The Generalized Septimalizer

In semaphore temperament, the following is true:

- If I add 7/4 to 4:5:6, it becomes an otonal 7-limit chord.
- If I add 7/4 to 10:12:15, it becomes a utonal 7-limit chord.

So 7/4 and 12/7 end up becoming the same thing. It's the Generalized
Septimalizer!

This is an extraordinarily useful, non-12 way to think about the
harmonic series. Furthermore, it's proven to also be the most stable
non-12-based way I have so far to experience the harmonic series.
Instead of hearing the 12-EDO "formula" to construct the harmonic
series, I can hear this "formula" instead. Instead of remembering that
there's a "new type of minor third" and a "new type of major second"
(7/6 and 8/7 respectively), I simply remember that there's the New
Generalized Septimalizer which gives me access to a bunch of hip new
sounds. Easy.

Porcupine lends itself to other useful theorems: if I have 3/2, I just
go down by the Magical Porcupine Interval and create 8:11:12,
8:10:11:12, and 8:9:10:11:12, in that order. And then if I go up from
1/1, I get this great maqamic sounding interval, and then a minor
third, and then a fourth. That's a theorem of how the Magical
Porcupine Interval works. And then if I take that, and I put 9/8 below
it, I now have 8:9:10:11:12. Now I'm building new theorems about how
the Magic Porcupine Interval "works" based on the old ones as lemmata.

So now, when I hear the harmonic series, it's pretty clear to me what
the underlying formula is. It's obviously made up of slight variations
of magical porcupine intervals!

Any other useful "theorems" anyone's worked out? This stuff seems to
be quite dramatically related to my categorical perception of
intervals.

-Mike

PS: I note that non-musicians typically don't exhibit very strong
categorical perception. Clearly a good dose of ear training would
teach them that harmonic series is obviously made up of slight
variations of magical porcupine intervals!

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> In semaphore temperament, the following is true:
>
> - If I add 7/4 to 4:5:6, it becomes an otonal 7-limit chord.
> - If I add 7/4 to 10:12:15, it becomes a utonal 7-limit chord.

Why do I care about 10:12:14:15, and what makes it utonal? I say it's otonal.

I mean adding 7/4 to 1/1 6/5 3/2.

-Mike

On Mar 22, 2012, at 12:02 PM, genewardsmith <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> In semaphore temperament, the following is true:
>
> - If I add 7/4 to 4:5:6, it becomes an otonal 7-limit chord.
> - If I add 7/4 to 10:12:15, it becomes a utonal 7-limit chord.

Why do I care about 10:12:14:15, and what makes it utonal? I say it's
otonal.

Anyway, if people don't get the point, this is what I'm saying: say
you're in 5-limit 12-EDO land. You're a budding microtonalist, and you
want to get into this new thing called "the 7-limit."

WHAT MOST PEOPLE DO IN THIS SITUATION:
- "It's clear that the essence of 7-limitness lies in there being more
than one kind of minor seventh. I need to learn the different
comma-shifted variations of the minor seventh now, and its inverse,
and compounds of this with familiar intervals!"

WHAT YOU COULD DO INSTEAD
- "It's obvious that there's this new interval, which lies right
between the major sixth and minor seventh, that turns normal triads
into extended 7-limit tetrads. I need to learn all of the different
comma-shifted variations of this new generic 7-limit interval now, and
its inverse, and compounds of this with familiar intervals!"

-Mike

On Thu, Mar 22, 2012 at 12:39 PM, Mike Battaglia <battaglia01@...>
wrote:
>
> I mean adding 7/4 to 1/1 6/5 3/2.
>
> -Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I mean adding 7/4 to 1/1 6/5 3/2.

Great. Now you have an approximate 11/8 between 7/4 and 12/5. Just how utonal is 1-3/2-7/4-12/5? What about 1-3/2-7/4-77/32?

On Thu, Mar 22, 2012 at 1:45 PM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > I mean adding 7/4 to 1/1 6/5 3/2.
>
> Great. Now you have an approximate 11/8 between 7/4 and 12/5. Just how
> utonal is 1-3/2-7/4-12/5? What about 1-3/2-7/4-77/32?

Please note that I wrote "In semaphore temperament, the following is
true" in my initial post.

-Mike

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

> PS: I note that non-musicians typically don't exhibit very strong
> categorical perception. Clearly a good dose of ear training would
> teach them that harmonic series is obviously made up of slight
> variations of magical porcupine intervals!

212edo has the great 3 and 5 limits of 53, plus a great 7 limit, plus it tempers out 385/384 so all of your septimalizing stuff works better. Ear train people in that.

Not sure why this is so ambiguous, but in case it's not clear, if
49/48 vanishes, 7/4 is the same thing as 12/7. So here are the two
options:

1) adding 7/4 to 1/1 5/4 3/2 gets you a 7-limit tetrad
2) adding 12/7 to 1/1 6/5 3/2 gets you a 7-limit tetrad

In semaphore temperament, 7/4 and 12/7 are The Same Thing. Therefore,
what it actually is is this:

QUOTE UNQUOTE LEMMA
1) adding The Same Thing to 1/1 5/4 3/2 gets you a 7-limit tetrad
2) adding The Same Thing to 1/1 6/5 3/2 gets you a 7-limit tetrad

Thus, the following is true:

QUOTE UNQUOTE THEOREM
1) In semaphore temperament, if I add The Same Thing to any 5-limit
triad, I end up with a 7-limit tetrad.

Then hence suggested that because of this "theorem," The Same Thing is
basically a "Generalized Septimalizer."

Then, I suggested that instead of learning to hear things like 7/4 and
12/7 as specific types of minor seventh and major sixth, that someone
can learn to hear them as specific types of Generalized Septimalizer.

Finally, I noted that these internalized "theorems" strongly influence
my categorical perception of musical intervals. I noted that I'm
growing aware of specific "theorems" about 12-EDO that I'm misapplying
everywhere. I gave the above as an example of a theorem in a new
tuning systems that don't apply in 12-EDO that I've hence learned.

-Mike

On Thu, Mar 22, 2012 at 1:47 PM, Mike Battaglia <battaglia01@...> wrote:
> On Thu, Mar 22, 2012 at 1:45 PM, genewardsmith <genewardsmith@...>
> wrote:
>>
>> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>> >
>> > I mean adding 7/4 to 1/1 6/5 3/2.
>>
>> Great. Now you have an approximate 11/8 between 7/4 and 12/5. Just how
>> utonal is 1-3/2-7/4-12/5? What about 1-3/2-7/4-77/32?
>
> Please note that I wrote "In semaphore temperament, the following is
> true" in my initial post.
>
> -Mike

On Thu, Mar 22, 2012 at 1:58 PM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > PS: I note that non-musicians typically don't exhibit very strong
> > categorical perception. Clearly a good dose of ear training would
> > teach them that harmonic series is obviously made up of slight
> > variations of magical porcupine intervals!
>
> 212edo has the great 3 and 5 limits of 53, plus a great 7 limit, plus it
> tempers out 385/384 so all of your septimalizing stuff works better. Ear
> train people in that.

What useful theorems are there that I should try to internalize about
385/384-tempering?

-Mike

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

> What useful theorems are there that I should try to internalize about
> 385/384-tempering?

(1) 5/4-6/5-7/6-8/7, permuted in any order, is a cool, consonant chord.

(2) The hexany is an 11-limit scale.

On 3/22/2012 11:53 AM, Mike Battaglia wrote:

> So now, when I hear the harmonic series, it's pretty clear to me what
> the underlying formula is. It's obviously made up of slight variations
> of magical porcupine intervals!

I briefly experimented with a "magic-porcupine" notation (back in 2004), although the reason I called it that is that since it had 22 notes, it supported both magic and porcupine temperaments. Basically I extended the porcupine A-G notation all the way up to V.

A P I B Q J C R K D S L E T M F U N G V O H

Dave Keenan preferred a system where A-G represents a chain of fifths, so I modified this notation first to 26 notes and then down to 24. But I think this 22-note version is more elegant in some ways.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Anyway, if people don't get the point, this is what I'm saying: say
> you're in 5-limit 12-EDO land. You're a budding microtonalist, and you
> want to get into this new thing called "the 7-limit."
>
> WHAT MOST PEOPLE DO IN THIS SITUATION:
> - "It's clear that the essence of 7-limitness lies in there being more
> than one kind of minor seventh. I need to learn the different
> comma-shifted variations of the minor seventh now, and its inverse,
> and compounds of this with familiar intervals!"
>

In my experience, most people recognize without hesitation simple ratios of 7 as "blues!".

On Thu, Mar 22, 2012 at 3:08 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What useful theorems are there that I should try to internalize about
> > 385/384-tempering?
>
> (1) 5/4-6/5-7/6-8/7, permuted in any order, is a cool, consonant chord.
>
> (2) The hexany is an 11-limit scale.

OK. That's interesting.

I think the problem is that my internalized sets of "theorems" end up
working from the other direction; i.e. I start with something like
rank-1 and then additional theorems move me up in dimensionality. So
I've already internalized 385/384 tempering because I understand that
The Thing I Call Seven Four plus The Thing I Call Five Four equals An
Octave plus A Neutral Second. But internalizing the 385/384 rank-4
temperament is different, because my current understanding is
necessarily too "coarse" for that.

I hope it's clear that I'm not talking about stuff I know
intellectually, but a deeper intuitive musical understanding of tuning
systems I'm trying to build. Call it "ear training," perhaps.

-Mike

-Mike

On Fri, Mar 23, 2012 at 4:28 AM, lobawad <lobawad@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Anyway, if people don't get the point, this is what I'm saying: say
> > you're in 5-limit 12-EDO land. You're a budding microtonalist, and you
> > want to get into this new thing called "the 7-limit."
> >
> > WHAT MOST PEOPLE DO IN THIS SITUATION:
> > - "It's clear that the essence of 7-limitness lies in there being more
> > than one kind of minor seventh. I need to learn the different
> > comma-shifted variations of the minor seventh now, and its inverse,
> > and compounds of this with familiar intervals!"
> >
>
> In my experience, most people recognize without hesitation simple ratios
> of 7 as "blues!".

OK, but that's unrelated to what I'm talking about here. I'm talking
about the process by which a 12-EDO musician learns to categorize new
musical intervals.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Mar 23, 2012 at 4:28 AM, lobawad <lobawad@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > Anyway, if people don't get the point, this is what I'm saying: say
> > > you're in 5-limit 12-EDO land. You're a budding microtonalist, and you
> > > want to get into this new thing called "the 7-limit."
> > >
> > > WHAT MOST PEOPLE DO IN THIS SITUATION:
> > > - "It's clear that the essence of 7-limitness lies in there being more
> > > than one kind of minor seventh. I need to learn the different
> > > comma-shifted variations of the minor seventh now, and its inverse,
> > > and compounds of this with familiar intervals!"
> > >
> >
> > In my experience, most people recognize without hesitation simple ratios
> > of 7 as "blues!".
>
> OK, but that's unrelated to what I'm talking about here. I'm talking
> about the process by which a 12-EDO musician learns to categorize new
> musical intervals.
>
> -Mike
>

At least two musicians I can recall (contrabass player, slide guitar player) immediately identified 7:4 specifically as "blue seventh". Untrained people just go by feel. I think the categorization of "7-limit" intervals for "12-tEt" musicians is simple and obvious: these intervals are "blue" versions of functional intervals on a more or less 12-tET grid.

Do you mean that you'd like 7-limit intervals not to be heard in this way, but in categories distinct from the usual? That's an interesting problem because, unlike say with the 11-limit, the differences between 7 and 3 or 5 limit intervals don't lend themselves to being perceived as step sizes; 36/35 and 49/48 for example are what we'd usually consider commas rather than steps.

I think it would be very difficult to get 12-tEt musicians not to hear the 7-limit in terms of known categories, and the 11 limit in terms of "quartertones".

And I think that the solution to getting these intervals to be heard on their own terms is obvious: they need categorical (scalar, functional) identities which are not dependent on specific precise tuning.

On Fri, Mar 23, 2012 at 9:19 AM, lobawad <lobawad@...> wrote:
>
> At least two musicians I can recall (contrabass player, slide guitar
> player) immediately identified 7:4 specifically as "blue seventh". Untrained
> people just go by feel. I think the categorization of "7-limit" intervals
> for "12-tEt" musicians is simple and obvious: these intervals are "blue"
> versions of functional intervals on a more or less 12-tET grid.

Correct. That's what I put under "what most people do." They get into
how there's "more than one type of seventh/third/second/etc," the
septimal/7-limit/blue/subminor/whatever one and the
5-limit/classical/meantone/whatever one. That's the usual approach.

> Do you mean that you'd like 7-limit intervals not to be heard in this way,
> but in categories distinct from the usual? That's an interesting problem
> because, unlike say with the 11-limit, the differences between 7 and 3 or 5
> limit intervals don't lend themselves to being perceived as step sizes;
> 36/35 and 49/48 for example are what we'd usually consider commas rather
> than steps.

Most 12-EDO listeners people perceive 36/35 as a "comma" and 49/48 as
a "step." 36/35 switches between different types of the same thing,
and 49/48 switches between a major second and a minor third, or a
major sixth and a minor seventh, etc.

I suggest that there's an alternate way to do things other than the
usual approach above: if one is in 24-EDO or 19-EDO, then 7/4 and 12/7
are the same thing, and they both correspond to an interval "of around
950 cents." This magical interval turns 5-limit triads into 7-limit
tetrads, regardless of the polarity of the original triad.

So an alternate way to conceive of the 7-limit is that there's this
singular new interval to learn, which is in between things that you
know, and that 12/7 and 7/4 are comma-shifted versions of that. Now
49/48's the comma, and 36/35's the step. Now you're thinking of the
7-limit in terms of "quarter tones" rather than thinking of the
11-limit in terms of quarter tones.

> I think it would be very difficult to get 12-tEt musicians not to hear the
> 7-limit in terms of known categories, and the 11 limit in terms of
> "quartertones".

I don't think it's that difficult, after a grand total of a half a
month with a 19-EDO guitar. Basically, just imagine you're not
learning "the 7-limit," but that you're instead making the jump from
meantone to semaphore.

It's almost the entire point of semaphore, as it pertains to JI, to
think of the 7-limit in the way that I said above - 8/7 and 7/6 become
the same thing, and hence 7/4 and 12/7 become the same thing. Someone
who's jumping from meantone to semaphore will quickly learn that
there's this new interval which turns any 5-limit triad they know into
the awesome 7-limit extended version of that triad. If someone's
really internalizing semaphore temperament, then that will be the
cognitive framework with which they understand harmony. Then when they
want to move to JI, 7/4 and 12/7 will just be comma-shifted variants
of the general-purpose septimalizer utility interval that they know so
well.

I think I'm going to start telling people that 7/4 is "the interval
that's about a quarter tone between a minor seventh and a major sixth"
from now on. I'm going to cruelly and evilly split my musician friends
into two groups and teach them totally differently, using my social
status to convince each group separately that the way I'm explaining
it to them is "the right way" to think about microtonal music. Then
I'll have them play music with one another and watch as nobody can
understand what the other is doing. God, I'm such a sadist.

> And I think that the solution to getting these intervals to be heard on
> their own terms is obvious: they need categorical (scalar, functional)
> identities which are not dependent on specific precise tuning.

Yes, this is what I've been trying to say. My whole point is, they can
simply learn that "7-limit stuff" emerges from the -general category-
of what Margo Schulter has called an "interseptimal interval." Then
7/4 and 12/7 emerge as comma-shifted variants of that.

I note also that it's not immediately intuitive for many people to
think of 11/8 as a quarter tone between 500 and 600 cents or whatever.
At first I thought of it as a flat #4, so like a flat version of 600
cents, or something like that. (Neutral thirds and seconds were
different though.)

-Mike

Mike Battaglia <battaglia01@...> wrote:

> Most 12-EDO listeners people perceive 36/35 as a "comma"
> and 49/48 as a "step." 36/35 switches between different
> types of the same thing, and 49/48 switches between a
> major second and a minor third, or a major sixth and a
> minor seventh, etc.

49/48???????

Graham

On Fri, Mar 23, 2012 at 3:40 PM, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
>
> > Most 12-EDO listeners people perceive 36/35 as a "comma"
> > and 49/48 as a "step." 36/35 switches between different
> > types of the same thing, and 49/48 switches between a
> > major second and a minor third, or a major sixth and a
> > minor seventh, etc.
>
> 49/48???????
>
> Graham

Yes. From the perspective of 12-EDO-based interval categories, 49/48
turns a major second (8/7) into a minor third (7/6), so it's an
intonation of diatonic half step. On the other hand, 36/35 turns a
minor third (7/6) into another intonation of minor third (6/5), so
from that perspective it's just a comma.

-Mike

More to the point, <7 11 16 20| sends 36/35 to 0 and
49/48 to 1. -C.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Yes. From the perspective of 12-EDO-based interval categories, 49/48
> turns a major second (8/7) into a minor third (7/6), so it's an
> intonation of diatonic half step. On the other hand, 36/35 turns a
> minor third (7/6) into another intonation of minor third (6/5), so
> from that perspective it's just a comma.
>
> -Mike
>

On Fri, Mar 23, 2012 at 4:24 PM, Carl Lumma <carl@...> wrote:
>
> More to the point, <7 11 16 20| sends 36/35 to 0 and
> 49/48 to 1. -C.

Yeah, there you go. That's a great way to put it. And so from that
perspective, it's a pretty useful alternate categorization structure
for 7-limit stuff to think about <7 11 16 19.5|, to allow for a slight
abuse of notation, instead of <7 11 16 20|. Maybe as I continue to get
more comfortable with this way of thinking, which 19-EDO very much
supports, I'll even start getting used to shopping at <14 22 32 39|
for my generic interval needs.

-Mike

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

>
> Yes. From the perspective of 12-EDO-based interval categories, 49/48
> turns a major second (8/7) into a minor third (7/6), so it's an
> intonation of diatonic half step. On the other hand, 36/35 turns a
> minor third (7/6) into another intonation of minor third (6/5), so
> from that perspective it's just a comma.
>
> -Mike
>

From the perspective of "5-limit common practice", 8/7 sounds like the third (diminished) and 7/6 sounds like the second (augmented). In both cases the pun sounds "natural", the comma being 225/224 (256/225 being the 5-limit c.p. diminished third, 75/64 the augmented second, for anyone following along but not familiar with the tradtional conception here).

But yeah, we know what you mean.

On Sat, Mar 24, 2012 at 6:03 PM, lobawad <lobawad@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> From the perspective of "5-limit common practice", 8/7 sounds like the
> third (diminished) and 7/6 sounds like the second (augmented). In both cases
> the pun sounds "natural", the comma being 225/224 (256/225 being the 5-limit
> c.p. diminished third, 75/64 the augmented second, for anyone following
> along but not familiar with the tradtional conception here).
>
> But yeah, we know what you mean.

Right, but see my recent post about 19-EDO. The augmented second and
minor third both share the 3\meantone[12] interval class, and the
diminished third and the major second both share the 2\meantone[12]
interval class. In this setup, 49/48 is a step and 36/35 is a comma.
Or, as per Carl's way of putting it, in <12 19 28 34|, 36/35 maps to 0
steps and 49/48 maps to 1 step.

-Mike

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

>
> I suggest that there's an alternate way to do things other than the
> usual approach above: if one is in 24-EDO or 19-EDO, then 7/4 and 12/7
> are the same thing, and they both correspond to an interval "of around
> 950 cents." This magical interval turns 5-limit triads into 7-limit
> tetrads, regardless of the polarity of the original triad.

>
> So an alternate way to conceive of the 7-limit is that there's this
> singular new interval to learn, which is in between things that you
> know, and that 12/7 and 7/4 are comma-shifted versions of that. Now
> 49/48's the comma, and 36/35's the step. Now you're thinking of the
> 7-limit in terms of "quarter tones" rather than thinking of the
> 11-limit in terms of quarter tones.

Yes, the link to "quartertones" was immediately apparant. It's a cool idea.

Another way to look at this is a way of reintroducing the idea of augmented sixth and second into a world in which they've been subsumed by minor seventh and third.

19 and 24 are (relatively) very mainstream, and maybe this is a neat link conceptual link between the two.

>
> It's almost the entire point of semaphore, as it pertains to JI, to
> think of the 7-limit in the way that I said above - 8/7 and 7/6 become
> the same thing, and hence 7/4 and 12/7 become the same thing. Someone
> who's jumping from meantone to semaphore will quickly learn that
> there's this new interval which turns any 5-limit triad they know into
> the awesome 7-limit extended version of that triad. If someone's
> really internalizing semaphore temperament, then that will be the
> cognitive framework with which they understand harmony. Then when they
> want to move to JI, 7/4 and 12/7 will just be comma-shifted variants
> of the general-purpose septimalizer utility interval that they know >so
> well.

That's wacked from a 5-limit c.p. point of view, making aug. 2 and dim. 3 comma-altered versions of the same version, but it makes a lot of sense from the viewpoint of harmony based on the (tempered) spectrum.

> I note also that it's not immediately intuitive for many people to
> think of 11/8 as a quarter tone between 500 and 600 cents or >whatever.
> At first I thought of it as a flat #4, so like a flat version of 600
> cents, or something like that. (Neutral thirds and seconds were
> different though.)

I think that might be generally true- 11/9 and 11/10 immediately apparent as "quartertones" but 11/8 sounds like "tritone". In context of course you get the "quartertoneness" of it but anyway, yeah.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 24, 2012 at 6:03 PM, lobawad <lobawad@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > From the perspective of "5-limit common practice", 8/7 sounds like the
> > third (diminished) and 7/6 sounds like the second (augmented). In both cases
> > the pun sounds "natural", the comma being 225/224 (256/225 being the 5-limit
> > c.p. diminished third, 75/64 the augmented second, for anyone following
> > along but not familiar with the tradtional conception here).
> >
> > But yeah, we know what you mean.
>
> Right, but see my recent post about 19-EDO. The augmented second and
> minor third both share the 3\meantone[12] interval class, and the
> diminished third and the major second both share the 2\meantone[12]
> interval class. In this setup, 49/48 is a step and 36/35 is a comma.
> Or, as per Carl's way of putting it, in <12 19 28 34|, 36/35 maps to 0
> steps and 49/48 maps to 1 step.
>
> -Mike
>

Yes, I was just pointing out the difference between 12-tET conception and 5-limit c.p. conception, which is backwards from a blued-notes-in-12-tET point of view.

On Sat, Mar 24, 2012 at 6:17 PM, lobawad <lobawad@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Another way to look at this is a way of reintroducing the idea of
> augmented sixth and second into a world in which they've been subsumed by
> minor seventh and third.
>
> 19 and 24 are (relatively) very mainstream, and maybe this is a neat link
> conceptual link between the two.

Yup, and that conceptual link is semaphore temperament. It's not
really that horrifically high in error, but I always ignored it
because it broke my categories (and yet I never minded dominant in
16-EDO). And now I still am slightly uneasy with it because it's hard
for me to fully get used to, but now I'm warming up to it a lot.

> That's wacked from a 5-limit c.p. point of view, making aug. 2 and dim. 3
> comma-altered versions of the same version, but it makes a lot of sense from
> the viewpoint of harmony based on the (tempered) spectrum.

You also said this in your later reply

> Yes, I was just pointing out the difference between 12-tET conception and
> 5-limit c.p. conception, which is backwards from a blued-notes-in-12-tET
> point of view.

Sorry, I messed up the math here; augmented seconds and -MINOR- thirds
are what share the generic interval class of 3\meantone[12]. The thing
you just said was about augmented seconds and -diminished- thirds.
They're different in meantone[12]; the augmented second is in the
generic class of 3\meantone[12] and the diminished third is in the
class of 2\meantone[12]. However, they're both in the same generic
interval class in meantone[19], which is 4\meantone[19].

And in 19-EDO, they're the same thing, which is really strange and
remarkable at first, because the meantone[12] scale itself is geared
to slightly break 12-EDO categories in a very subtle way.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 24, 2012 at 6:17 PM, lobawad <lobawad@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Another way to look at this is a way of reintroducing the idea of
> > augmented sixth and second into a world in which they've been subsumed by
> > minor seventh and third.
> >
> > 19 and 24 are (relatively) very mainstream, and maybe this is a neat link
> > conceptual link between the two.
>
> Yup, and that conceptual link is semaphore temperament. It's not
> really that horrifically high in error, but I always ignored it
> because it broke my categories (and yet I never minded dominant in
> 16-EDO). And now I still am slightly uneasy with it because it's hard
> for me to fully get used to, but now I'm warming up to it a lot.
>
> > That's wacked from a 5-limit c.p. point of view, making aug. 2 and dim. 3
> > comma-altered versions of the same version, but it makes a lot of sense from
> > the viewpoint of harmony based on the (tempered) spectrum.
>
> You also said this in your later reply

Yes, but I didn't make the distinction between 5-limit c.p. and harmony based on the (tempered) spectrum, which I think is very important.

>
> > Yes, I was just pointing out the difference between 12-tET conception and
> > 5-limit c.p. conception, which is backwards from a blued-notes-in-12-tET
> > point of view.
>
> Sorry, I messed up the math here; augmented seconds and -MINOR- thirds
> are what share the generic interval class of 3\meantone[12]. The thing
> you just said was about augmented seconds and -diminished- thirds.
> They're different in meantone[12]; the augmented second is in the
> generic class of 3\meantone[12] and the diminished third is in the
> class of 2\meantone[12]. However, they're both in the same generic
> interval class in meantone[19], which is 4\meantone[19].
>
> And in 19-EDO, they're the same thing, which is really strange and
> remarkable at first, because the meantone[12] scale itself is geared
> to slightly break 12-EDO categories in a very subtle way.

"However, they're both in the same generic
> interval class in meantone[19], which is 4\meantone[19]." is what I took to be your main point anyway.

By the way, there's another place were 8/7 and 7/6 can be of the same interval class in a "natural" way- if you take for example the tetrachord 1/1, 16/15, 8/7, 4/3, it'll be a tense chromatic tetrachord, and 1/1, 16/15, 7/6, 4/3 a diatonic tetrachord, but in both 8/7 and 7/6 are "middle finger", or "Eb", or whatever you want to call it (the ancient Greek name escapes my memory, have to look it up).

BTW you can do the "generalized septamalizer" in 34-edo. It sounds more "metastable", or something like that, to me than it does "blues!" (i.e. really 7th-partial sounding).

Mike, although I doubt you'll pay any attention to the advice, and this group has not, I would advise against using "utonal" and "otonal" except when specifically discussing things Partchian.

The reason is that the concepts were originally methods of creating a practical analogy, or homology, of a functional major/minor dichotomy.
When used outside of this original context, problems arise. Stick to things like harmonic/subharmonic, is my advice.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I mean adding 7/4 to 1/1 6/5 3/2.
>
> -Mike
>
> On Mar 22, 2012, at 12:02 PM, genewardsmith <genewardsmith@...>
> wrote:
>
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > In semaphore temperament, the following is true:> >
> > - If I add 7/4 to 4:5:6, it becomes an otonal 7-limit chord.
> > - If I add 7/4 to 10:12:15, it becomes a utonal 7-limit chord.
>
> Why do I care about 10:12:14:15, and what makes it utonal? I say it's
> otonal.
>

On Sat, Mar 24, 2012 at 7:22 PM, lobawad <lobawad@...> wrote:
>
> BTW you can do the "generalized septamalizer" in 34-edo. It sounds more
> "metastable", or something like that, to me than it does "blues!" (i.e.
> really 7th-partial sounding).

I note that if you sharpen the generalized septimalizer by a few
cents, it sounds sweeter, kind of how if you flatten the major third
by a few cents it also sounds sweeter.

-Mike

You mean towards 7/6?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Mar 24, 2012 at 7:22 PM, lobawad <lobawad@...> wrote:
> >
> > BTW you can do the "generalized septamalizer" in 34-edo. It sounds more
> > "metastable", or something like that, to me than it does "blues!" (i.e.
> > really 7th-partial sounding).
>
> I note that if you sharpen the generalized septimalizer by a few
> cents, it sounds sweeter, kind of how if you flatten the major third
> by a few cents it also sounds sweeter.
>
> -Mike
>

On Sun, Mar 25, 2012 at 12:31 PM, lobawad <lobawad@...> wrote:
>
> Mike, although I doubt you'll pay any attention to the advice, and this
> group has not, I would advise against using "utonal" and "otonal" except
> when specifically discussing things Partchian.
>
> The reason is that the concepts were originally methods of creating a
> practical analogy, or homology, of a functional major/minor dichotomy.
> When used outside of this original context, problems arise. Stick to
> things like harmonic/subharmonic, is my advice.

I don't get it. Why?

-Mike

On Sun, Mar 25, 2012 at 12:38 PM, lobawad <lobawad@...> wrote:
>
> You mean towards 7/6?

I was thinking more that you meant 7/4, but yeah, I guess towards 7/6
if that's what you care about.

-Mike

Take a look at the Wikipedia article for an example.

http://en.wikipedia.org/wiki/Otonality_and_Utonality

The conceptual error here is glaring.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 25, 2012 at 12:31 PM, lobawad <lobawad@...> wrote:
> >
> > Mike, although I doubt you'll pay any attention to the advice, and this
> > group has not, I would advise against using "utonal" and "otonal" except
> > when specifically discussing things Partchian.
> >
> > The reason is that the concepts were originally methods of creating a
> > practical analogy, or homology, of a functional major/minor dichotomy.
> > When used outside of this original context, problems arise. Stick to
> > things like harmonic/subharmonic, is my advice.
>
> I don't get it. Why?
>
> -Mike
>

On Sun, Mar 25, 2012 at 12:48 PM, lobawad <lobawad@...> wrote:
>
> Take a look at the Wikipedia article for an example.
>
> http://en.wikipedia.org/wiki/Otonality_and_Utonality
>
> The conceptual error here is glaring.

Sorry, I'm not understanding. I think that useful mathematical
definitions can remain useful even if the original author sometimes
applied them in modeling music cognition the wrong way, however. A
mathematical definition is just a definition, no more, no less.

-Mike

I didn't say that there is anything wrong with the original otonal/utonal concept, it worked for Partch. It's when the concept is applied elsewhere that you get inchoherent things.

Look at the Wikipedia article.

"Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples) or "undertones" (divisors) of a given fixed tone. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,...."

Groovy, you can build a homology of the functional major/minor dichotomy from here.

Now look below:

"The 5-limit Otonality is simply a just major chord, and the 5-limit Utonality is a just minor chord. Thus Otonality and Utonality can be viewed as extensions of major and minor tonality respectively. However, whereas standard music theory views a minor chord as being built up from the root with a minor third and a perfect fifth, an Utonality is viewed as descending from what's normally considered the "fifth" of the chord, so the correspondence is not perfect. This corresponds with the dualistic theory of Hugo Riemann."

But the imperfection of the correspondence is greater than that. The example presents (let's call it) I and i as a pair. In order to do this, the otonality is built on C and the utonality on G. What happened to our "given fixed tone"?

If we insist on coherence, the otonal/utonal pair as applied to common practice here would not be I-i, but I-iv.

You can dismiss this as idiotic boring semantics in classic tuning-list style, but it goes to the heart of some very important compositional distinctions.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 25, 2012 at 12:48 PM, lobawad <lobawad@...> wrote:
> >
> > Take a look at the Wikipedia article for an example.
> >
> > http://en.wikipedia.org/wiki/Otonality_and_Utonality
> >
> > The conceptual error here is glaring.
>
> Sorry, I'm not understanding. I think that useful mathematical
> definitions can remain useful even if the original author sometimes
> applied them in modeling music cognition the wrong way, however. A
> mathematical definition is just a definition, no more, no less.
>
> -Mike
>

On Sun, Mar 25, 2012 at 1:12 PM, lobawad <lobawad@...> wrote:
>
> I didn't say that there is anything wrong with the original otonal/utonal
> concept, it worked for Partch. It's when the concept is applied elsewhere
> that you get inchoherent things.
>
> Look at the Wikipedia article.
>
> "Otonality and Utonality are terms introduced by Harry Partch to describe
> chords whose notes are the overtones (multiples) or "undertones" (divisors)
> of a given fixed tone. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,...."
>
> Groovy, you can build a homology of the functional major/minor dichotomy
> from here.

Right, so I'm saying, I think the whole homology thing is a bit silly.
I think there's more to "minor" than utonalness, although I do admit
that at least some of the properties I enjoy about major chords are
captured in the psychoacoustic phenomena that otonal chords cause. But
I don't have to accept that those claims are true in order to just use
the mathematical definition of "utonal" as a term meaning only what
it's exactly defined to mean.

-Mike

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

>
> Right, so I'm saying, I think the whole homology thing is a bit >silly.

Maybe it is, but it worked for Partch, and I think it's cool, and significant, that he concieved of a functional "dichotomy" other than major/minor.

> I think there's more to "minor" than utonalness, although I do admit
> that at least some of the properties I enjoy about major chords are
> captured in the psychoacoustic phenomena that otonal chords cause. But
> I don't have to accept that those claims are true in order to just use
> the mathematical definition of "utonal" as a term meaning only what
> it's exactly defined to mean.

Sure, but you do see that the Wikipedia article is inaccurate in its application of the concept outside of the Partchian, don't you?

On Sun, Mar 25, 2012 at 1:32 PM, lobawad <lobawad@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> >
> > Right, so I'm saying, I think the whole homology thing is a bit >silly.
>
> Maybe it is, but it worked for Partch, and I think it's cool, and
> significant, that he concieved of a functional "dichotomy" other than
> major/minor.

OK, yeah, it's cool if you use those chords artistically to MEAN that
sort of thing, as part of a language you're constructing, but I don't
think that there's some magic psychoacoustic power that makes
upside-down harmonic series chords sound minor. 6:7:9 is pretty minor
to me, whereas 1/1-3/2-12/7 isn't minor to me at all. It's really
interesting and exciting and shiny, I think.

> > I think there's more to "minor" than utonalness, although I do admit
> > that at least some of the properties I enjoy about major chords are
> > captured in the psychoacoustic phenomena that otonal chords cause. But
> > I don't have to accept that those claims are true in order to just use
> > the mathematical definition of "utonal" as a term meaning only what
> > it's exactly defined to mean.
>
> Sure, but you do see that the Wikipedia article is inaccurate in its
> application of the concept outside of the Partchian, don't you?

No, I'm totally lost... Utonal chords are derived from a so-called
"subharmonic series." OK, what's wrong with that?

-Mike

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

> Sorry, I'm not understanding. I think that useful mathematical
> definitions can remain useful even if the original author sometimes
> applied them in modeling music cognition the wrong way, however. A
> mathematical definition is just a definition, no more, no less.

Of course, the Wikipedia article did not give a mathematical definition. If you want one, look here:

http://xenharmonic.wikispaces.com/Otonality+and+utonality

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 25, 2012 at 1:32 PM, lobawad <lobawad@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > >
> > > Right, so I'm saying, I think the whole homology thing is a bit >silly.
> >
> > Maybe it is, but it worked for Partch, and I think it's cool, and
> > significant, that he concieved of a functional "dichotomy" other than
> > major/minor.
>
> OK, yeah, it's cool if you use those chords artistically to MEAN that
> sort of thing, as part of a language you're constructing,

Yes, that's what Partch did and that's cool. Same with "field of attraction" (gravity, magnetism, etc., he uses different words). The alleged psychoacoustic part of that, which gets carried religiously over into tuning theory here, is just part of what the concept means. The principal import is compositional. Partch goes out of his way to point out that he is NOT making universal psychoacoustic claims, yet that's what endures here. Ever seen Life of Brian?

>but I don't
> think that there's some magic psychoacoustic power that makes
> upside-down harmonic series chords sound minor. 6:7:9 is pretty minor
> to me, whereas 1/1-3/2-12/7 isn't minor to me at all. It's really
> interesting and exciting and shiny, I think.

Definitely.

>
> No, I'm totally lost... Utonal chords are derived from a so-called
> "subharmonic series." OK, what's wrong with that?

I pointed out the error in the Wikipedia article here:

/tuning/topicId_104161.html#104212

...and that article is indicative, not unique.

On Sun, Mar 25, 2012 at 1:47 PM, lobawad <lobawad@...> wrote:
>
> Yes, that's what Partch did and that's cool. Same with "field of attraction" (gravity, magnetism, etc., he uses different words). The alleged psychoacoustic part of that, which gets carried religiously over into tuning theory here, is just part of what the concept means. The principal import is compositional. Partch goes out of his way to point out that he is NOT making universal psychoacoustic claims, yet that's what endures here. Ever seen Life of Brian?

I thought you were saying he was. I haven't read much Partch and don't
consider it to be of much but historical interest at this point; I
don't claim to be much of an expert on what he said.

> > No, I'm totally lost... Utonal chords are derived from a so-called
> > "subharmonic series." OK, what's wrong with that?
>
> I pointed out the error in the Wikipedia article here:
>
> /tuning/topicId_104161.html#104212
>
> ...and that article is indicative, not unique.

OK, you're saying that the utonal form of C major would be F minor, or
something to that effect. OK, that's true if you keep the
"fundamental" fixed, but in general C major is an "otonal" chord, and
C minor is a "utonal" chord, assuming a 5-limit tuning. You're correct
that the fundamental changes between the two.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 25, 2012 at 1:47 PM, lobawad <lobawad@...> wrote:
> >
> > Yes, that's what Partch did and that's cool. Same with "field of attraction" (gravity, magnetism, etc., he uses different words). The alleged psychoacoustic part of that, which gets carried religiously over into tuning theory here, is just part of what the concept means. The principal import is compositional. Partch goes out of his way to point out that he is NOT making universal psychoacoustic claims, yet that's what endures here. Ever seen Life of Brian?
>
> I thought you were saying he was. I haven't read much Partch and don't
> consider it to be of much but historical interest at this point; I
> don't claim to be much of an expert on what he said.

He's worth reading at the very least to see some of the thinking of a "Gesamtkunstwerk perfomance artist", and to see how his theoretical ideas are misapplied.

>
> > > No, I'm totally lost... Utonal chords are derived from a so-called
> > > "subharmonic series." OK, what's wrong with that?
> >
> > I pointed out the error in the Wikipedia article here:
> >
> > /tuning/topicId_104161.html#104212
> >
> > ...and that article is indicative, not unique.
>
> OK, you're saying that the utonal form of C major would be F minor, or
> something to that effect. OK, that's true if you keep the
> "fundamental" fixed, but in general C major is an "otonal" chord, and
> C minor is a "utonal" chord, assuming a 5-limit tuning. You're correct
> that the fundamental changes between the two.

But keeping the fundamental fixed is essential to the concept. It is a misapplication to present a major/minor dichotomy in the way it is presented in the article.

On Sun, Mar 25, 2012 at 2:10 PM, lobawad <lobawad@...> wrote:
>
> But keeping the fundamental fixed is essential to the concept. It is a
> misapplication to present a major/minor dichotomy in the way it is presented
> in the article.

It's not a misapplication if you simply say that major chords in
general are otonal, and that minor chords in general are utonal,
without referencing what the so-called fundamental of the various
series are supposed to be.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Mar 25, 2012 at 2:10 PM, lobawad <lobawad@...> wrote:
> >
> > But keeping the fundamental fixed is essential to the concept. It is a
> > misapplication to present a major/minor dichotomy in the way it is presented
> > in the article.
>
> It's not a misapplication if you simply say that major chords in
> general are otonal, and that minor chords in general are utonal,
> without referencing what the so-called fundamental of the various
> series are supposed to be.
>
> -Mike
>

"If you take the concept out of its original useful and interesting context and use it an obscure synonym for harmonic/subharmonic in general..."

Sure, and that's what the Wikipedia article does. The problem is that, as clearly illustrated by the image in the article, you are taking a new concept and strapping it right back onto the old procrustean bed of major/minor thinking.