Paul's new names

Here are some comma/temperament names introduced on the dualzoomer series. I give the name, the comma, (weighted) complexity, rms error and badness in that order.

dicot 25/24 1.597771 28.851897 117.684239

unicorn 16875/16384 4.719204 5.942563 624.568220

diaschizoid 6561/6250 5.526963 8.492497 1433.821424

schizoid 262144/253125 6.027920 5.487501 1201.924424

valentine 1990656/1953125 7.198353 2.983296 1112.745097

shibboleth 1953125/1889568 8.141716 4.245388 2291.212088

sycamore 48828125/47775744 8.794602 2.796055 1901.928533

vulture 10485760000/10460353203 13.577520 .153767 384.880223

crazy 9010162353515625/9007199254740992 17.879417 .017725 101.309796

tricot 68719476736000/68630377364883 19.249427 .057500 410.130475

"Dicot" I've been calling "neutral thirds", which might be confusing.
"Unicorn" I called "negrisma"--did I get that wrong? "Crazy" seems to be a mutant form of my "quasiseptima", but where "vulture" comes from I have no idea. It's not as bad as it sounds, though!

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> Here are some comma/temperament names introduced on the dualzoomer
series. I give the name, the comma, (weighted) complexity, rms error
and badness in that order.
>
> dicot 25/24 1.597771 28.851897 117.684239
>
> unicorn 16875/16384 4.719204 5.942563 624.568220
>
> diaschizoid 6561/6250 5.526963 8.492497 1433.821424
>
> schizoid 262144/253125 6.027920 5.487501 1201.924424
>
> valentine 1990656/1953125 7.198353 2.983296 1112.745097
>
> shibboleth 1953125/1889568 8.141716 4.245388 2291.212088
>
> sycamore 48828125/47775744 8.794602 2.796055 1901.928533
>
> vulture 10485760000/10460353203 13.577520 .153767 384.880223
>
> crazy 9010162353515625/9007199254740992 17.879417 .017725 101.309796
>
> tricot 68719476736000/68630377364883 19.249427 .057500 410.130475
>
> "Dicot" I've been calling "neutral thirds", which might be
confusing.
> "Unicorn" I called "negrisma"--did I get that wrong?

you can find the real negri on my graph -- it's where 9, 10, and 19
intersect.

>"Crazy" seems to be a mutant form of my "quasiseptima",

how?

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> >"Crazy" seems to be a mutant form of my "quasiseptima",

> how?

Quasi --> crazy, as in "quasi wabbit!"

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> > "Unicorn" I called "negrisma"--did I get that wrong?
>
> you can find the real negri on my graph -- it's where 9, 10, and 19
> intersect.

Here's the real unicorn:

comma 1594323/1562500
complexity 8.939708723
rms error 2.173108990
badness 1552.571517

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > >"Crazy" seems to be a mutant form of my "quasiseptima",
>
> > how?
>
> Quasi --> crazy, as in "quasi wabbit!"

h yes -- ara and i say that all the time -- so i propose we spell
it "kwazy"!

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > > "Unicorn" I called "negrisma"--did I get that wrong?
> >
> > you can find the real negri on my graph -- it's where 9, 10, and
19
> > intersect.
>
> Here's the real unicorn:
>
> comma 1594323/1562500
> complexity 8.939708723
> rms error 2.173108990
> badness 1552.571517

the list of commas came from monz's et page (with a few ridonculous
ones deleted) . . . i'd like to know what is missing, based on some
log-flat badness measure.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> h yes -- ara and i say that all the time -- so i propose we spell
> it "kwazy"!

I love it; "kwazy" it is.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> the list of commas came from monz's et page (with a few ridonculous
> ones deleted) . . . i'd like to know what is missing, based on some
> log-flat badness measure.

Here's my personal 5-limit comma list, which I've been using for some time; unweighted rms badness is less than 500, and complexity less than 50.

17763568394002504646778106689453125/
17763086495282268024161967871623168

381520424476945831628649898809/381469726562500000000000000000

22300745198530623141535718272648361505980416/
22297583945629639856633730232715606689453125

162285243890121480027996826171875/162259276829213363391578010288128

450359962737049600/450283905890997363

444089209850062616169452667236328125/
444002166576103304796646509039845376

116450459770592056836096/
116415321826934814453125

9010162353515625/9007199254740992

2475880078570760549798248448/2474715001881122589111328125

7629394531250/7625597484987

50031545098999707/50000000000000000

274877906944/274658203125

582076609134674072265625/581595589965365114830848

32805/32768, 19073486328125/19042491875328,
6115295232/6103515625, 1224440064/1220703125, 1600000/1594323,
15625/15552, 2109375/2097152, 393216/390625, 78732/78125, 2048/2025, 81/80, 3125/3072, 128/125, 250/243, 648/625, 25/24, 135/128, 16/15, 27/25

Could someone give me the formulas that are used to calculate complexity,
rms error and badness? I'll also check monz's dictionary.

Thanks

Paul

wallyesterpaulrus
<wallyesterpaulrus To: tuning-math@yahoogroups.com
@yahoo.com> cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Paul's new names
11/24/2002 10:20
PM
Please respond to
tuning-math

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > > "Unicorn" I called "negrisma"--did I get that wrong?
> >
> > you can find the real negri on my graph -- it's where 9, 10, and
19
> > intersect.
>
> Here's the real unicorn:
>
> comma 1594323/1562500
> complexity 8.939708723
> rms error 2.173108990
> badness 1552.571517

the list of commas came from monz's et page (with a few ridonculous
ones deleted) . . . i'd like to know what is missing, based on some
log-flat badness measure.

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--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...>
wrote:
>
> Could someone give me the formulas that are used to calculate
complexity,
> rms error and badness? I'll also check monz's dictionary.
>
> Thanks
>
> Paul

there are several different versions of complexity. rms error is
simply the root-mean-square error of the distinct consonant intervals
in the odd limit in question (in this case, it's 5-limit, so it's the
root-mean square error, in cents, of 3/2, 5/4, and 5/3). badness has
different versions, but in general it's a product of complexity (to
some power) and rms error (to some power).

>there are several different versions of complexity.

Has there been any discussion on where we disagree on
complexity? What are the main formualtions, and how
desirable is it that we pick one?

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >there are several different versions of complexity.
>
> Has there been any discussion on where we disagree on
> complexity? What are the main formualtions, and how
> desirable is it that we pick one?
>
> -Carl

i'd be happy with an unweighted complexity measure for the purposes
of these graphs, if that's what people prefer. ultimately, though, to
repeat myself, i'd like to figure out a complexity measure that
agrees with the size of the numbers in the comma (at least in the 5-
limit case). it seems kees's lattice does this, but . . .

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> What are the main formualtions,

we have graham's (unweighted minimax), unweighted rms, weighted rms,
geometric (both unweighted and weighted??) . . .

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> > >there are several different versions of complexity.
> >
> > Has there been any discussion on where we disagree on
> > complexity? What are the main formualtions, and how
> > desirable is it that we pick one?
> >
> > -Carl
>
> i'd be happy with an unweighted complexity measure for the purposes
> of these graphs, if that's what people prefer.

I like geometric complexity because it is a uniform system applicable to any regular temperament.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> > > >there are several different versions of complexity.
> > >
> > > Has there been any discussion on where we disagree on
> > > complexity? What are the main formualtions, and how
> > > desirable is it that we pick one?
> > >
> > > -Carl
> >
> > i'd be happy with an unweighted complexity measure for the
purposes
> > of these graphs, if that's what people prefer.
>
> I like geometric complexity because it is a uniform system
>applicable to any regular temperament.

including equal temperaments? can you give a sorting of ETs in some
limit? does GC reduce to something simpler for ETs?

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> including equal temperaments? can you give a sorting of ETs in some
> limit? does GC reduce to something simpler for ETs?

Indeed it does--for any et n, the geometric complexity is proportional to n.

>What are the main formualtions,
>
>we have graham's (unweighted minimax), unweighted rms,
>weighted rms, geometric (both unweighted and weighted??) . . .

For complexity? What are they?

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >What are the main formualtions,
> >
> >we have graham's (unweighted minimax), unweighted rms,
> >weighted rms, geometric (both unweighted and weighted??) . . .
>
> For complexity?

yup, doze R dem.

> What are they?

dem. doze.

>>>we have graham's (unweighted minimax), unweighted rms,
>>>weighted rms, geometric (both unweighted and weighted??) . . .
>>
>> For complexity?
>
>yup, doze R dem.
>
>> What are they?
>
>dem. doze.

Minimax, rms, etc. of what? The numbers in the map?

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>>we have graham's (unweighted minimax), unweighted rms,
> >>>weighted rms, geometric (both unweighted and weighted??) . . .
> >>
> >> For complexity?
> >
> >yup, doze R dem.
> >
> >> What are they?
> >
> >dem. doze.
>
> Minimax, rms, etc. of what? The numbers in the map?
>
> -Carl

the number of generators comprising each consonant interval.

>>Minimax, rms, etc. of what? The numbers in the map?
>
>the number of generators comprising each consonant interval.

Ok, thanks; as I thought.

Conceptually, if we're thinking in terms of Partchian limits,
I prefer simply the number of generators needed to span all
the identities (consonant intervals). This can be 'weighted'
by simply dividing by the number of identities. Reason being,
I view the choice of a complete Partchian limit as a statement
that all the identities in that limit will be treated as
consonances in the music, and thus are all equally important
in that sense. If we're not talking about Partchian limits,
we can just omit the identities that make the range bad. Why
we would want to smooth such bad approximations out with
something like rms I cannot guess.

I can see weighted error, but not weighted complexity, where
the weighting proporational to the identity. Why should we
expect more generators to be required to render 7 than 5?

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>Minimax, rms, etc. of what? The numbers in the map?
> >
> >the number of generators comprising each consonant interval.
>
> Ok, thanks; as I thought.

It doesn't cover geometric complexity, for which you should see my postings on this list.

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>Minimax, rms, etc. of what? The numbers in the map?
> >
> >the number of generators comprising each consonant interval.
>
> Ok, thanks; as I thought.
>
> Conceptually, if we're thinking in terms of Partchian limits,
> I prefer simply the number of generators needed to span all
> the identities (consonant intervals).

that's what i meant by "minimax" -- it's graham's way.

> This can be 'weighted'
> by simply dividing by the number of identities.

that has no effect on the rankings.

> Reason being,
> I view the choice of a complete Partchian limit as a statement
> that all the identities in that limit will be treated as
> consonances in the music, and thus are all equally important
> in that sense.

**but if modulation is usually accomplished by making a small number
of "chromatic" changes to the basic "diatonic" scale, shouldn't extra
points be awarded if the modulations more often move one by the
*simpler* consonances, particularly the 3-limit ones?

> If we're not talking about Partchian limits,
> we can just omit the identities that make the range bad. Why
> we would want to smooth such bad approximations out with
> something like rms I cannot guess.

smooth such bad approximations out? i'm not sure what you mean. rms
is similar to graham's method, but takes into account the second-
longest, third-longest, etc. chains of generators to a small extent
too. that seems like a good thing to me.

> I can see weighted error, but not weighted complexity, where
> the weighting proporational to the identity. Why should we
> expect more generators to be required to render 7 than 5?

see ** above.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> smooth such bad approximations out? i'm not sure what you mean. rms
> is similar to graham's method, but takes into account the second-
> longest, third-longest, etc. chains of generators to a small extent
> too. that seems like a good thing to me.

It really depends on whether or not you are interested in incomplete n-limit chords; normally, I think we would be.

>>Conceptually, if we're thinking in terms of Partchian limits,
>>I prefer simply the number of generators needed to span all
>>the identities (consonant intervals).
>
>that's what i meant by "minimax" -- it's graham's way.

Thought so. Isn't minimax a bad term for this?

>>This can be 'weighted' by simply dividing by the number of
>>identities.
>
> that has no effect on the rankings.

But it allows you to compare different limits. Actually I
haven't checked if something stronger than division would
be needed, but you get the idea.

>**but if modulation is usually accomplished by making a small
>number of "chromatic" changes to the basic "diatonic" scale,
>shouldn't extra points be awarded if the modulations more often
>move one by the *simpler* consonances, particularly the 3-limit
>ones?

Not in my view. I'm thinking we should not even assume tonal
composition at this level.

-Carl

>>smooth such bad approximations out? i'm not sure what you mean.
>>rms is similar to graham's method, but takes into account the
>>second-longest, third-longest, etc. chains of generators to a
>>small extent too. that seems like a good thing to me.
>
>It really depends on whether or not you are interested in
>incomplete n-limit chords; normally, I think we would be.

If I'm interested in complete 7-limit tetrads on every beat,
have a temperament with really simple 3s and 7s but complex 5s,
rms will punish this less than I'd like. At least, the history
of Western music seems to assert that for music employing
n-limit harmony, < n-limit harmony sounds too different to
fall back on for any length of time.

-Carl

>>>the number of generators comprising each consonant interval.
>>
>>Ok, thanks; as I thought.
>
>It doesn't cover geometric complexity, for which you should
>see my postings on this list.

Msg. 4533 is the one, I'm guessing. Very cool. But I don't
have the technique for choosing the defining commas. Is
there any way this can be defined in terms of the map?

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> If I'm interested in complete 7-limit tetrads on every beat,
> have a temperament with really simple 3s and 7s but complex 5s,
> rms will punish this less than I'd like. At least, the history
> of Western music seems to assert that for music employing
> n-limit harmony, < n-limit harmony sounds too different to
> fall back on for any length of time.

Then you want Graham complexity, which is nice, because it is very easy to compute.

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> Msg. 4533 is the one, I'm guessing. Very cool. But I don't
> have the technique for choosing the defining commas. Is
> there any way this can be defined in terms of the map?

My program extracts commas from the wedgie and then reduces them. Another way would be to run down a comma list using the map, and see which are mapped to [0,0].

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>Conceptually, if we're thinking in terms of Partchian limits,
> >>I prefer simply the number of generators needed to span all
> >>the identities (consonant intervals).
> >
> >that's what i meant by "minimax" -- it's graham's way.
>
> Thought so. Isn't minimax a bad term for this?

yup, sorry -- it's just "max".

> >**but if modulation is usually accomplished by making a small
> >number of "chromatic" changes to the basic "diatonic" scale,
> >shouldn't extra points be awarded if the modulations more often
> >move one by the *simpler* consonances, particularly the 3-limit
> >ones?
>
> Not in my view. I'm thinking we should not even assume tonal
> composition at this level.

i agree. but even on the level of modal composition, i feel this
weighting has some appeal, for the same kinds of reasons.

> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
>
> > If I'm interested in complete 7-limit tetrads on every beat,
> > have a temperament with really simple 3s and 7s but complex 5s,
> > rms will punish this less than I'd like. At least, the history
> > of Western music seems to assert that for music employing
> > n-limit harmony, < n-limit harmony sounds too different to
> > fall back on for any length of time.

the incomplete chords need not be lower-limit. for your 7-limit
example, how about 1:5:7, 1:3:7, etc.? in western music, you have
plenty of two-part music (especially bach) which uses incomplete
triads (that is, dyads) *all* the time, but typically these are still
5-limit.

> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
>
> > Msg. 4533 is the one, I'm guessing. Very cool

i like the idea of preserving some features of tenney's harmonic
distance function . . . but i'm not sure i can fully visualize
geometric complexity yet . . .

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >
> > > If I'm interested in complete 7-limit tetrads on every beat,
> > > have a temperament with really simple 3s and 7s but complex 5s,
> > > rms will punish this less than I'd like. At least, the history
> > > of Western music seems to assert that for music employing
> > > n-limit harmony, < n-limit harmony sounds too different to
> > > fall back on for any length of time.

> the incomplete chords need not be lower-limit. for your 7-limit
> example, how about 1:5:7, 1:3:7, etc.?

I think Saxfare, which does not use 5, sounds like music.