Yahoo Tuning Groups Ultimate Backup tuning Regular Temperament puzzle that needs to be addressed

While trying to explain regular temperament to Michael, I realized that there's something odd that really needs to be addressed. It's been bugging me for ages but now that the xenwiki is really substantial and good, I think it's long past time to deal with it.

The generator of "miracle" temperament is half an 8/7, or one-sixth a 3/2. The defining commas are listed on the xenwiki as 1029/1024, 225/224. That is, the temperament tempering out the difference between 16/15 and 15/14, as well as the comma between 3*(8/7) and 3/2. The must be one of the slickest temperament schemes ever, great.

Here's the problem: among the equal divisions of the octave to which we may tune "miracle" temperament well, we find both 31 and 72 equal divisions of the octave. But, the sixth root of 3/2 and the sixth root of fourth-root-of-5 are deeply different matters altogether. The same temperament tempering out 81/80 here and not tempering it out there is simply not a concept that is going to fly well- not when the comma is as important as 81/80. Not when 3 and 5 are vital aspects of the temperament.

I realize that the changing number of commas, while retaining the characteristic commas, is a feature, not a flaw, of regular temperaments as concieved and exercised here. But here is an example, right at the historical heart of the regular temperament paradigm, that needs to corrected or further explained, or something.

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 6:04 AM, lobawad <lobawad@...> wrote:
>
> While trying to explain regular temperament to Michael, I realized that there's something odd that really needs to be addressed. It's been bugging me for ages but now that the xenwiki is really substantial and good, I think it's long past time to deal with it.
>
> The generator of "miracle" temperament is half an 8/7, or one-sixth a 3/2. The defining commas are listed on the xenwiki as 1029/1024, 225/224. That is, the temperament tempering out the difference between 16/15 and 15/14, as well as the comma between 3*(8/7) and 3/2. The must be one of the slickest temperament schemes ever, great.
>
> Here's the problem: among the equal divisions of the octave to which we may tune "miracle" temperament well, we find both 31 and 72 equal divisions of the octave. But, the sixth root of 3/2 and the sixth root of fourth-root-of-5 are deeply different matters altogether. The same temperament tempering out 81/80 here and not tempering it out there is simply not a concept that is going to fly well- not when the comma is as important as 81/80. Not when 3 and 5 are vital aspects of the temperament.
>
> I realize that the changing number of commas, while retaining the characteristic commas, is a feature, not a flaw, of regular temperaments as concieved and exercised here. But here is an example, right at the historical heart of the regular temperament paradigm, that needs to corrected or further explained, or something.

Well, first off, you're looking at the 7-limit version of miracle,
which is silly, because the 11-limit one is the real winner here. But
okay, let's work with the 7-limit version, because it might make it
easier to explain.

Think back to Fokker. Let's take a break from miracle and say you're
in 7-limit JI: there are four dimensions that represent every note in
the system: 2, 3, 5, and 7. If you find some comma and temper it, you
will distort the axes of the system so that there are only 3
remaining, each of which represents some combination of a tempered 2,
3, 5, and 7. You now have a rank 3, or a "planar" temperament. If you
temper out one more, you will only have two axes remaining, and now
you're at rank 2, which leads to wonderful things like MOS scales.
Temper out one more and you end up at rank 1, which is an equal
temperament.

An important theorem that results from this is, if you're in a 5-limit
system and you temper out two commas, you will end up at a single
equal temperament. Or, in other words, the only equal temperament that
eliminates 81/80 and 128/125 is 12-equal. If you're starting at the
7-limit, and you temper out three commas, you still just end up at a
single equal temperament. If you're in the 7-limit and you temper out
250/243, 64/63, and 50/49, you will end up with 22-equal and only
22-equal.

So in the case of 7-limit miracle temperament, there are two commas
vanishing - 1029/1024, 225/224 - and you're at rank 2. But, when you
look at different equal temperaments that "support" miracle
temperament, there will necessarily be another comma that vanishes on
top of those, which was the comma that brought you down to rank 1. In
the case of 31-equal, 81/80 vanishes. With 41-equal, which also
supports miracle, the schisma vanishes. With 72-equal, it's the
pythagorean comma.

The "additional" comma actually ends up turning into an infinite
series of commas, for example - in 31-equal, it's not only 81/80 that
vanishes, but also 126/125 as well, since 126/125 and 81/80 are
separated by 225/224, which also vanishes within the miracle system.

The real point is that these temperaments define scales, and in the
case of miracle some important scales defined are the 10, 21, 31 note
MOS's - and their MODMOS's, which have yet to be explored - and these
scales exist whether or not 81/80 vanishes and aren't much affected
either way if it does. I think the 81/80 pun pops up only one time in
the 31-tet version of blackjack, because there ends up being a string
of four fifths, which in 31-tet is equated to 10/3.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 10:16 AM, Mike Battaglia <battaglia01@...> wrote:
>
> The real point is that these temperaments define scales, and in the
> case of miracle some important scales defined are the 10, 21, 31 note
> MOS's - and their MODMOS's, which have yet to be explored - and these
> scales exist whether or not 81/80 vanishes and aren't much affected
> either way if it does. I think the 81/80 pun pops up only one time in
> the 31-tet version of blackjack, because there ends up being a string
> of four fifths, which in 31-tet is equated to 10/3.

Another point is that, someday, I hope these temperaments won't just
define scales, but define entire tonal structures. Right now, they
don't, because we aren't sure what features really define a tonal
structure yet, so it's a bit hit or miss. But hopefully, once we
figure it out, we'll be able to have a miracle-oriented tonality, a
meantone-oriented tonality which we already know, a porcupine-oriented
tonality, etc. And once you're "thinking" in a miracle tonal system,
whether or not the chosen miracle tuning also makes 81/80 vanish will
be sort of a side thing that isn't as important, just like how the
12-tet tuning for meantone makes 128/125 vanish isn't as important if
you're thinking diatonically (but does pop up occasionally).

But we're not there yet, because the regular mapping paradigm isn't
complete yet. I think that a more thorough search of the MODMOS's of
each temperament will help us get there. I also think that Petr's
recent work with comma pumps, and Gene's recent development of the
"minimal" comma pump for each temperament, will also help us get
there. I tend to think that tonality is more a feature of harmony
itself than of scales, but that certain scales can compactly
encapsulate the things that make tonality work, and that the diatonic
scale does this nicely. Anyway, if you have more insight into this,
I'm all ears.

One last point is that I feel that your mentioning of 81/80 vanishing
in the 31-tet version of miracle is one of the drawbacks of thinking
in purely rank 2 systems when you're in a rank 1 tuning. For example,
128/125 and 648/625 vanish in the 12-tet version of meantone, and this
provides additional musical structure outside of the meantone
structure itself. For example, there are very much certain cases where
the 128/125 unison vector is utilized in modern meantone-oriented
music, like the 12-tet altered scale - C Db Eb Fb Gb Ab Bb C - where
Fb and E end up being the same thing in 12-tet. So you have,
effectively C-E-Bb in the 12-tet altered scale, and this scale is used
over dominant 7 chords all the time in jazz. Consequently, if you were
to try and use this scale in 19-tet, the C-Fb would now be something
on the order of 450 cents, and the whole concept wouldn't work as
well.

Another example is if you start with lydian - C D E F# G A B C - and
you sharp the 2: C D# E F# G A B C. Now you have lydian #2, which is a
mode of harmonic minor and a beautiful scale. Now let's say, instead,
that you flat the 7: C D E F# G A Bb C - you get lydian dominant,
which is a mode of melodic minor, and another beautiful scale.
Finally, let's say you want to both sharp the 2 AND flat the 7: you
get C D# E F# G A Bb C. In 12-tet, this is a 7-note subset of
diminished[8]: C Db D# E F# G A Bb C - so it lets you mix
diminished[8] in with diatonic harmony, and suddenly you have this
mixing of temperaments and tonal structures together. This is yet
another technique that is used all the time in jazz (and, at one
point, was used in popular music too).

Analogously to how rank 2 meantone techniques "don't quite work" in
rank 3 5-limit JI, the above rank 1 12-meantone technique "doesn't
quite work" in rank 2 meantone. Nor does it work in 19-meantone or
31-meantone, where 128/125 and 648/625 don't vanish. And this isn't
something you'd realize if you're thinking only in rank-2 systems like
meantone, but it's been a crucial feature of a huge portion of the
music written in the last 100 years. Which is why I prefer to work in
rank-2 "modalities" but in rank-1 tunings, because it leads to an
awesome jumping around of temperaments and tonal structures like that.

-Mike

🔗lobawad <lobawad@...>

LOL. First of all, I'm not looking at 7-limit, but 11:

11-limit
Commas: 1029/1024, 225/224, 243/242

Minimax tuning:
[|1 0 0 0 0>, |25/19 12/19 -6/19 0 0>,
|50/19 -14/19 7/19 0 0>, |55/19 -4/19 2/19 0 0>,
|53/19 30/19 -15/19 0 0>]
Eigenmonzos: 2, 10/9

POTE generator: 116.633
Algebraic generator: Secor59

Map: [<1 1 3 3 2|, <0 6 -7 -2 15|]
EDOs: 10, 31, 41, 72
Badness: 0.0107

See where it says EDOs: 10, 31, 41, 72?

So...eh, I see, I didn't mention 243/242, so whoops. Nevertheless 11-limit miracle can be tuned to 31-edo.

I already understand the content of your response- I specifically said "I realize that the changing number of commas, while retaining the characteristic commas, is a feature, not a flaw, of regular temperaments as concieved and exercised here." In order to obviate such a non-answer (though you wrote it very well, nice).

You're still not addressing the real-life problem of the musically very significant fact that you're tuning a temperament in different ways such that in one way the generator is the sixth root of the fourth root of 5 and in another way the sixth root of 1.5. This needs a footnote at least.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Apr 12, 2011 at 6:04 AM, lobawad <lobawad@...> wrote:
> >
> > While trying to explain regular temperament to Michael, I realized that there's something odd that really needs to be addressed. It's been bugging me for ages but now that the xenwiki is really substantial and good, I think it's long past time to deal with it.
> >
> > The generator of "miracle" temperament is half an 8/7, or one-sixth a 3/2. The defining commas are listed on the xenwiki as 1029/1024, 225/224. That is, the temperament tempering out the difference between 16/15 and 15/14, as well as the comma between 3*(8/7) and 3/2. The must be one of the slickest temperament schemes ever, great.
> >
> > Here's the problem: among the equal divisions of the octave to which we may tune "miracle" temperament well, we find both 31 and 72 equal divisions of the octave. But, the sixth root of 3/2 and the sixth root of fourth-root-of-5 are deeply different matters altogether. The same temperament tempering out 81/80 here and not tempering it out there is simply not a concept that is going to fly well- not when the comma is as important as 81/80. Not when 3 and 5 are vital aspects of the temperament.
> >
> > I realize that the changing number of commas, while retaining the characteristic commas, is a feature, not a flaw, of regular temperaments as concieved and exercised here. But here is an example, right at the historical heart of the regular temperament paradigm, that needs to corrected or further explained, or something.
>
> Well, first off, you're looking at the 7-limit version of miracle,
> which is silly, because the 11-limit one is the real winner here. But
> okay, let's work with the 7-limit version, because it might make it
> easier to explain.
>
> Think back to Fokker. Let's take a break from miracle and say you're
> in 7-limit JI: there are four dimensions that represent every note in
> the system: 2, 3, 5, and 7. If you find some comma and temper it, you
> will distort the axes of the system so that there are only 3
> remaining, each of which represents some combination of a tempered 2,
> 3, 5, and 7. You now have a rank 3, or a "planar" temperament. If you
> temper out one more, you will only have two axes remaining, and now
> you're at rank 2, which leads to wonderful things like MOS scales.
> Temper out one more and you end up at rank 1, which is an equal
> temperament.
>
> An important theorem that results from this is, if you're in a 5-limit
> system and you temper out two commas, you will end up at a single
> equal temperament. Or, in other words, the only equal temperament that
> eliminates 81/80 and 128/125 is 12-equal. If you're starting at the
> 7-limit, and you temper out three commas, you still just end up at a
> single equal temperament. If you're in the 7-limit and you temper out
> 250/243, 64/63, and 50/49, you will end up with 22-equal and only
> 22-equal.
>
> So in the case of 7-limit miracle temperament, there are two commas
> vanishing - 1029/1024, 225/224 - and you're at rank 2. But, when you
> look at different equal temperaments that "support" miracle
> temperament, there will necessarily be another comma that vanishes on
> top of those, which was the comma that brought you down to rank 1. In
> the case of 31-equal, 81/80 vanishes. With 41-equal, which also
> supports miracle, the schisma vanishes. With 72-equal, it's the
> pythagorean comma.
>
> The "additional" comma actually ends up turning into an infinite
> series of commas, for example - in 31-equal, it's not only 81/80 that
> vanishes, but also 126/125 as well, since 126/125 and 81/80 are
> separated by 225/224, which also vanishes within the miracle system.
>
> The real point is that these temperaments define scales, and in the
> case of miracle some important scales defined are the 10, 21, 31 note
> MOS's - and their MODMOS's, which have yet to be explored - and these
> scales exist whether or not 81/80 vanishes and aren't much affected
> either way if it does. I think the 81/80 pun pops up only one time in
> the 31-tet version of blackjack, because there ends up being a string
> of four fifths, which in 31-tet is equated to 10/3.
>
> -Mike
>

🔗lobawad <lobawad@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Apr 12, 2011 at 10:16 AM, Mike Battaglia <battaglia01@...> wrote:
> >
> > The real point is that these temperaments define scales, and in the
> > case of miracle some important scales defined are the 10, 21, 31 note
> > MOS's - and their MODMOS's, which have yet to be explored - and these
> > scales exist whether or not 81/80 vanishes and aren't much affected
> > either way if it does.

I realize that proper "strict miracle" thinking doesn't assume anything about 81/80, why do you think I stopped at 3*(8/7) when introducing the temperament and didn't bring in any relation of 3/2 and 5/4?

But think about this: if we're looking at 3*(8/7) equally "3/2" as a vital and defining element, then we're also looking at modalities of 26-edo which are related to "miracle" in this way.

>
> Another point is that, someday, I hope these temperaments won't just
> define scales, but define entire tonal structures. Right now, they
> don't, because we aren't sure what features really define a tonal
> structure yet, so it's a bit hit or miss.

The first step of "defining tonal structures" is accepting how much of it is about modality, not just the materials involved. The verbs not just the nouns.

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 11:11 AM, lobawad <lobawad@...> wrote:
>
> See where it says EDOs: 10, 31, 41, 72?
>
> So...eh, I see, I didn't mention 243/242, so whoops. Nevertheless 11-limit miracle can be tuned to 31-edo.

You can also tune it to 21, which would serve as a doable miracle
temperament were you stuck on a desert island for the rest of your
life with nothing but a 21-tone bamboo flute.

> I already understand the content of your response- I specifically said "I realize that the changing number of commas, while retaining the characteristic commas, is a feature, not a flaw, of regular temperaments as concieved and exercised here." In order to obviate such a non-answer (though you wrote it very well, nice).
>
> You're still not addressing the real-life problem of the musically very significant fact that you're tuning a temperament in different ways such that in one way the generator is the sixth root of the fourth root of 5 and in another way the sixth root of 1.5. This needs a footnote at least.

Take a square. Now let's say that you round the edges of the square -
you now have an altered version of the original object. It's "a
square," but "with rounded edges." If you only round them 0.01%, you'd
never know it wasn't a square! But, if you round them too much, you
might instead feel like you now have a "circle" that has been "cut
off" at the sides. On the other hand, if you round the edges 99.9%,
you'd never know it wasn't a circle. Or, maybe you would insist that
neither the 0.001% rounded square, nor the 99.99% rounded pretty much
circle, are really a square or a circle at all. You'd insist that both
of these are their own, unique objects, and that you can learn to
distinguish them from a square and a circle.

All of these are analogous to different "mappings" that you can apply
to a "tempered" square. And you are free to apply whatever mapping you
want. The point of "regular mapping" isn't to force judgments about
when a square stops being a square or when it starts being a circle -
it's just a mathematical structure to formalize whatever judgments you
choose to make. It isn't psychoacoustics, it isn't harmonic entropy,
it's pure mathematics. It enables you to make your own judgments, and
then use secret esoteric voodoo witchcraft/mathematical tools to see
what the structure is that those judgments imply. It allows you to see
whatever the 17th-order logical implications of whatever judgments you
made are, because unless you're a genius you aren't going to
immediately see them.

You can map anything to anything you want. You can map the 0.01%
rounded square to a square, which is probably a logical choice. Or,
you could say that the 0.01% rounded square is just a really
straightened out circle... if you wanted. Likewise, if you really
want, you can map 14 cents to 9/7, and the mathematics is there to
help you do it. You can also define the 5-limit rank 2 temperament
that eliminates 5/3 as a "comma," if you want, and there will be a
logically consistent structure that results - and here it is:

http://x31eq.com/cgi-bin/rt.cgi?ets=1_2&error=843.409&limit=5&invariant=1_1_1_0_0

I see that nobody has decided to name this temperament... I wonder
why. Anyway, whether that structure is useful enough to make music is
up to you, which is when you have to leave the realm of pure
mathematics and have to start to making musical decisions, which could
be informed by models like harmonic entropy, or informed instead by an
internal concept that has more to do with JI, and also with your own
varied musical experiences, and so on.

So, back to your example. In 31, the nearest interval to 3/2 is the
697 interval, so most people choose to "map" the 697 cent interval to
3/2. That is, we say it's a 3/2 with some rounded edges. But, maybe
you see things differently. Maybe you feel that in between a square
and a circle lies an object that is its own fundamental entity - and
that the 697 cent interval is its own animal, not 3/2 at all, and that
it is more accurately described as the fourth root of five. You could
do that if you wanted, which would put you in a 2.5^(1/4)-limit
system. Or maybe you'd like to frame things where we're all in a
2.3.5^(1/4) system, but everyone else likes to "temper" out the
"comma" between 5^(1/4) and 3/2, and you don't like to. That is also
fine. People might think your ideas are a bit eccentric, and you might
get into some arguments with people, but the math allows you to do
whatever you want.

The list's psychoacoustic ideas are admittely not as developed as its
mathematical ideas, and that's why I think you shouldn't let your
criticism of harmonic entropy hinder you from learning about regular
mapping. Gene isn't the biggest fan of harmonic entropy either.
There's just an infinite landscape of new musical territory out there,
and the math allows us to narrow things down and find scales and
temperaments that we think will be useful, or have the features we
want for the piece we're writing. The debate over which psychoacoustic
parameters should inform "usefulness" is a side issue which I think
comes up a bit too often on this list - do what you want, use the
math, write music. That's all.

Phew.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 11:36 AM, lobawad <lobawad@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Tue, Apr 12, 2011 at 10:16 AM, Mike Battaglia <battaglia01@...> wrote:
> > >
> > > The real point is that these temperaments define scales, and in the
> > > case of miracle some important scales defined are the 10, 21, 31 note
> > > MOS's - and their MODMOS's, which have yet to be explored - and these
> > > scales exist whether or not 81/80 vanishes and aren't much affected
> > > either way if it does.
>
> I realize that proper "strict miracle" thinking doesn't assume anything about 81/80, why do you think I stopped at 3*(8/7) when introducing the temperament and didn't bring in any relation of 3/2 and 5/4?
>
> But think about this: if we're looking at 3*(8/7) equally "3/2" as a vital and defining element, then we're also looking at modalities of 26-edo which are related to "miracle" in this way.

The temperament that divides 3/2 into three 8/7's with no further
subdivision is called "gamelismic temperament" - miracle requires it
to be divided into 6 parts. But yes - you're right: 26 technically
supports gamelismic temperament, and as you are aware, it's a pretty
crappy tuning for it. So does 16, and that's even worse. 5-equal and
21-equal are some other crappy tunings.

If you're really sadistic, the closest approximation to 3/2 in
11-equal is also split into 3 parts, and each part is the closest
approximation to 8/7, so technically 11-equal is also a gamelismic
temperament, but that's a bit silly.

Since it is a bit silly, it's at this point that you might want to
consider just not mapping 3/2 to anything in 11-equal at all, and
change the prime-limit that you're working within to have a hole in it
at 3 - a subgroup temperament. 11-equal has no useful 3 or 5 to speak
of, but it has a somewhat passable 9/8, albeit not ideal, and so you
can map it as a 2.7.9.11-limit system. It actually works really well
as a 2.7.9.11 subgroup system - it's about as accurate with 4:7:9:11
as 12-equal is with 4:5:6.

So if you apply that mapping to 11-equal instead, then it's not a
gamelismic temperament anymore. And if you think that the fifth of
26-equal is too flat to count as a 3/2, then with your mapping 26
isn't a gamelismic temperament either. On the other hand, I love
26-equal even with its flat fifths, so I don't mind using it like
that.

> > Another point is that, someday, I hope these temperaments won't just
> > define scales, but define entire tonal structures. Right now, they
> > don't, because we aren't sure what features really define a tonal
> > structure yet, so it's a bit hit or miss.
>
> The first step of "defining tonal structures" is accepting how much of it is about modality, not just the materials involved. The verbs not just the nouns.

Wait... that's what I'm saying, because I thought you were saying the opposite.

-Mike

🔗genewardsmith <genewardsmith@...>

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

> Here's the problem: among the equal divisions of the octave to which we may tune "miracle" temperament well, we find both 31 and 72 equal divisions of the octave. But, the sixth root of 3/2 and the sixth root of fourth-root-of-5 are deeply different matters altogether. The same temperament tempering out 81/80 here and not tempering it out there is simply not a concept that is going to fly well- not when the comma is as important as 81/80. Not when 3 and 5 are vital aspects of the temperament.

The basic problem here is that if you add 81/80 to 225/224 and 1029/1024, you don't get the same temperament, you get a different temperament, namely 7-limit 31et. This is an instance of a general phenomenon, which is that a temperament of lower rank can support a temperament of higher rank. If the tuning doesn't change much, the two may be closely identified. This isn't the case here; in 72, a better tuning for miracle than 32, the 81/80 comma remains comma-sized. Tempering it out isn't something a optimized tuning of miracle would suggest. On the other hand for the rank three prodigy temperament, tempering out 225/224 and 441/440, an optimized tuning will shrink 1029/1024 (and 385/384, 243/242 etc.) down to a tiny interval of a fraction of a cent, so we may as well add it, getting miracle. Prodigy is then seen as a sort of 2D layout for miracle, in place of secors.

Another way of looking at it is in terms of complexity. In miracle, the complexity of 81/80 is 31, so it is a complex interval we won't run into in smaller scales like Blackjack. In meantone, 1029/1024 has complexity 31, and again you aren't likely to trip across it.

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 12:59 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > Another point is that, someday, I hope these temperaments won't just
> > define scales, but define entire tonal structures. Right now, they
> > don't, because we aren't sure what features really define a tonal
> > structure yet, so it's a bit hit or miss.
>
> Oh, please. We can say a lot more about it than "Duh".

Yes, but we aren't at the point where we can go to the xenharmonic
wiki, and for every temperament there's a list of not only the MOS
scales but the MODMOS's that have the secret tonal x-factor that makes
the world go round.

Like I said, my personal take is that tonality doesn't really have
much to do with scales, but exists outside of any particular scale
structure. However, I think that the diatonic scale serves as a neat
way to "encapsulate" the useful features of tonality. In the early
20th century, as folks started to experiment more and more with
extended harmonies, switching to diminished[8] for dominant 7 chords
sometimes, or making use of 128/125 as a unison vector other times, we
got further away from needing the simple diatonic "tonal scale"
abstraction and got closer to ascalar tonality. That is, clinging to
the diatonic scale as the source of tonal structure became less and
less important, because folks started to figure out how to still
remain tonal outside of it.

But it was useful, at first, before people realized that those
techniques were there, to treat the diatonic scale as a dumbed down
source of tonal enlightenment to enable someone to write tonal chord
progressions really easily. This is still useful for basic musicians
who don't know anything about extended harmony, but know about I, IV,
and V.

So I wish that, for every temperament, like say miracle, in addition
to listing the MOS's, we also listed the similarly dumbed down "tonal"
MODMOS's that had the same features that make tonality work. Then we
can extend away from them later.

-Mike

🔗Petr Parízek <petrparizek2000@...>

Mike wrote:

> Another point is that, someday, I hope these temperaments won't just
> define scales, but define entire tonal structures. Right now, they
> don't, because we aren't sure what features really define a tonal
> structure yet, so it's a bit hit or miss. But hopefully, once we
> figure it out, we'll be able to have a miracle-oriented tonality, a
> meantone-oriented tonality which we already know, a porcupine-oriented
> tonality, etc.

This is exactly one of the reasons I was so so so much interested in comma pumps characteristic for particular temperaments for almost all the year of 2009.

> But we're not there yet, because the regular mapping paradigm isn't
> complete yet. I think that a more thorough search of the MODMOS's of
> each temperament will help us get there. I also think that Petr's
> recent work with comma pumps, and Gene's recent development of the
> "minimal" comma pump for each temperament, will also help us get
> there.

Gene's development? How's it possible that I didn't know that something like this ever happened? What do you know about it? -- Moreover, not sure how "recent" Gene's work is but my systematic exploration of the shortest possible triadic comma pumps began to penetrate into actual music at the end of 2008. At that time, Carl expressed his interest in the topic so I sent him (offlist, I confess) an unfinished article describing the mathematical procedure to find them. I'll finish it soon, I hope.

Petr

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 1:22 PM, Mike Battaglia <battaglia01@...> wrote:
>
> So I wish that, for every temperament, like say miracle, in addition
> to listing the MOS's, we also listed the similarly dumbed down "tonal"
> MODMOS's that had the same features that make tonality work. Then we
> can extend away from them later.

I should add that I still don't think I fully understand what all of
the necessary features are. I think a high tetrachordality rate may be
one of them.

A high tetrachordality rate is great, but what if you're in a 2.7.9.11
subgroup system? It seems to me that if you want to get away from 3/2
and 4/3, the concept of omnitetrachordality can be generalized to
scale pseudo-periodicity - there are truncated, repeating,
sub-periodic structures within the larger period of the octave.
Graham's tripod stuff with magic temperament was a really interesting
take on this. It's sort of a generalization of the fractional period
octave... sort of.

And then there's propriety, which I have a love/hate relationship with
-- sometimes it seems like it describes my perception 100% perfectly,
and sometimes I feel like it falls completely short of the mark.
Hemiennealimmal[18] is proper, but might as well not be.

MOS is nice because it means that you have major (consonant), minor
(less consonant), and diminished (dissonant) chords sharing the same
triad class, but I haven't found another albitonic MOS scale that has
things work out nicely like that, even for triadic subgroups like
2.7.11. Maybe certain MODMOS's will do the trick.

Aside from that, it also seems to have something to do with leading
tones, which I am starting to think more and more have some
psychoacoustic origin. It also seems to have some other unknown thing
to do with harmony, meaning I don't know why V has to resolve to I.

I think that V7 is basically a really detuned 4:5:6:7, and in jazz
it's sometimes preceded by a II7#11, which is an even more poorly
detuned 4:5:6:7:9:11. I've noticed this pattern of detuned
higher-limit sonorities resolving to lower-limit ones, but I don't
know if it's significant. The detuning makes sense, because you want
to create dissonance on a chord that wants to resolve, but I'm not
sure if the higher limit -> lower limit thing is significant in some
sense or just a coincidence. It might be that the detuned 4:5:6:7:9:11
is like ATTENTION! for a certain fundamental, and then it moves to a
weaker fundamental a 3/2 away, and finally resolves to the I chord
another 3/2 away... ah. How nice. Maybe we could have detuned
4:7:11:13 chords resolving to 4:7:11 chords a 3/2 away or something,
and it'll still work. I dunno.

Anyway, I maybe have a rough idea of how some aspects of it work, but
I don't think I really "get it" yet and I haven't seen it clearly
defined on this list. If you understand it better, I'm all ears. But
if we could figure it out I think all of this would jump mainstream,
because we could tell musicians "here's a scale, here's how it works,
this chord resolves there, and technically it's part of orwell
temperament" and they could jump right in.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Tue, Apr 12, 2011 at 1:29 PM, Petr Parízek <petrparizek2000@...> wrote:
>
> This is exactly one of the reasons I was so so so much interested in comma
> pumps characteristic for particular temperaments for almost all the year of
> 2009.

Yeah, I'm obsessed with them now too. I'll give you a new one to play
with, which I haven't seen posted: load up porcupine[8], all
generators going down, and play 8:14:22:33. Then move around a bit.
Lots of fun stuff to do there.

> > But we're not there yet, because the regular mapping paradigm isn't
> > complete yet. I think that a more thorough search of the MODMOS's of
> > each temperament will help us get there. I also think that Petr's
> > recent work with comma pumps, and Gene's recent development of the
> > "minimal" comma pump for each temperament, will also help us get
> > there.
>
> Gene's development? How's it possible that I didn't know that something like
> this ever happened? What do you know about it?

He just posted this yesterday -

/tuning/topicId_97812.html#97823

Now we can just algorithmically compute the minimal comma pump for any
temperament and use that as a starting point. Maybe if we could get a
list of the pumps by minimality, that would be even better. I still
don't get the algorithm because I haven't studied the concept of the
spectrum of a temperament yet, but the results seem to make sense.

> -- Moreover, not sure how
> "recent" Gene's work is but my systematic exploration of the shortest
> possible triadic comma pumps began to penetrate into actual music at the end
> of 2008. At that time, Carl expressed his interest in the topic so I sent
> him (offlist, I confess) an unfinished article describing the mathematical
> procedure to find them. I'll finish it soon, I hope.
>
> Petr

Can you send it to me offlist too? That would be awesome.

-Mike

🔗Petr Parízek <petrparizek2000@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗Petr Parízek <petrparizek2000@...>

Mike wrote:

> Yeah, I'm obsessed with them now too. I'll give you a new one to play
> with, which I haven't seen posted: load up porcupine[8], all
> generators going down, and play 8:14:22:33. Then move around a bit.

Not sure if I've understood where there's a pump in it.

>Lots of fun stuff to do there.

> He just posted this yesterday -
>
> /tuning/topicId_97812.html#97823

Yeah, this is a crucial step forward. I was doing the same last year but only within 5-limit tempering.

> Now we can just algorithmically compute the minimal comma pump for any
> temperament and use that as a starting point.

This raises the question how far you want to go. Gene's octave-reduced sequence offers a fast ragisma pump but doesn't include a single fifth or fourth. The two "raw" (non-reduced) sequences I posted are significantly longer but include some obvious 7-limit tetrads.

> Maybe if we could get a
> list of the pumps by minimality, that would be even better.

You're losing me now.

> I still
> don't get the algorithm because I haven't studied the concept of the
> spectrum of a temperament yet, but the results seem to make sense.

Neither have I.

> Can you send it to me offlist too? That would be awesome.

If plain text is fine for you, I'll dig it out of there somewhere on my laptop.

Petr

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗chrisvaisvil@...

🔗Daniel Nielsen <nielsed@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗lobawad <lobawad@...>

If you guys look at what you've been saying in this thread, I think you must agree that the definition and identity of temperaments in the regualr temperament paradigm require what amount to specific compositional rules.

This is a good thing in my opinion, it just needs to be clearly stated, and explained on the xenwiki.

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> Mike wrote:
>
> > Yeah, I'm obsessed with them now too. I'll give you a new one to play
> > with, which I haven't seen posted: load up porcupine[8], all
> > generators going down, and play 8:14:22:33. Then move around a bit.
>
> Not sure if I've understood where there's a pump in it.
>
> >Lots of fun stuff to do there.
>
> > He just posted this yesterday -
> >
> > /tuning/topicId_97812.html#97823
>
> Yeah, this is a crucial step forward. I was doing the same last year but
> only within 5-limit tempering.
>
> > Now we can just algorithmically compute the minimal comma pump for any
> > temperament and use that as a starting point.
>
> This raises the question how far you want to go. Gene's octave-reduced
> sequence offers a fast ragisma pump but doesn't include a single fifth or
> fourth. The two "raw" (non-reduced) sequences I posted are significantly
> longer but include some obvious 7-limit tetrads.
>
> > Maybe if we could get a
> > list of the pumps by minimality, that would be even better.
>
> You're losing me now.
>
> > I still
> > don't get the algorithm because I haven't studied the concept of the
> > spectrum of a temperament yet, but the results seem to make sense.
>
> Neither have I.
>
> > Can you send it to me offlist too? That would be awesome.
>
> If plain text is fine for you, I'll dig it out of there somewhere on my
> laptop.
>
> Petr
>

🔗Mike Battaglia <battaglia01@...>

🔗lobawad <lobawad@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Apr 13, 2011 at 12:25 PM, lobawad <lobawad@...> wrote:
> >
> > If you guys look at what you've been saying in this thread, I think you must agree that the definition and identity of temperaments in the regualr temperament paradigm require what amount to specific compositional rules.
> >
> > This is a good thing in my opinion, it just needs to be clearly stated, and explained on the xenwiki.
>
> Feel free to help figure out what those rules might be.
>
> -Mike
>

My recent posts have been rhetorical, as you must have realized by now. Obviously I very well understand the distinguishing principles between miracle and meantone-7, for example, having recently posted a calculated demonstration of non-meantone use of 19-edo on MMM. The point I'm trying to emphasize the importance of is that the xenwiki absolutely needs, in the introduction to regular temperament, a statement to effect "the definition and identity of temperaments in the regular temperament paradigm require what amount to specific compositional rules".

"Compositional rules" sounds bad. I propose "distinguishing modalities". This needs to be explained in the introduction, and distinguishing modalities included as part of the definition of a temperament.

I propose this:

"The definition and identity of a temperament includes the identification and application of modalities which distingish the temperament from other temperaments which may share the same raw tuning material. That is, a temperament is more than the material
with which it is realized, it is also the way these materials are used in musical practice."

This is accurate and necessary- miracle tuned to 31-edo and meantone-7 cannot be distinguished in conception or practice without this understanding. Gene's example of the blackjack scale making the distinction between miracle and meantone-7, even if both are tuned to 31-edo, is an example of a structure which has distinguishing modality "built in".