Yahoo Tuning Groups Ultimate Backup makemicromusic 11-EDO and Godzilla

I was bored in class today and decided to investigate the flame-wars at MMM, and was pleasantly surprised to instead find discussions of 11-EDO as well as Godzilla temperament! Two of my favorite, uh, "intonational systems"(?)!

To share my happiness at seeing MMM restored to its music-oriented glory, I thought I'd link to the two (old) pieces I've written in 11-EDO, one on "Map of an Internal Landscape" and the other on "Early Microtonal Works":

"She is My Lilac-Hued Obsession"
http://tinyurl.com/4ozpvzs

"Breakdown Hives"
http://tinyurl.com/4sro4k9

The first one uses the 6-note MOS of "Machine" temperament, though it does not do much harmonically. In the 2nd one, the first half is in 22-EDO treated as two chains of 11-EDO an approximate 11/10 apart, so two arpeggios of 11-EDO chords are harmonized in parallel neutral 2nds and "quartertones". Then the fuzz pedal goes on and it's just one chain of 11-EDO, playing in (IIRC) the 7-note MOS of LLsLsLs, which I don't know what kind of temperament it is but the generator is 11-EDO's sharp minor 3rd.

Also, another link from "MoaIL", to the piece in 19-EDO's Godzilla[9]:

"Change is on the Wind"
http://tinyurl.com/4z648ft

This one really takes advantage of the fact that in Godzilla[9], all the triads have two types of 3rd (as is really the case in all 9-note MOS scales). I really, really love Godzilla in 19. It's melodically awesome, harmonically pleasant but exotic, and structurally fascinating. I'm coming to love Negri for the same reasons; both of these are great modalities for 19 that take it far, far away from the bland meantone territory that it is mostly known for. Once I get finished with all my "max dissonance" stuff, I'm going to get to work on an album that is focused on 16 and 19 (they make great contrast with each other), and I plan on really going to town with Negri and Godzilla. Are there any compositions in Negri floating around out there? I'd love to hear them if there are.

And of course I must give kudos to JLS--Jaunt is awesome and I love the way you are using these sounds. This is an excellent example of how to appropriately use these kinds of sounds. It's great that you're really just "going with it" and making it an aesthetic, and it's an aesthetic I can really relate to (as I've gone in similar directions myself). Once you've written maybe 6 or 7 more tracks, you should put 'em together as an album!

And to Cambert Bobrawad--like that you are posting music. Your identity could not be more clear. More, more, more!

-Igs

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

🔗genewardsmith <genewardsmith@...>

--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Can somebody tell me what Godzilla temperament is? There are mentions
> of 9 and 19 but 9&19 is Negri.

What do I need to add to the discussion here:

http://xenharmonic.wikispaces.com/Meantone+family

to make it clearer? That tells you that Godzilla tempers out 49/48 and 81/80, with mapping [<1 0 -4 2|, <0 2 8 1|] and wedgie <<2 8 1 8 -4 -20||, and that the generator can be taken to be 4\19. On this page

http://xenharmonic.wikispaces.com/Optimal+patent+val

it tells you that Godzilla is 7-limit rank two, define by patent vals 5&19. It doesn't say it can be extended to the 13-limit in a couple of ways, the details of which are not very relevant as you are going to tune it in 19edo anyway.

By the way, if someone can tell me how the header on this page

http://xenharmonic.wikispaces.com/Regular+Temperaments

is created it would help, as I could do the same on other pages.

🔗Aaron Krister Johnson <aaron@...>

🔗Carl Lumma <carl@...>

🔗Aaron Krister Johnson <aaron@...>

This kind of answer basically makes me want to reiterate that we need a
'tuning math for dummies' page in the xenwiki.

Since these terms ("monzo", "val", "patent val", "mapping"---well okay,
that's not yours, but in the context of tuning, it is) are neologisms
largely made up by Gene Ward Smith, not to mention the entire family naming
system (you and Graham and maybe Paul Erlich---sorry if I'm leaving anyone
out, but I'm not up on this history); everyone else who is curious feels
like it's a private club whose means of obtaining membership is to ask and
ask again for clearer definitions. I'm curious enough that I would like to
follow, but even the xenwiki "let's break this down" definitions give me a
headache. One suggestion might be to make visuals (a clock is an excellent
visual of pitch space, where generators and periods are no-brainers) that
relate these concepts non-verbally as much as possible.

By this, I mean that you've documented this stuff very very well, but not
from the standpoint of anyone who wants to really "get it" without getting a
PhD in higher mathematics (you've routinely casually tossed around terms
like 'field', 'Lie group', etc. as if most people would immediately
understand, where I can guarantee that maybe 5% or fewer folks really do)

Anyway, I don't mean to rant, but another point---maybe this very
interesting conversation should be happening over on the tuning- or
tuning-math lists anyway? :) Jon has repeatedly made appeals to all to keep
this forum more strictly related to discussions around making and composing
the stuff that may or may not use these newly crafted pitch spaces.

Best,
AKJ

On Wed, Mar 9, 2011 at 11:09 AM, genewardsmith
<genewardsmith@...>wrote:

>
> --- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> > Can somebody tell me what Godzilla temperament is? There are mentions
> > of 9 and 19 but 9&19 is Negri.
>
> What do I need to add to the discussion here:
>
> http://xenharmonic.wikispaces.com/Meantone+family
>
> to make it clearer? That tells you that Godzilla tempers out 49/48 and
> 81/80, with mapping [<1 0 -4 2|, <0 2 8 1|] and wedgie <<2 8 1 8 -4 -20||,
> and that the generator can be taken to be 4\19. On this page
>
> http://xenharmonic.wikispaces.com/Optimal+patent+val
>
> it tells you that Godzilla is 7-limit rank two, define by patent vals 5&19.
> It doesn't say it can be extended to the 13-limit in a couple of ways, the
> details of which are not very relevant as you are going to tune it in 19edo
> anyway.
>
> By the way, if someone can tell me how the header on this page
>
> http://xenharmonic.wikispaces.com/Regular+Temperaments
>
> is created it would help, as I could do the same on other pages.
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

[Non-text portions of this message have been removed]

🔗Aaron Krister Johnson <aaron@...>

🔗Carl Lumma <carl@...>

Are you following Tuning at all? -C.

At 12:47 PM 3/9/2011, you wrote:
>You're joking, right?
>
>On Wed, Mar 9, 2011 at 2:41 PM, Carl Lumma <carl@...> wrote:
>
>> Aaron: You couldn't possibly understand, sorry. Just use the
>> scale and accept it. -Carl
>>
>> At 12:36 PM 3/9/2011, you wrote:
>> >While we are at it, what does the '&' do in such notations: e.g. '9&19'
>> >
>> >AKJ
>> >
>> >On Wed, Mar 9, 2011 at 5:05 AM, Graham Breed <gbreed@...> wrote:
>> >
>> >> On 9 March 2011 00:06, cityoftheasleep <igliashon@...> wrote:
>> >> > I was bored in class today and decided to investigate the flame-wars
>> at
>> >> MMM, and was pleasantly surprised to instead find discussions of 11-EDO
>> as
>> >> well as Godzilla temperament! Two of my favorite, uh, "intonational
>> >> systems"(?)!
>> >>
>> >> Can somebody tell me what Godzilla temperament is? There are mentions
>> >> of 9 and 19 but 9&19 is Negri.
>> >>
>> >>
>> >> Graham
>>
>>
>>
>> ------------------------------------
>>
>> Yahoo! Groups Links
>>
>>
>>
>>
>
>
>--
>Aaron Krister Johnson
>http://www.akjmusic.com
>http://www.untwelve.org
>
>
>[Non-text portions of this message have been removed]
>
>
>
>------------------------------------
>
>Yahoo! Groups Links
>
>
>

🔗Aaron Krister Johnson <aaron@...>

No Carl, I'm not as much following the big tuning list these days.

My problem(s) are:

1) that the discoveries of these pitch spaces, largely by GWS, etc., are
much much much further ahead than the decent music being produced in them.
However for reasons below, I don't instinctually feel they hold the promise
of systems I already grok pretty well...let me continue:

2) These families and family names are much much much less meaningful to me
than good old fashioned "this is the period, generator", or "this is the MOS
scale that happens when you have N iterations of this generator within this
EDO", etc. etc.

3) I've been unimpressed with the few random samples of scales that are up
on the scalesmith pages (so far). I remain open, but often, I don't find
that they are immediately musically compelling and "calling" to me. Of
course, this is now, and me now, that might change.

4) It's enough for more than one lifetime to continue to explore a "basic"
good old-fashioned system like _boring_ 19-equal in all its shades moods and
modalities (e.g. whispers of a non-meantone way of looking at it like
"magic" or "negri")

To summarize, I haven't been following tuning b/c I'm afraid my time is
limited, and my limited sniffing leads me to believe its a timesink at best
(lord knows this list is!), and a compositional blind alley at worst, and I
have to make instinctual priorities. That said, it's interesting enough that
I want to make sure I don't shut the door on really potentially interesting
and practical stuff for music making.

Fair enough?

AKJ

On Wed, Mar 9, 2011 at 2:50 PM, Carl Lumma <carl@...> wrote:

> Are you following Tuning at all? -C.
>
> At 12:47 PM 3/9/2011, you wrote:
> >You're joking, right?
> >
> >On Wed, Mar 9, 2011 at 2:41 PM, Carl Lumma <carl@...> wrote:
> >
> >> Aaron: You couldn't possibly understand, sorry. Just use the
> >> scale and accept it. -Carl
> >>
> >> At 12:36 PM 3/9/2011, you wrote:
> >> >While we are at it, what does the '&' do in such notations: e.g.
> '9&19'
> >> >
> >> >AKJ
> >> >
> >> >On Wed, Mar 9, 2011 at 5:05 AM, Graham Breed <gbreed@...> wrote:
> >> >
> >> >> On 9 March 2011 00:06, cityoftheasleep <igliashon@...>
> wrote:
> >> >> > I was bored in class today and decided to investigate the
> flame-wars
> >> at
> >> >> MMM, and was pleasantly surprised to instead find discussions of
> 11-EDO
> >> as
> >> >> well as Godzilla temperament! Two of my favorite, uh, "intonational
> >> >> systems"(?)!
> >> >>
> >> >> Can somebody tell me what Godzilla temperament is? There are
> mentions
> >> >> of 9 and 19 but 9&19 is Negri.
> >> >>
> >> >>
> >> >> Graham
> >>
> >>
> >>
> >> ------------------------------------
> >>
> >> Yahoo! Groups Links
> >>
> >>
> >>
> >>
> >
> >
> >--
> >Aaron Krister Johnson
> >http://www.akjmusic.com
> >http://www.untwelve.org
> >
> >
> >[Non-text portions of this message have been removed]
> >
> >
> >
> >------------------------------------
> >
> >Yahoo! Groups Links
> >
> >
> >
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

[Non-text portions of this message have been removed]

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

--- In MakeMicroMusic@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:
>
> No Carl, I'm not as much following the big tuning list these days.
>
> My problem(s) are:
>
> 1) that the discoveries of these pitch spaces, largely by GWS, etc., are
> much much much further ahead than the decent music being produced in them.

An unsurprising situation considering they've only recently been developed.

> 2) These families and family names are much much much less meaningful to me
> than good old fashioned "this is the period, generator", or "this is the MOS
> scale that happens when you have N iterations of this generator within this
> EDO", etc. etc.

This more or less says that any rank three or higher scale, including the ones in just intonation, are not of interest to you, but it hardly means they won't be of use or interest to anyone else.

> 3) I've been unimpressed with the few random samples of scales that are up
> on the scalesmith pages (so far).

And other people have been effusive in their praise. To each their own.

🔗cityoftheasleep <igliashon@...>

--- In MakeMicroMusic@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:
>
> This kind of answer basically makes me want to reiterate that we need a
> 'tuning math for dummies' page in the xenwiki.

It's pretty simple when you break it down to brass tacks, actually. Start with JI. Imagine a scale where the only note is a 1/1--no octaves, even. This is a 0-dimensional scale. We can't get any other notes because 1 to any power is always 1. Now add a 2nd interval, any interval will do but let's say 2/1 for convention's sake. We now have a 1-dimensional scale, because we can go up or down from 1/1 by powers of 2/1. Adding another interval---3/1, for instance--gets us to a 2-dimensional tuning-space where we can move along either or both of two axes (powers of 2 and/or powers of 3). Adding another interval--5/1, say--gets us to a 3-dimensional space. Every time we add an interval that is not expressible as some combination of the intervals we already have, we are adding a dimension to tuning space.

In JI, all the dimensions of tuning space are open and "flat" and infinite--you can move as far as you want along any combination of them and you'll never return to the origin. When we temper, we are essentially "folding" or "curling" tuning space in such a way that it becomes possible to return to the origin of 1/1 by moving a finite number of steps along some dimensions/axes. This happens because we equate an interval (usually a comma) with 1/1 so that that comma ceases to exist--it becomes "one" with the unison. I think of this in terms of "fudging" the rules of math so that two distinct numbers become equated--it's like we're defining a universe in which 2+2=5. What this also does is equate movement along one dimension with movement along other dimensions. IOW, with the example of meantone temperament, we can now get to places on the 5 axis by moving only along the 2 and 3 axes.

This where mappings come in: depending on the comma we temper out, we will create different rules for movement in our folded space. A mapping is nothing more than a set of directions that tells you how far you far you have to go along the available axes of your tempered tuning space to get to the places that existed in the untempered tuning space.

In a rank-2 temperament, we have only two axes: the period and the generator. When we give a mapping for the temperament, as determined by the comma, one way to give it is by giving a val for each axis such that the two vals together are coordinates that correspond (usually) to the successive primes. A rank-2 temperament can be derived from any level of JI dimensionality, but we will have to temper out more commas in order to reduce higher dimensions down to two. For instance, we can give 5-limit meantone with the two vals <1 1 0] and <0 1 4], where the first val is the period, the second is the generator. Taking the first number from each val, we have the coordinate (1, 0) or "one period up, zero generators up", and that is how we get to 2/1. The second number from each val gives (1, 1), or "one period up, one generator up" and that gets us to 3/1. The third numbers give (0, 4) or "zero periods up, four generators up" and that gets us to 5/1.

We can reduce rank-2 temperaments to rank-1 temperaments, which are equal tunings (though not necessarily EDOs), by further tempering out another comma. Any time we temper out a comma, we reduce the dimensionality of tuning-space. Doing this means we can specify the mapping with a single val, and our generator is always going to be one step of the equal scale. So the 5-limit val for 12-TET would be <12 19 28]; a rank-1 val is also known as a "Breed". Of course, we can give mappings for tunings that are not the most accurate mappings; for 7-EDO, we could map 5/1 to its perfect 4th rather than its neutral 3rd, and give a val of <7 11 17]. A "patent val" is the val that corresponds to the most accurate mapping of JI to steps of an ET.

Does that help?

Also, the whole random naming thing--this used to bother the crap out of me, but I've since come to accept that with the plethora of things that need naming, there's no good clear logical way to go about it.

-Igs

🔗Aaron Krister Johnson <aaron@...>

Igs,

I'm cc'ing this to tuning, because I think we should continue anything from
this thread over there......but I appreciate your taking the time to lay
this out here.

Much of this I already knew, but I think this is a nice executive summary
for newbies and seasoned experts nonetheless. I didn't know the last part
about a "Breed", but the dimesionality part (an N-dimensional space
corresponding to given primes) is of course old hat to me. You also managed
to make the val concept much less ambiguous in such a way that it will
really stick this time.

But more importantly, this shows that you are engaged in making people
understand things in as intuitive a way as possible, which is a big plus in
our field of work. I think using what the brain tends to do best---visual
and geometric intuition, is crucial to get these concepts across.

Anyway, Igs--you or Carl I think are excellent candidates for bridging the
"comprehension gap" in the prose at the xenharmonic wiki---i.e., bringing it
down to the mere mortal level.

Best,
AKJ

On Wed, Mar 9, 2011 at 4:46 PM, cityoftheasleep <igliashon@...>wrote:

> --- In MakeMicroMusic@yahoogroups.com, Aaron Krister Johnson <aaron@...>
> wrote:
> >
> > This kind of answer basically makes me want to reiterate that we need a
> > 'tuning math for dummies' page in the xenwiki.
>
> It's pretty simple when you break it down to brass tacks, actually. Start
> with JI. Imagine a scale where the only note is a 1/1--no octaves, even.
> This is a 0-dimensional scale. We can't get any other notes because 1 to
> any power is always 1. Now add a 2nd interval, any interval will do but
> let's say 2/1 for convention's sake. We now have a 1-dimensional scale,
> because we can go up or down from 1/1 by powers of 2/1. Adding another
> interval---3/1, for instance--gets us to a 2-dimensional tuning-space where
> we can move along either or both of two axes (powers of 2 and/or powers of
> 3). Adding another interval--5/1, say--gets us to a 3-dimensional space.
> Every time we add an interval that is not expressible as some combination
> of the intervals we already have, we are adding a dimension to tuning space.
>
> In JI, all the dimensions of tuning space are open and "flat" and
> infinite--you can move as far as you want along any combination of them and
> you'll never return to the origin. When we temper, we are essentially
> "folding" or "curling" tuning space in such a way that it becomes possible
> to return to the origin of 1/1 by moving a finite number of steps along some
> dimensions/axes. This happens because we equate an interval (usually a
> comma) with 1/1 so that that comma ceases to exist--it becomes "one" with
> the unison. I think of this in terms of "fudging" the rules of math so that
> two distinct numbers become equated--it's like we're defining a universe in
> which 2+2=5. What this also does is equate movement along one dimension
> with movement along other dimensions. IOW, with the example of meantone
> temperament, we can now get to places on the 5 axis by moving only along the
> 2 and 3 axes.
>
> This where mappings come in: depending on the comma we temper out, we will
> create different rules for movement in our folded space. A mapping is
> nothing more than a set of directions that tells you how far you far you
> have to go along the available axes of your tempered tuning space to get to
> the places that existed in the untempered tuning space.
>
> In a rank-2 temperament, we have only two axes: the period and the
> generator. When we give a mapping for the temperament, as determined by the
> comma, one way to give it is by giving a val for each axis such that the two
> vals together are coordinates that correspond (usually) to the successive
> primes. A rank-2 temperament can be derived from any level of JI
> dimensionality, but we will have to temper out more commas in order to
> reduce higher dimensions down to two. For instance, we can give 5-limit
> meantone with the two vals <1 1 0] and <0 1 4], where the first val is the
> period, the second is the generator. Taking the first number from each val,
> we have the coordinate (1, 0) or "one period up, zero generators up", and
> that is how we get to 2/1. The second number from each val gives (1, 1), or
> "one period up, one generator up" and that gets us to 3/1. The third
> numbers give (0, 4) or "zero periods up, four generators up" and that gets
> us to 5/1.
>
> We can reduce rank-2 temperaments to rank-1 temperaments, which are equal
> tunings (though not necessarily EDOs), by further tempering out another
> comma. Any time we temper out a comma, we reduce the dimensionality of
> tuning-space. Doing this means we can specify the mapping with a single
> val, and our generator is always going to be one step of the equal scale.
> So the 5-limit val for 12-TET would be <12 19 28]; a rank-1 val is also
> known as a "Breed". Of course, we can give mappings for tunings that are
> not the most accurate mappings; for 7-EDO, we could map 5/1 to its perfect
> 4th rather than its neutral 3rd, and give a val of <7 11 17]. A "patent
> val" is the val that corresponds to the most accurate mapping of JI to steps
> of an ET.
>
> Does that help?
>
> Also, the whole random naming thing--this used to bother the crap out of
> me, but I've since come to accept that with the plethora of things that need
> naming, there's no good clear logical way to go about it.
>
> -Igs
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

[Non-text portions of this message have been removed]

🔗Herman Miller <hmiller@...>

🔗hstraub64 <straub@...>

🔗Daniel Forró <dan.for@...>

Sorry to say that this is quite unusable from didactic point of view for somebody who wants to learn something about tuning. I say this as a long-year and not unexperienced language and music teacher. It can't help at all.

You just wrote some rules, starting with JI (why?) but you didn't explain any of basic terms used there and the reason what's behind, WHY it should be so and why we should accept it as a truth, fact, and start to work with it and continue to derive something from it. Each more clever student will ask such questions from curiosity and for better understanding. Teaching/learning is a dialogue and understanding the principles behind, not memorizing some dogmas and rules.
Also WHO invented those rules, WHEN, WHY, why we need them, for what it's good, what it offers for music, how it can be reflected and used in music... And then students need more examples, music examples, audio examples, pictures, graphs...

Besides there are some sentences which don't give any sense - for example why you call "a scale which has only one note" a scale (or in other words - who needs "0-dimensional scale", for what it's usable, and why it's called "scale"), and similar. Some words are used in a different way than in common music language, like temperament, tempering, period, generator...

So this is still too complex for potential student of microtonality. Starting themes should be more simple, and done really didactically, not like this.

Such explanating primers and articles must be done in cooperation with a layman, or any person who will ask such "stupid" questions and need to explain them. It can't be written just by some microtuning expert, who is too far and deep in his world that he is unable to imagine that somebody can think in a different way, more simple and straight.

Daniel Forro

On Mar 10, 2011, at 7:46 AM, cityoftheasleep wrote:

> --- In MakeMicroMusic@yahoogroups.com, Aaron Krister Johnson > <aaron@...> wrote:
>>
>> This kind of answer basically makes me want to reiterate that we >> need a
>> 'tuning math for dummies' page in the xenwiki.
>
> It's pretty simple when you break it down to brass tacks, > actually. Start with JI. Imagine a scale where the only note is a > 1/1--no octaves, even. This is a 0-dimensional scale. We can't > get any other notes because 1 to any power is always 1. Now add a > 2nd interval, any interval will do but let's say 2/1 for > convention's sake. We now have a 1-dimensional scale, because we > can go up or down from 1/1 by powers of 2/1. Adding another > interval---3/1, for instance--gets us to a 2-dimensional tuning-> space where we can move along either or both of two axes (powers of > 2 and/or powers of 3). Adding another interval--5/1, say--gets us > to a 3-dimensional space. Every time we add an interval that is > not expressible as some combination of the intervals we already > have, we are adding a dimension to tuning space.
>
> In JI, all the dimensions of tuning space are open and "flat" and > infinite--you can move as far as you want along any combination of > them and you'll never return to the origin. When we temper, we are > essentially "folding" or "curling" tuning space in such a way that > it becomes possible to return to the origin of 1/1 by moving a > finite number of steps along some dimensions/axes. This happens > because we equate an interval (usually a comma) with 1/1 so that > that comma ceases to exist--it becomes "one" with the unison. I > think of this in terms of "fudging" the rules of math so that two > distinct numbers become equated--it's like we're defining a > universe in which 2+2=5. What this also does is equate movement > along one dimension with movement along other dimensions. IOW, > with the example of meantone temperament, we can now get to places > on the 5 axis by moving only along the 2 and 3 axes.
>
> This where mappings come in: depending on the comma we temper out, > we will create different rules for movement in our folded space. A > mapping is nothing more than a set of directions that tells you how > far you far you have to go along the available axes of your > tempered tuning space to get to the places that existed in the > untempered tuning space.
>
> In a rank-2 temperament, we have only two axes: the period and the > generator. When we give a mapping for the temperament, as > determined by the comma, one way to give it is by giving a val for > each axis such that the two vals together are coordinates that > correspond (usually) to the successive primes. A rank-2 > temperament can be derived from any level of JI dimensionality, but > we will have to temper out more commas in order to reduce higher > dimensions down to two. For instance, we can give 5-limit meantone > with the two vals <1 1 0] and <0 1 4], where the first val is the > period, the second is the generator. Taking the first number from > each val, we have the coordinate (1, 0) or "one period up, zero > generators up", and that is how we get to 2/1. The second number > from each val gives (1, 1), or "one period up, one generator up" > and that gets us to 3/1. The third numbers give (0, 4) or "zero > periods up, four generators up" and that gets us to 5/1.
>
> We can reduce rank-2 temperaments to rank-1 temperaments, which are > equal tunings (though not necessarily EDOs), by further tempering > out another comma. Any time we temper out a comma, we reduce the > dimensionality of tuning-space. Doing this means we can specify > the mapping with a single val, and our generator is always going to > be one step of the equal scale. So the 5-limit val for 12-TET > would be <12 19 28]; a rank-1 val is also known as a "Breed". Of > course, we can give mappings for tunings that are not the most > accurate mappings; for 7-EDO, we could map 5/1 to its perfect 4th > rather than its neutral 3rd, and give a val of <7 11 17]. A > "patent val" is the val that corresponds to the most accurate > mapping of JI to steps of an ET.
>
> Does that help?
>
> Also, the whole random naming thing--this used to bother the crap > out of me, but I've since come to accept that with the plethora of > things that need naming, there's no good clear logical way to go > about it.
>
> -Igs

🔗jonszanto <jszanto@...>

🔗Carl Lumma <carl@...>

🔗Graham Breed <gbreed@...>

Aaron Krister Johnson <aaron@...> wrote:
> While we are at it, what does the '&' do in such
> notations: e.g. '9&19'

It means you combine two equal temperaments to get a rank 2
temperament class. The symbol & is related to addition.
Meantone could be 7&12 and 7+12=19 is also a meantone
division. (I'm not sure if 28 works for Negri.) But also,
& is related to union. Negri is a union of 9, 19, and an
infinite number of other equal temperaments.

Why 5&19 gives Semaphore/Godzilla/Hemifourths/Semifourths
instead of Meantone is a more subtle question. Try looking
at the code.

With my old Python library, & is an operator that acts on
equal temperament objects:

>>> import temper
>>> h5 = temper.BestET(5, temper.limit7)
>>> h19 = temper.BestET(19, temper.limit7)
>>> h5&h19

5/24, 251.7 cent generator

basis:
(1.0, 0.20975898813907889)

mapping by period and generator:
[(1, 0), (2, -2), (4, -8), (3, -1)]

mapping by steps:
[(19, 5), (30, 8), (44, 12), (53, 14)]

highest interval width: 8
complexity measure: 8 (9 for smallest MOS)
highest error: 0.017114 (20.537 cents)
>>>

Graham

🔗cityoftheasleep <igliashon@...>

--- In MakeMicroMusic@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
> Sorry to say that this is quite unusable from didactic point of view
> for somebody who wants to learn something about tuning. I say this as
> a long-year and not unexperienced language and music teacher. It
> can't help at all.

Aaron is not a beginner, and I did not endeavor to make my reply to his question a comprehensive introduction for the complete neophyte. I am, however, writing a book to precisely that purpose, and am almost finished with the introduction. It is (so far) approximately 25 pages, and I explain first how we determine whether two notes will sound "in tune" or not (i.e. the basic principles of JI, including a discussion of partials and how they can be harmonic or inharmonic depending on the instrument), then I explain how to expand that concept to determine whether more than two notes will sound in tune, and in both places I give tables to show all the basic ratios that have the audible property of sounding beatless. Then I explain how the ear and the mind are not interested in precise numerology and that irrational frequency relationships (and more complex rational relationships) can achieve a close enough effect via "approximation".

Then I move into temperament by showing that the JI major scale doesn't work for regular music because it provides only 5 in-tune fifths, and I introduce the syntonic comma. I then introduce the Pythagorean major scale and demonstrate how it solves the problem of "not enough fifths" but develops the problem of "out of tune thirds and sixths". Then I introduce meantone temperament and show how it solves both problems, and discuss the varieties of meantone (1/4 and 1/3 comma, specifically) and develop the distinction between a "tuning", a "temperament", and a "scale". I go on to discuss how the concept of meantone temperament can be generalized and that there are other temperaments we can produce to meet either the same harmonic goals (4:5:6 triads) or else different harmonic goals (any of the other JI triads that sound beatless). To illustrate this, I introduce Magic temperament (which aims for the same 5-limit harmonies as meantone) and Superpyth (which I treat as having a 6:7:9 triadic basis). I then introduce the concept of mapping, and concepts like period, generator, complexity and error in evaluating temperaments.

Then I move on to Distributionally-Even scales and talk about the properties that make them useful, and how they can be constructed by "stacking" a generator over and over and wrapping within the period until there are two step-sizes. I explain that some generators eventually produce an equal scale, while other generators can go on dividing the period indefinitely, and what determines this is whether the generator is an integer fraction of the period or not (in terms of cents values). I demonstrate this by constructing a couple different scales, one of which is the octatonic scale from 12-TET.

Then I clarify that temperaments produce DE scales but that the difference between them is that a temperament maps JI ratios to some number of generators and periods, whereas a DE scale does not specify a mapping but only a pattern of large and small steps. Then I talk about how both are tuning-inspecific (they don't specify a tuning, but only a range of tunings).

THEN I move on to equal divisions of the octave, discussing their advantages (simplicity, compatibility with multiple temperaments, etc.) and illustrate how 12-TET can be interpreted as a variety of different temperaments by showing the different mappings it supports.

I may at this point also include a discussion of how beating is not necessarily a bad thing, and that accurate temperaments are not always what is desirable in music, and that simplicity may be a more important concern.

Then I move on to individual EDOs, showing what Just dyads and triads they approximate decently for their size, and then showing the simplest temperaments that support those triads (i.e. the ones that produce the most triads for the smallest DE scale). I start at 5-EDO and work my way up to 41-EDO. Anything I didn't clarify enough in the introduction, I clarify when it comes up as an issue in the various EDOs.

I don't introduce anymore terminology than is absolutely necessary, I try to be informal and humorous whenever possible, and I offer frequent encouragement and acknowledgement that this material is unusual and seems to have little to do with "normal" music theory.

Once I've finished the 1st draft of the introduction, I plan on posting the file at Tuning and asking for feedback/proof-reading/additional input/etc. I want both the "experts" and the "laymen", as well as those with teaching backgrounds, to read it and rip it apart so that I can make it the best and most accessible text on the subject that's ever been written. I know it's going to need more visual aids--I'll be asking for them as well, since I don't have any good sources of, say, pictures of sine waves interacting or the waveforms of various musical instruments.

At that point, Daniel, I will more than welcome your criticism.

-Igs

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

🔗Daniel Forró <dan.for@...>

That looks very well, I'm looking forward to see it. As I'm far from
being expert in many fields you describe, don't expect please
criticism from my side. I'm sure there are men of knowledge here to
do this. But I hope to find enough time to go through carefully and
ask more simple questions. Maybe it will help you to make it really
comprehensive.

Good luck!

Daniel Forro

On Mar 11, 2011, at 2:10 AM, cityoftheasleep wrote:

> --- In MakeMicroMusic@yahoogroups.com, Daniel Forró <dan.for@...>
> wrote:
>>
>> Sorry to say that this is quite unusable from didactic point of view
>> for somebody who wants to learn something about tuning. I say this as
>> a long-year and not unexperienced language and music teacher. It
>> can't help at all.
>
> Aaron is not a beginner, and I did not endeavor to make my reply to
> his question a comprehensive introduction for the complete
> neophyte. I am, however, writing a book to precisely that purpose,
> and am almost finished with the introduction. It is (so far)
> approximately 25 pages, and I explain first how we determine
> whether two notes will sound "in tune" or not (i.e. the basic
> principles of JI, including a discussion of partials and how they
> can be harmonic or inharmonic depending on the instrument), then I
> explain how to expand that concept to determine whether more than
> two notes will sound in tune, and in both places I give tables to
> show all the basic ratios that have the audible property of
> sounding beatless. Then I explain how the ear and the mind are not
> interested in precise numerology and that irrational frequency> relationships (and more complex rational relationships) can achieve
> a close enough effect via "approximation".
>
> Then I move into temperament by showing that the JI major scale
> doesn't work for regular music because it provides only 5 in-tune
> fifths, and I introduce the syntonic comma. I then introduce the
> Pythagorean major scale and demonstrate how it solves the problem
> of "not enough fifths" but develops the problem of "out of tune
> thirds and sixths". Then I introduce meantone temperament and show
> how it solves both problems, and discuss the varieties of meantone
> (1/4 and 1/3 comma, specifically) and develop the distinction > between a "tuning", a "temperament", and a "scale". I go on to
> discuss how the concept of meantone temperament can be generalized
> and that there are other temperaments we can produce to meet either
> the same harmonic goals (4:5:6 triads) or else different harmonic
> goals (any of the other JI triads that sound beatless). To
> illustrate this, I introduce Magic temperament (which aims for the
> same 5-limit harmonies as meantone) and Superpyth (which I treat as
> having a 6:7:9 triadic basis). I then introduce the concept of
> mapping, and concepts like period, generator, complexity and error
> in evaluating temperaments.
>
> Then I move on to Distributionally-Even scales and talk about the
> properties that make them useful, and how they can be constructed
> by "stacking" a generator over and over and wrapping within the
> period until there are two step-sizes. I explain that some
> generators eventually produce an equal scale, while other
> generators can go on dividing the period indefinitely, and what
> determines this is whether the generator is an integer fraction of
> the period or not (in terms of cents values). I demonstrate this
> by constructing a couple different scales, one of which is the
> octatonic scale from 12-TET.
>
> Then I clarify that temperaments produce DE scales but that the
> difference between them is that a temperament maps JI ratios to
> some number of generators and periods, whereas a DE scale does not
> specify a mapping but only a pattern of large and small steps.> Then I talk about how both are tuning-inspecific (they don't
> specify a tuning, but only a range of tunings).
>
> THEN I move on to equal divisions of the octave, discussing their
> advantages (simplicity, compatibility with multiple temperaments,
> etc.) and illustrate how 12-TET can be interpreted as a variety of
> different temperaments by showing the different mappings it supports.
>
> I may at this point also include a discussion of how beating is not
> necessarily a bad thing, and that accurate temperaments are not
> always what is desirable in music, and that simplicity may be a
> more important concern.
>
> Then I move on to individual EDOs, showing what Just dyads and
> triads they approximate decently for their size, and then showing
> the simplest temperaments that support those triads (i.e. the ones
> that produce the most triads for the smallest DE scale). I start
> at 5-EDO and work my way up to 41-EDO. Anything I didn't clarify
> enough in the introduction, I clarify when it comes up as an issue
> in the various EDOs.
>
> I don't introduce anymore terminology than is absolutely necessary,
> I try to be informal and humorous whenever possible, and I offer
> frequent encouragement and acknowledgement that this material is
> unusual and seems to have little to do with "normal" music theory.
>
> Once I've finished the 1st draft of the introduction, I plan on
> posting the file at Tuning and asking for feedback/proof-reading/
> additional input/etc. I want both the "experts" and the "laymen",
> as well as those with teaching backgrounds, to read it and rip it
> apart so that I can make it the best and most accessible text on
> the subject that's ever been written. I know it's going to need
> more visual aids--I'll be asking for them as well, since I don't
> have any good sources of, say, pictures of sine waves interacting
> or the waveforms of various musical instruments.
>
> At that point, Daniel, I will more than welcome your criticism.
>
> -Igs