deepens. But, is
> it possible to create an oscillating pattern that, if you worked out
> the average deviation from 3/2, somehow ended up being less than phi
> would be, despite the oscillation? That's what trips me up.

Note that, at the beginning of my post, I specified the statement I was going to prove, and it had nothing to do with the "average deviation". I proved that L/s = phi minimizes the maximum deviation.

I suspect that the version with the average is also true, but that would require a totally different proof.

Keenan

P.S. Should we kick ourselves off the tuning list and go to tuning-math at some point? I'm sure a lot of people don't care about this.