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Message: 9575 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 10:28:05

Subject: Re: What the numbers mean

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>>> The complexity is defined by the weighted value for 5/3, which 
is 
>> the

>>> worst case (as, in fact, it clearly is.)
>> 

>> Meaning what, exactly?
> 

> It's hard to find a reasonable take on septimal miracle which 
doesn't 
> have it that 5/3 is the most complex consonance.
> 

>> Can you show this process in action for the simpler, 3-limit and 
5-
>> limit cases? And why do we take the worst case, instead of some 
> sort 

>> of product (which would appear to get you your spacial measure 
once 
>> you've orthogonalized)?
> 

> "Orthogonalized" is one of those words which is holding up 
> communication, as I can only interpret that in terms of a L2 norm.


What if I just talk about "projections"? It seems the wedgie gives us 
the size of the projection of the commas' bimonzo onto the 
unit 'orthogonal' bimonzos of the lattice. If we use L_inf, two 
temperaments with the same largest projection have the same 
complexity. But clearly it requires more notes to fill the bimonzo if 
its second-largest projection is nonzero. And in fact, in the taxicab 
lattice, no matter how you shape it, you can't construct a bivector 
with fewer notes than what you get by constructing the projections 
and sticking them together like two walls and a floor in the 3D 
case . . . So I see pretty clearly now that the L_1 norm of the 
multimonzos is appropriate; unfortunately I think I just calculated 
them wrong, so they might not actually disagree with:


> The vals, if we assume the Tenney metric, have an L_inf norm, and I 
> am regarding wedgies as multivals,

Message: 9576 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 20:21:32

Subject: Cross-check for TOP 5-limit 12-equal

From: Paul Erlich

Wedgie norm for 12-equal:

 Take the two unison vectors
 
 |7 0 -3>
 |-4 4 -1>

 Now find the determinant, and the "area" it represents, in each of 
 the basis planes:
 
 |7 0| = 28*(e23) -> 28/lg2(5) = 12.059
 |-4 4|
 
 |7 -3| = -19*(e25) -> 19/lg2(3) = 11.988
 |-4 -1|

 |0  -3| = 12*(e35) -> 12 = 12
 |4 -1|

sum = 36.047

If I just use the maximum (L_inf = 12.059) as a measure of notes per 
acoustical octave, then I "predict" tempered octaves of 1194.1 cents. 
If I use the sum (L_1), dividing by the "mystery constant" 3, 
I "predict" tempered octaves of 1198.4 cents. Neither one is the TOP 
value . . . :( . . . but what sorts of error criteria, if any, *do* 
they optimize?

So the cross-checking I found for the 3-limit case in "Attn: Gene 2"
Yahoo groups: /tuning-math/message/8799 * [with cont.] 
doesn't seem to work in the 5-limit ET case for either the L_1 or 
L_inf norms.

However, if I just add the largest and smallest values above:

28/lg2(5)+19/lg2(3)

I do predict the correct tempered octave (aside from a factor of 2),

1197.67406985219 cents.

So what sort of norm, if any, did I use to calculate complexity this 
time? It's related to how we temper for TOP . . .

Message: 9577 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 00:49:24

Subject: Re: Crunch algorithm

From: Paul Erlich

This is similar, if not identical, to Viggo Brun's algorithm that 
Kraig is always referring to . . . See Mandelbaum's book for a full 
exposition . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> Suppose we have a list of rational numbers greater than one, sorted 
in
> ascending order, the elements of which should be independent. Define
> the crunch function as follows: take the quotient of the last with 
the
> next-to-last, and then place it in its proper location in the list,
> and return the new sorted list. If we start from an independent set 
of
> numbers, and in particular a basis for the p-limit, we crunch down 
to
> a basis with smaller elements.
> 
> Here is crunch, starting out on [2, 3, 5, 7]. The next column is the
> top row of the inverse matrix for the monzos, which is a val matrix;
> so these are scale divisions.
> 
> I thought of this because Oldlyzko pointed me to an unpublished 
paper
> of his, which suggested the problem of finding very large scale
> divisions (in the context of the Riemann zeta function, not music!) 
is
> harder than I believe it to be. I think I'll send him something and
> mention Paul's discovery that 2401/2400 dominantes up to 100000 or 
so
> for 7-limit divisions, translated of course into zeta language.
> 
> [2, 3, 5, 7] [1, 0, 0, 0]
> [7/5, 2, 3, 5] [0, 1, 0, 0]
> [7/5, 5/3, 2, 3] [0, 0, 1, 0]
> [7/5, 3/2, 5/3, 2] [0, 0, 0, 1]
> [6/5, 7/5, 3/2, 5/3] [1, 0, 0, 1]
> [10/9, 6/5, 7/5, 3/2] [1, 1, 0, 1]
> [15/14, 10/9, 6/5, 7/5] [1, 1, 1, 1]
> [15/14, 10/9, 7/6, 6/5] [1, 1, 1, 2]
> [36/35, 15/14, 10/9, 7/6] [2, 1, 1, 3]
> [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4]
> [36/35, 28/27, 21/20, 15/14] [2, 4, 3, 5]
> [50/49, 36/35, 28/27, 21/20] [5, 2, 4, 8]
> [81/80, 50/49, 36/35, 28/27] [8, 5, 2, 12]
> [245/243, 81/80, 50/49, 36/35] [12, 8, 5, 14]
> [126/125, 245/243, 81/80, 50/49] [14, 12, 8, 19]
> [4000/3969, 126/125, 245/243, 81/80] [19, 14, 12, 27]
> [19683/19600, 4000/3969, 126/125, 245/243] [27, 19, 14, 39]
> [4375/4374, 19683/19600, 4000/3969, 126/125] [39, 27, 19, 53]
> [250047/250000, 4375/4374, 19683/19600, 4000/3969] [53, 39, 27, 72]
> [250047/250000, 4375/4374, 1600000/1594323, 19683/19600] [53, 39, 
72, 99]
> 
> 
> [53, 39, 99, 171]
> [53, 39, 270, 171]
> [53, 39, 441, 171]
> [53, 39, 612, 171]
> [53, 39, 783, 171]
> [53, 171, 39, 954]
> [53, 171, 993, 954]
> [53, 171, 954, 1947]
> [1947, 53, 171, 2901]
> [1947, 2901, 53, 3072]
> [3072, 1947, 2901, 3125]
> [3072, 1947, 6026, 3125]
> [3072, 1947, 9151, 3125]
> [3072, 1947, 12276, 3125]
> [3072, 1947, 15401, 3125]
> [3072, 1947, 18526, 3125]
> [3072, 1947, 21651, 3125]
> [3072, 1947, 24776, 3125]
> [3072, 1947, 27901, 3125]
> [3072, 1947, 31026, 3125]

Message: 9578 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 10:29:26

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> While I think it would be nice to name this after Larry Hanson (and
> I certainly agreeable to the idea), my preference is to keep 
kleismic,
> since it tells the name of the comma involved, and has a fairly-well
> established body of use.  What say everybody?


I like kleismic, but which version of kleismic did Hanson like?

Message: 9579 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 15:08:19

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Herman Miller

On Sat, 31 Jan 2004 08:59:44 -0000, "Paul Erlich" <perlich@xxx.xxxx.xxx>
wrote:


>In the 5-limit linear case, it would be really easy to do this if we 
>didn't want to go out to the complexity of schismic and 
>kleismic/hanson (the only argument that would arise would be whether 
>the father and beep couple should be in or out, leading to a grand 
>total of 11 or 9 5-limit LTs).


Father is really only useful up to 8 notes, and beep up to 9, but father
could potentially be of interest as a "warping" of traditional melody and
harmony, since tempering out semitones drastically changes the effect of
both melodies and harmonic progressions. As a "temperament" in the sense of
approximating JI, it isn't worth considering. Beep could theoretically be
of similar use as a "warping" of Bohlen-Pierce-type harmony, since 27/25 is
the BP-scale's equivalent of a semitone, but in other respects it seems
less interesting than other 5-limit tunings.


> Unfortunately, the low error of 
>schismic has proved tantalizing enough for a few musicians to 
>construct instruments capable of playing the large extended scales 
>that its approximations require. If we consider such complexity 
>justifiable, it seems we should be interested in 15 to 17 5-limit 
>LTs, or 17 to 19 if we include father and beep.


Schismic and kleismic/hanson start being useful (barely) around 12 notes,
but the tiny size of schismic steps beyond 12 notes is a drawback until you
get to around 41 notes when the steps are a bit more evenly spaced.
Kleismic[15] and kleismic[19] are usable and have reasonably sized steps.
So the complexity of kleismic in a musically useful sense isn't really
comparable to schismic; this is one thing that the horagrams are useful
for.


> The couple residing 
>in "the middle of the road" is 2187;2048 and 3126;2916. With Herman, 
>we could split the difference and select only the better of the pair, 
>2187;2048 (Dave, have you *heard* Blackwood's 21-equal suite?) . . . 


Star Wars fans will recognize 2187 as Princess Leia's cell number, if
that's of any help in assigning a name to this temperament. The TOP tuning
for these:

[-11 7] (2187;2048) [1205.145343, 1893.799825]
[-2 -6 5] (3125;2916) [1205.183460, 1910.170591, 2774.278093]

Certainly, [-11 7] is useful as a 3-limit temperament, and this might be a
useful way to think of 7-ET in Thai and Burmese music. Since it's basically
a 3-limit temperament like Blackwood or Aristoxenan, its 5:1 can be tuned
just, resulting in 10.3 cent flat major thirds (slightly better than the
13.7 cent sharp thirds of 21-ET).


>I don't think anyone's talked about the 20480;19683 system. But if 
>schismic, and certainly if semisixths, is not too complex to be a 
>useful alternative to strict JI, why shouldn't this system merit some 
>attention from musicians too? I don't think near-JI triads sound 
>enough better than chords in this system (which are purer than those 
>of augmented, porcupine, or diminished) to merit a much higher 
>allowed complexity to generate them linearly.


This is [12 -9 1], TOP tuning: [1197.596121, 1905.765059, 2780.732080]. It
goes through 22 and 27 on the temperament chart, between porcupine and
diaschismic. The tiny steps are just barely big enough to be perceived as
melodic steps rather than commas, around the size of 22-ET steps. It might
be worth looking into.

-- 
see my music page --->   ---<The Music Page * [with cont.]  (Wayb.)>--
hmiller (Herman Miller)   "If all Printers were determin'd not to print any
@io.com  email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" /  there would be very little printed." -Ben Franklin

Message: 9580 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 00:52:29

Subject: Re: Crunch algorithm

From: Paul Erlich

Please see %PDF-1.3 * [with cont.]  (Wayb.) . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> This is similar, if not identical, to Viggo Brun's algorithm that 
> Kraig is always referring to . . . See Mandelbaum's book for a full 
> exposition . . .
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:

>> Suppose we have a list of rational numbers greater than one, 
sorted 
> in

>> ascending order, the elements of which should be independent. 
Define
>> the crunch function as follows: take the quotient of the last 
with 
> the

>> next-to-last, and then place it in its proper location in the 
list,
>> and return the new sorted list. If we start from an independent 
set 
> of

>> numbers, and in particular a basis for the p-limit, we crunch 
down 
> to

>> a basis with smaller elements.
>> 
>> Here is crunch, starting out on [2, 3, 5, 7]. The next column is 
the
>> top row of the inverse matrix for the monzos, which is a val 
matrix;
>> so these are scale divisions.
>> 
>> I thought of this because Oldlyzko pointed me to an unpublished 
> paper

>> of his, which suggested the problem of finding very large scale
>> divisions (in the context of the Riemann zeta function, not 
music!) 
> is

>> harder than I believe it to be. I think I'll send him something 
and
>> mention Paul's discovery that 2401/2400 dominantes up to 100000 
or 
> so

>> for 7-limit divisions, translated of course into zeta language.
>> 
>> [2, 3, 5, 7] [1, 0, 0, 0]
>> [7/5, 2, 3, 5] [0, 1, 0, 0]
>> [7/5, 5/3, 2, 3] [0, 0, 1, 0]
>> [7/5, 3/2, 5/3, 2] [0, 0, 0, 1]
>> [6/5, 7/5, 3/2, 5/3] [1, 0, 0, 1]
>> [10/9, 6/5, 7/5, 3/2] [1, 1, 0, 1]
>> [15/14, 10/9, 6/5, 7/5] [1, 1, 1, 1]
>> [15/14, 10/9, 7/6, 6/5] [1, 1, 1, 2]
>> [36/35, 15/14, 10/9, 7/6] [2, 1, 1, 3]
>> [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4]
>> [36/35, 28/27, 21/20, 15/14] [2, 4, 3, 5]
>> [50/49, 36/35, 28/27, 21/20] [5, 2, 4, 8]
>> [81/80, 50/49, 36/35, 28/27] [8, 5, 2, 12]
>> [245/243, 81/80, 50/49, 36/35] [12, 8, 5, 14]
>> [126/125, 245/243, 81/80, 50/49] [14, 12, 8, 19]
>> [4000/3969, 126/125, 245/243, 81/80] [19, 14, 12, 27]
>> [19683/19600, 4000/3969, 126/125, 245/243] [27, 19, 14, 39]
>> [4375/4374, 19683/19600, 4000/3969, 126/125] [39, 27, 19, 53]
>> [250047/250000, 4375/4374, 19683/19600, 4000/3969] [53, 39, 27, 
72]
>> [250047/250000, 4375/4374, 1600000/1594323, 19683/19600] [53, 39, 
> 72, 99]
>> 
>> 

>> [53, 39, 99, 171]
>> [53, 39, 270, 171]
>> [53, 39, 441, 171]
>> [53, 39, 612, 171]
>> [53, 39, 783, 171]
>> [53, 171, 39, 954]
>> [53, 171, 993, 954]
>> [53, 171, 954, 1947]
>> [1947, 53, 171, 2901]
>> [1947, 2901, 53, 3072]
>> [3072, 1947, 2901, 3125]
>> [3072, 1947, 6026, 3125]
>> [3072, 1947, 9151, 3125]
>> [3072, 1947, 12276, 3125]
>> [3072, 1947, 15401, 3125]
>> [3072, 1947, 18526, 3125]
>> [3072, 1947, 21651, 3125]
>> [3072, 1947, 24776, 3125]
>> [3072, 1947, 27901, 3125]
>> [3072, 1947, 31026, 3125]

Message: 9581 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 10:43:34

Subject: Re: the choice of wedgie-norm greatly impacts miracle's ranking

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> oops -- I may have done that all wrong. The scaling factors for the 
> elements of the wedgie, the ones that you divide by to calculate 
the  
> multival norm -- do you have to *multiply* by them when you 
calculate 
> the multimonzo norm?


You rescale the monzos by multiplying, but it all comes out the same 
up to a constant factor, so the real difference is L1 vs L_inf. 
There's something to be said for L1 in that comparison.

Message: 9582 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 02:43:52

Subject: Re: kleismic v. hanson

From: Carl Lumma

>> >hile I think it would be nice to name this after Larry Hanson (and
>> I certainly agreeable to the idea), my preference is to keep
>> kleismic, since it tells the name of the comma involved, and has a
>> fairly-well established body of use.  What say everybody?
>

>I like kleismic, but which version of kleismic did Hanson like?


AFAIK, mainly the 53-note version.

-Carl

Message: 9583 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 01:04:58

Subject: Re: 60 for Dave

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>>>> Complexity, I should think, should definitely be punished 
more 
>>>>> than 1:1.  Using 8 notes instead of 7 notes would seem to 
demand
>>>>> more than eight 7ths the mental energy.
>>>> 

>>>> We could use a quadratic penalty on the complexity too.
>>> 

>>> If we square both terms, doesn't this give the same ranking?
>> 

>> No, because following Dave, we're adding the terms, not 
multiplying 
>> them. Dave restated Gene's product as a sum of logs.
> 

> Oh.  Why do that?


So that he could understand Gene's badness and my linear badness in 
the same form, and propose a compromise.


>>> In March '02 I wrote, "I think I'd rather have a smooth pain
>>> function, like ms, and a stronger exponent on complexity."
>> 

>> In response to what?
> 

> Dunno, but by "stronger" I meant "stronger than whatever we use
> on error".  And "exponent" maybe shouldn't be taken literally...
> I just meant to say that I'm willing to accept lots of error for
> a small savings in notes.


Well, according to log-flat badness, this will only happen at very 
low complexity values. At high complexity values, log-flat badness is 
essentially "flat" in error.


>>> By the way, when doing ms error, if an error is less than a cent
>>> it will get *smaller* when squared.
>> 

>> No, you can't compare cents to cents-squared. These quantities do 
not 
>> have the same dimension.
>> 

>>> Do you see this as a good
>>> thing, should we be ceilinging these to 1 before squarring, 
or...?
>> 

>> To 1 cent? Definitely not -- there's no justification for treating 
1 
>> cent as a special error size.
>> 
>> Besides, the errors Gene gave are only in units of cents if you're 
>> looking at the error of the octave -- other intervals have 
different 
>> units, since it's minimax Tenney-weighed error we're looking at.
> 

> This was a more general question.


Agreed -- in fact, rms is used in all kinds of fields, including 
experimental error analysis, electrical engineering, and acoustics.


> When calculating the rms error of
> an equal temperament, as we used to do, we just allow things less
> than 1 to get smaller


Again, they don't *really* get smaller, because they don't have the 
same units.


> and influence the mean?

Yes.

> It seems one is special
> whether or not we do anything.


Incorrect. If you change to a different system of units (say, 
millicents) so that nothing's smaller than 1, perform the rms 
calculation, and then change back to the original units, you get the 
same answer. Try it!

Message: 9584 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 10:44:36

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 

>> While I think it would be nice to name this after Larry Hanson 
(and
>> I certainly agreeable to the idea), my preference is to keep 
> kleismic,

>> since it tells the name of the comma involved, and has a fairly-
well
>> established body of use.  What say everybody?
> 

> I like kleismic, but which version of kleismic did Hanson like?


5-limit -- 19-, 34-, 53-, and 72-equal.

Message: 9585 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 10:48:21

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> 5-limit -- 19-, 34-, 53-, and 72-equal.


Any music available?

Message: 9586 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 02:09:52

Subject: Re: Crunch algorithm

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> This is similar, if not identical, to Viggo Brun's algorithm that 
> Kraig is always referring to . . . 


I'm not so sure--I think possibly Brun's algorithm simply sets out to 
find integer relations, not unimodular matricies. I've never seem an 
exposition of it, just references to it, and don't know how closely 
what people have said about in tuning connections matches what Brun 
actually did, which is described as an integer relation algorthm or 
something along the lines of Jacbobi-Perron when I read about it 
elsewhere. However, what I did is identical to the Erv Wilson method--
he *is* using it to find unimodular matricies, and then inverting 
them.

See Mandelbaum's book for a full 

> exposition . . .


You'll have to do better than that--is this Joel Mandelbaum, or some 
other Mandelbaum, and what book?

Message: 9587 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 11:01:36

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>> 5-limit -- 19-, 34-, 53-, and 72-equal.
> 

> Any music available?


Neil Haverstick used Hanson's 34-equal guitars on his CDs . . .

Message: 9588 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 02:12:20

Subject: Re: 60 for Dave

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> Ah.  Is yours the one from the Attn: Gene post?  It involves taking
> determinants, which I haven't fully learned how to do yet.


You stick it into your computer, and press the button. :)

Message: 9589 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 11:03:21

Subject: hey gene . . .

From: Paul Erlich

I'm trying to re-rank your top 10 using multimonzo L_1 norm (see next 
post). Could you do this too to provide an independent check?

Message: 9590 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 04:48:48

Subject: Twenty beepy commas

From: Gene Ward Smith

Here are twenty commas with relative error less than 0.015 and
epimericity less than 0.75, divisible only by 3, 5, and 7. 

[405/343, 729/625, 625/567, 2401/2187, 375/343, 49/45, 18225/16807, 
27/25, 16807/15625, 6561/6125, 546875/531441, 15625/15309,
60025/59049, 5859375/5764801, 40353607/39858075, 3125/3087, 245/243,
823543/820125, 16875/16807, 13841287201/13839609375]

Message: 9591 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 11:10:16

Subject: So MIRACLE is really #3, Pajara #2 in log-flat?

From: Paul Erlich

This time, I'll multiply, instead of dividing, the elements of the 
wedgie by the relevant "unit areas" . . . still using L_1 . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> This time I'll try L_1 (multimonzo interpretation?) instead of 
> L_infinity (multival interpretation?) to get complexity from the 
> wedgie. Let's see how it affects the rankings -- we don't need to 
> worry about scaling because Gene's badness measure is 
> multiplicative . . .
> 
> The top 10 get re-ordered as follows, though this is probably not 
the 
> new top 10 overall . . .

1.
> 1.

>> Number 1 Ennealimmal
>> 
>> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
>> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
>> TOP generators [133.3373752, 49.02398564]
>> bad: 4.918774 comp: 11.628267 err: .036377
> 

> 39.8287 -> bad = 57.7058


464.95 -> bad = 7864

2.

> 7.

>> Number 6 Pajara
>> 
>> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>> TOP generators [598.4467109, 106.5665459]
>> bad: 27.754421 comp: 2.988993 err: 3.106578
> 

> 10.4021 -> bad = 336.1437


130.6 -> bad = 52987

3.

> 3.

>> Number 9 Miracle
>> 
>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>> TOP generators [1200.631014, 116.7206423]
>> bad: 29.119472 comp: 6.793166 err: .631014
> 

> 21.1019 --> bad = 280.9843


310.15 -> bad = 60699

4.

> 2.

>> Number 2 Meantone
>> 
>> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>> TOP generators [1201.698520, 504.1341314]
>> bad: 21.551439 comp: 3.562072 err: 1.698521
> 

> 11.7652 -> bad = 235.1092


189.73 -> bad = 61144

5.

> 6.

>> Number 4 Beep
>> 
>> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>> TOP generators [1194.642673, 254.8994697]
>> bad: 23.664749 comp: 1.292030 err: 14.176105
> 

> 4.7295 -> bad = 317.0935


69.852 -> bad = 69170

6.

> 9.

>> Number 8 Schismic
>> 
>> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>> TOP generators [1200.760624, 498.1193303]
>> bad: 28.818558 comp: 5.618543 err: .912904
> 

> 20.2918 --> bad = 375.8947


291.09 -> bad = 77353

7.

> 5.

>> Number 3 Magic
>> 
>> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>> TOP generators [1201.276744, 380.7957184]
>> bad: 23.327687 comp: 4.274486 err: 1.276744
> 

> 15.5360 -> bad = 308.1642


265.95 -> bad = 90301

8.

> 8.

>> Number 10 Orwell
>> 
>> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
>> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
>> TOP generators [1199.532657, 271.4936472]
>> bad: 30.805067 comp: 5.706260 err: .946061
> 

> 19.9797 -> bad = 377.6573


324.9486 -> bad = 99896

9.

> 10.

>> Number 5 Augmented
>> 
>> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
>> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>> TOP generators [399.9922103, 107.3111730]
>> bad: 27.081145 comp: 2.147741 err: 5.870879
> 

> 8.3046 -> bad = 404.8933


143.07 -> bad = 1.2017e+005

10.

> 4.

>> Number 7 Dominant Seventh
>> 
>> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
>> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
>> TOP generators [1195.228951, 495.8810151]
>> bad: 28.744957 comp: 2.454561 err: 4.771049
> 

> 7.9560 -> bad = 301.9952


162.2 -> bad = 1.2552e+005

Message: 9592 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 05:06:24

Subject: Nonoctave scales in your livingroom

From: Gene Ward Smith

If you don't want to go to the bother of obtaining a musical
instrument based on 3^(1/13), you might try the temperament with TM
basis {3125/3087, 6561/6125}. If you take pure tritaves, you get a
generator of 3^(1/19), or 100.103 cents. If you object that after
twelve generator steps you get to a pretty good octave of 1201.235
cents, my reply to you is that 41-et has octaves also. Just ignore it.
Pretend it isn't there, and see what happens.

Message: 9593 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 11:12:40

Subject: Re: the choice of wedgie-norm greatly impacts miracle's ranking

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:

>> oops -- I may have done that all wrong. The scaling factors for 
the 
>> elements of the wedgie, the ones that you divide by to calculate 
> the  

>> multival norm -- do you have to *multiply* by them when you 
> calculate 

>> the multimonzo norm?
> 

> You rescale the monzos by multiplying, but it all comes out the 
same 
> up to a constant factor,


I don't think so! The element that you were formerly multiplying by 
the largest factor, you're now dividing by the largest factor, and 
vice versa!

Message: 9594 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 11:33:46

Subject: Re: the choice of wedgie-norm greatly impacts miracle's ranking

From: Paul Erlich

Whoops -- I forgot that the order of the elements changes too!

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
>> wrote:

>>> oops -- I may have done that all wrong. The scaling factors for 
> the 

>>> elements of the wedgie, the ones that you divide by to 
calculate 
>> the  

>>> multival norm -- do you have to *multiply* by them when you 
>> calculate 

>>> the multimonzo norm?
>> 

>> You rescale the monzos by multiplying, but it all comes out the 
> same 

>> up to a constant factor,
> 

> I don't think so! The element that you were formerly multiplying by 
> the largest factor, you're now dividing by the largest factor, and 
> vice versa!

Message: 9595 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 11:52:27

Subject: I guess Pajara's not #2

From: Paul Erlich

Here's the calculation for Pajara, the way I now understand it should 
be done:

take the two unison vectors

|1  0 2 -2>
|6 -2 0 -1>

Now find the determinant, and the "area" it represents, in each of 
the basis planes:

|1 0| = -2*(e23) -> 2*lg2(3) = 3.1699
|6 -2|

|1 2| = -12*(e25) -> 12*lg2(5) = 27.863
|6 0|

|1 -2| = 11*(e27) -> 11*lg2(7) = 30.881
|6 -1|

|0  2| = 4*(e35) -> 4*lg2(3)*lg2(5) = 14.721
|-2 0|

|0  -2| = -4*(e37) => 4*lg2(3)*lg2(7) = 17.798
|-2 -1|

|2  -2| = -2*(e57) => 2*lg2(5)*lg2(7) = 13.037
|0 -1|

The sum is 107.47.

So the below was wrong. I forgot that you reverse the order of the 
elements to convert a multival wedgie into a multimonzo wedgie! Doing 
so would, indeed, give the same rankings as my original L_1 
calculation. But that's gotta be the right norm. The Tenney lattice 
is set up to measure complexity, and the norm we always associate 
with it is the L_1 norm. Isn't that right? The L_1 norm on the monzo 
is what I've been using all along to calculate complexity for the 
codimension-1 case, in my graphs and in the "Attn: Gene 2" post . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> This time, I'll multiply, instead of dividing, the elements of the 
> wedgie by the relevant "unit areas" . . . still using L_1 . . .
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:

>> This time I'll try L_1 (multimonzo interpretation?) instead of 
>> L_infinity (multival interpretation?) to get complexity from the 
>> wedgie. Let's see how it affects the rankings -- we don't need to 
>> worry about scaling because Gene's badness measure is 
>> multiplicative . . .
>> 
>> The top 10 get re-ordered as follows, though this is probably not 
> the 

>> new top 10 overall . . .
> 
> 1.
>> 1.

>>> Number 1 Ennealimmal
>>> 
>>> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
>>> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
>>> TOP generators [133.3373752, 49.02398564]
>>> bad: 4.918774 comp: 11.628267 err: .036377
>> 

>> 39.8287 -> bad = 57.7058
> 

> 464.95 -> bad = 7864
> 
> 2.
>> 7.

>>> Number 6 Pajara
>>> 
>>> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>>> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>>> TOP generators [598.4467109, 106.5665459]
>>> bad: 27.754421 comp: 2.988993 err: 3.106578
>> 

>> 10.4021 -> bad = 336.1437
> 

> 130.6 -> bad = 52987
> 
> 3.
>> 3.

>>> Number 9 Miracle
>>> 
>>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>>> TOP generators [1200.631014, 116.7206423]
>>> bad: 29.119472 comp: 6.793166 err: .631014
>> 

>> 21.1019 --> bad = 280.9843
> 

> 310.15 -> bad = 60699
> 
> 4.
>> 2.

>>> Number 2 Meantone
>>> 
>>> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>>> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>>> TOP generators [1201.698520, 504.1341314]
>>> bad: 21.551439 comp: 3.562072 err: 1.698521
>> 

>> 11.7652 -> bad = 235.1092
> 

> 189.73 -> bad = 61144
> 
> 5.
>> 6.

>>> Number 4 Beep
>>> 
>>> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>>> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>>> TOP generators [1194.642673, 254.8994697]
>>> bad: 23.664749 comp: 1.292030 err: 14.176105
>> 

>> 4.7295 -> bad = 317.0935
> 

> 69.852 -> bad = 69170
> 
> 6.
>> 9.

>>> Number 8 Schismic
>>> 
>>> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>>> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>>> TOP generators [1200.760624, 498.1193303]
>>> bad: 28.818558 comp: 5.618543 err: .912904
>> 

>> 20.2918 --> bad = 375.8947
> 

> 291.09 -> bad = 77353
> 
> 7.
>> 5.

>>> Number 3 Magic
>>> 
>>> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>>> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>>> TOP generators [1201.276744, 380.7957184]
>>> bad: 23.327687 comp: 4.274486 err: 1.276744
>> 

>> 15.5360 -> bad = 308.1642
> 

> 265.95 -> bad = 90301
> 
> 8.
>> 8.

>>> Number 10 Orwell
>>> 
>>> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
>>> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
>>> TOP generators [1199.532657, 271.4936472]
>>> bad: 30.805067 comp: 5.706260 err: .946061
>> 

>> 19.9797 -> bad = 377.6573
> 

> 324.9486 -> bad = 99896
> 
> 9.
>> 10.

>>> Number 5 Augmented
>>> 
>>> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
>>> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>>> TOP generators [399.9922103, 107.3111730]
>>> bad: 27.081145 comp: 2.147741 err: 5.870879
>> 

>> 8.3046 -> bad = 404.8933
> 

> 143.07 -> bad = 1.2017e+005
> 
> 10.
>> 4.

>>> Number 7 Dominant Seventh
>>> 
>>> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
>>> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
>>> TOP generators [1195.228951, 495.8810151]
>>> bad: 28.744957 comp: 2.454561 err: 4.771049
>> 

>> 7.9560 -> bad = 301.9952
> 

> 162.2 -> bad = 1.2552e+005

Message: 9596 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 06:43:35

Subject: Re: 60 for Dave

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>>>>> We could use a quadratic penalty on the complexity too.
>>>>> 

>>>>> If we square both terms, doesn't this give the same ranking?
>>>> 

>>>> No, because following Dave, we're adding the terms, not
>>>> multiplying them. Dave restated Gene's product as a sum of logs.
>>> 

>>> Oh.  Why do that?
>> 

>> So that he could understand Gene's badness and my linear badness 
in 
>> the same form, and propose a compromise.
> 

> Ah.  Is yours the one from the Attn: Gene post?


No, it was the toy "Hermanic" example.


> That's good to know, but the above is just my value judgement, and
> as you point out log-flat badness frees us from those, in a sense.


But it results in an infinite number of temperaments, or none at all, 
depending on what level of badness you use as your cutoff.

Message: 9597 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 19:23:07

Subject: The true top 32 in log-flat?

From: Paul Erlich

I re-ranked Gene's top 64 using L_1 and got the following top 32. 
Anything missing?

1.

> Ennealimmal
>
> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> TOP generators [133.3373752, 49.02398564]
> bad: 4.918774 comp: 11.628267 err: .036377


39.8287 -> bad = 57.7058

2.

> Meantone (Huygens)
>
> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> TOP generators [1201.698520, 504.1341314]
> bad: 21.551439 comp: 3.562072 err: 1.698521


11.7652 -> bad = 235.1092

3.

> Miracle
>
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014


21.1019 --> bad = 280.9843

4.

> Hemiwuerschmidt
> 
> [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> TOP generators [1199.692003, 193.8224275]
> bad: 31.386908 comp: 10.094876 err: .307997


31.212 -> bad = 300.04

5.

> Dominant Seventh
>
> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> TOP generators [1195.228951, 495.8810151]
> bad: 28.744957 comp: 2.454561 err: 4.771049


7.9560 -> bad = 301.9952

6.

> Blackwood
> 
> [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
> TOP generators [239.1786927, 83.83059859]
> bad: 34.210608 comp: 2.173813 err: 7.239629


6.4749 -> bad = 303.52

7.

> Magic
>
> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> TOP generators [1201.276744, 380.7957184]
> bad: 23.327687 comp: 4.274486 err: 1.276744


15.5360 -> bad = 308.1642

8.

> Beep
>
> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> TOP generators [1194.642673, 254.8994697]
> bad: 23.664749 comp: 1.292030 err: 14.176105


4.7295 -> bad = 317.0935

9.

> Pajara
>
> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 106.5665459]
> bad: 27.754421 comp: 2.988993 err: 3.106578


10.4021 -> bad = 336.1437

10.

> Semisixths
> 
> [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
> TOP generators [1198.389531, 443.1602931]
> bad: 34.533812 comp: 4.630693 err: 1.610469


14.459 -> bad = 336.67

11.

> Catakleismic
> 
> [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
> TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
> TOP generators [1200.536355, 316.9063960]
> bad: 32.938503 comp: 7.836558 err: .536356


25.127 -> bad = 338.65

12.

> Diminished
> 
> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> TOP generators [298.5321149, 101.4561401]
> bad: 37.396767 comp: 2.523719 err: 5.871540


7.917 -> bad = 368.02

13.

> Schismic
>
> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> TOP generators [1200.760624, 498.1193303]
> bad: 28.818558 comp: 5.618543 err: .912904


20.2918 --> bad = 375.8947

14.

> Orwell
>
> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> TOP generators [1199.532657, 271.4936472]
> bad: 30.805067 comp: 5.706260 err: .946061


19.9797 -> bad = 377.6573

15.

> Hemififths
> 
> [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
> TOP tuning [1199.700353, 1902.429930, 2785.617954, 3368.041901]
> TOP generators [1199.700353, 351.3647888]
> bad: 34.737019 comp: 10.766914 err: .299647


35.677 -> bad = 381.41

16.

> Father
> 
> [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> TOP generators [1185.869125, 447.3863410]
> bad: 33.256527 comp: 1.534101 err: 14.130876


5.2007 -> bad = 382.2

17.

> Amity
> 
> [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
> TOP tuning [1199.723894, 1902.392618, 2786.717797, 3369.601033]
> TOP generators [1199.723894, 339.3558130]
> bad: 37.532790 comp: 11.659166 err: .276106


38.128 -> bad = 401.39

18.

> Augmented
>
> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> TOP generators [399.9922103, 107.3111730]
> bad: 27.081145 comp: 2.147741 err: 5.870879


8.3046 -> bad = 404.8933

19.

> Parakleismic
> 
> [13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]]
> TOP tuning [1199.738066, 1902.291445, 2786.921905, 3368.090564]
> TOP generators [1199.738066, 315.1076065]
> bad: 40.713036 comp: 12.467252 err: .261934


39.586 -> bad = 410.46

20.

> Tripletone
> 
> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> TOP generators [399.0200131, 92.45965769]
> bad: 48.112067 comp: 4.045351 err: 2.939961


12.125 -> bad = 432.24

21.

> {21/20, 28/27}
> 
> [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
> TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
> TOP generators [1214.253642, 509.4012304]
> bad: 42.300772 comp: 1.722706 err: 14.253642


5.5723 -> bad = 442.58

22.

> Decimal
> 
> [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
> TOP generators [603.8288989, 250.6116362]
> bad: 48.773723 comp: 2.523719 err: 7.657798


7.6792 -> bad = 451.58

23.

> Hemifourths
> 
> [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
> TOP generators [1203.668841, 252.4803582]
> bad: 43.552336 comp: 3.445412 err: 3.668842


11.204 -> bad = 460.59

24.

> Negri
> 
> [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
> TOP generators [1203.187309, 124.8419629]
> bad: 46.125886 comp: 3.804173 err: 3.187309


12.125 -> bad = 468.55

25.

> Nonkleismic
> 
> [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
> TOP generators [1198.828458, 309.8926610]
> bad: 46.635848 comp: 6.309298 err: 1.171542


20.326 -> bad = 484

26.

> Kleismic
> 
> [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
> TOP generators [1203.187309, 317.8344609]
> bad: 45.676063 comp: 3.785579 err: 3.187309


12.409 -> bad = 490.77

27.

> Dicot
> 
> [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
> TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
> TOP generators [1204.048159, 356.3998255]
> bad: 42.920570 comp: 2.137243 err: 9.396316


7.2314 -> bad = 491.37

28.

> Superpythagorean
> 
> [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
> TOP generators [1197.596121, 489.4271829]
> bad: 50.917015 comp: 4.602303 err: 2.403879


14.431 -> bad = 500.61

29.

> Injera
> 
> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
> TOP generators [600.8889070, 93.60982493]
> bad: 42.529834 comp: 3.445412 err: 3.582707


11.918 -> bad = 508.85

30.

> {25/24, 81/80} Jamesbond?
> 
> [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
> TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
> TOP generators [172.7759159, 86.69241190]
> bad: 58.637859 comp: 2.493450 err: 9.431411


7.4202 -> bad = 519.28

31.

> Quartaminorthirds
> 
> [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
> TOP generators [1199.792743, 77.83315314]
> bad: 47.721352 comp: 6.742251 err: 1.049791


22.397 -> bad = 526.59

32.

> Pelogic
> 
> [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
> TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
> TOP generators [1209.734056, 532.9412251]
> bad: 39.824125 comp: 2.022675 err: 9.734056


7.426 -> bad = 536.78






**********************************************************************















> Number 43
> 
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 162.3159606]
> bad: 57.621529 comp: 4.306766 err: 3.106578


13.19 -> bad = 540.44


> Number 36 Supersupermajor
> 
> [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
> TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
> TOP generators [1200.231587, 234.3804692]
> bad: 52.638504 comp: 7.670504 err: .894655


24.923 -> bad = 555.72


> Number 47
> 
> [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]]
> TOP tuning [1200.284965, 1901.503343, 2786.975381, 3369.219732]
> TOP generators [600.1424823, 83.17776441]
> bad: 61.101493 comp: 14.643003 err: .284965


44.37 -> bad = 561


> Number 46 Hemithirds
> 
> [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
> TOP tuning [1200.363229, 1901.194685, 2787.427555, 3367.479202]
> TOP generators [1200.363229, 193.3505488]
> bad: 60.573479 comp: 11.237086 err: .479706


34.589 -> bad = 573.94


> Number 44 Octacot
> 
> [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
> TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
> TOP generators [1199.031259, 88.05739491]
> bad: 58.217715 comp: 7.752178 err: .968741


24.394 -> bad = 576.47


> Number 35 Supermajor seconds
> 
> [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
> TOP generators [1201.698520, 232.5214630]
> bad: 51.806440 comp: 5.522763 err: 1.698521


18.448 -> bad = 578.06


> Number 25 Waage? Compton? Duodecimal?
> 
> [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
> TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
> TOP generators [100.0514209, 16.55882096]
> bad: 45.097159 comp: 8.548972 err: .617051


30.795 -> bad = 585.17


> Number 55
> 
> [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
> TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
> TOP generators [99.80617249, 24.58395811]
> bad: 65.630949 comp: 4.295482 err: 3.557008


12.84 -> bad = 586.43


> Number 48 Flattone
> 
> [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
> TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
> TOP generators [1202.536419, 507.1379663]
> bad: 61.126418 comp: 4.909123 err: 2.536420


15.376 -> bad = 599.67


> Number 38 {3136/3125, 5120/5103} Misty
> 
> [3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]]
> TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021]
> TOP generators [399.8871550, 96.94420930]
> bad: 53.622498 comp: 12.585536 err: .338535


42.92 -> bad = 623.63


> Number 41 {28/27, 50/49}
> 
> [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
> TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
> TOP generators [595.7998193, 127.8698005]
> bad: 56.092257 comp: 2.584059 err: 8.400361


8.701 -> bad = 635.97


> Number 63
> 
> [8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]]
> TOP tuning [1198.975478, 1900.576277, 2788.692580, 3365.949709]
> TOP generators [1198.975478, 62.17183489]
> bad: 68.767371 comp: 8.192765 err: 1.024522


25.137 -> bad = 647.35


> Number 49 Diaschismic
> 
> [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
> TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
> TOP generators [599.3662015, 103.7870123]
> bad: 61.527901 comp: 6.966993 err: 1.267597


22.629 -> bad = 649.07


> Number 57
> 
> [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> TOP generators [1185.869125, 223.6931705]
> bad: 66.774944 comp: 2.173813 err: 14.130876


6.7795 -> bad = 649.47


> Number 59
> 
> [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
> TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
> TOP generators [1193.415676, 158.1468146]
> bad: 67.670842 comp: 3.205865 err: 6.584324


9.9461 -> bad = 651.35


> Number 56
> 
> [2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
> TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
> TOP generators [1204.567524, 355.9419091]
> bad: 66.522610 comp: 2.696901 err: 9.146173


8.4704 -> bad = 656.21


> Number 26 Wizard
> 
> [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
> TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
> TOP generators [600.3197857, 216.7702531]
> bad: 45.381303 comp: 8.423526 err: .639571


32.407 -> bad = 671.69


> Number 37 {6144/6125, 10976/10935} Hendecatonic?
> 
> [11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]]
> TOP tuning [1199.662182, 1902.490429, 2787.098101, 3368.740066]
> TOP generators [109.0601984, 48.46705632]
> bad: 53.458690 comp: 12.579627 err: .337818


44.677 -> bad = 674.3


> Number 42 Porcupine
> 
> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> TOP generators [1196.905960, 162.3176609]
> bad: 57.088650 comp: 4.295482 err: 3.094040


14.796 -> bad = 677.35


> Number 33 {1029/1024, 4375/4374}
> 
> [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]]
> TOP tuning [1200.421488, 1901.286959, 2785.446889, 3367.642640]
> TOP generators [600.2107440, 183.2944602]
> bad: 50.004574 comp: 10.892116 err: .421488


40.255 -> bad = 683


> Number 39 {1728/1715, 4000/3993}
> 
> [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
> TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
> TOP generators [1199.083445, 45.17026643]
> bad: 55.081549 comp: 7.752178 err: .916555


28.441 -> bad = 741.38


> Number 62
> 
> [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> TOP generators [592.7342285, 146.7842660]
> bad: 68.668284 comp: 2.173813 err: 14.531543


7.1855 -> bad = 750.29


> Number 40 {36/35, 160/147} Hystrix?
> 
> [3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]]
> TOP tuning [1187.933715, 1892.564743, 2758.296667, 3402.700250]
> TOP generators [1187.933715, 161.1008955]
> bad: 55.952057 comp: 2.153383 err: 12.066285


8.0882 -> bad = 789.37


> Number 61 Hemikleismic
> 
> [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]]
> TOP tuning [1199.411231, 1902.888178, 2785.151380, 3370.478790]
> TOP generators [1199.411231, 158.5740148]
> bad: 68.516458 comp: 10.787602 err: .588769


36.649 -> bad = 790.81


> Number 52 Tritonic
> 
> [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]]
> TOP tuning [1201.023211, 1900.333250, 2785.201472, 3365.953391]
> TOP generators [1201.023211, 580.7519186]
> bad: 63.536850 comp: 7.880073 err: 1.023211


27.923 -> bad = 797.81


> Number 50 Superkleismic
> 
> [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
> TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
> TOP generators [1201.371918, 322.3731369]
> bad: 62.364585 comp: 6.742251 err: 1.371918


24.524 -> bad = 825.11


> Number 54
> 
> [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
> TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
> TOP generators [1202.659696, 82.97467050]
> bad: 64.556006 comp: 4.306766 err: 3.480440


15.623 -> bad = 849.49


> Number 53
> 
> [1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]]
> TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869]
> TOP generators [1199.680495, 497.2520023]
> bad: 64.536886 comp: 14.212326 err: .319505


51.639 -> bad = 851.99


> Number 51
> 
> [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
> TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
> TOP generators [1201.135545, 387.5841360]
> bad: 62.703297 comp: 6.411729 err: 1.525246


23.841 -> bad = 866.91


> Number 60
> 
> [3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
> TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
> TOP generators [397.8052253, 76.63521863]
> bad: 68.337269 comp: 3.221612 err: 6.584324


11.571 -> bad = 881.53


> Number 64
> 
> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
> TOP generators [1202.900537, 570.4479508]
> bad: 69.388565 comp: 4.891080 err: 2.900537


17.521 -> bad = 890.45


> Number 58
> 
> [5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
> TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
> TOP generators [1194.335372, 99.13879319]
> bad: 67.244049 comp: 3.445412 err: 5.664628


12.818 -> bad = 930.67

Message: 9598 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 06:54:13

Subject: Re: pelogic and kleismic/hanson

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:

> On Sat, 31 Jan 2004 00:37:28 -0000, "Paul Erlich" <perlich@a...>
> wrote:
> 
>> See
>> 
>> http://www.anaphoria.com/keygrid.PDF - Type Ok * [with cont.]  (Wayb.)
>> 

>> page 7 seems to be using some pelog terminology; anyone familiar 
with 
>> it?
> 

> I'm not familiar with this terminology, but the keyboard is clearly 
based
> on a generator of 5 steps of 23-ET, while the generator of pelogic
> temperament is 10 steps of 23. In other words, this is the scale 
I've been
> calling "superpelog", with the basic 9-note MOS subset used as the 
basis
> for a system of notation.
> 
> I'll add a reference to this paper on my superpelog page:
> 
> Superpelog tuning * [with cont.]  (Wayb.)

Kyewl!

Message: 9599 - Contents - Hide Contents

Date: Sat, 31 Jan 2004 19:28:13

Subject: Re: the choice of wedgie-norm greatly impacts miracle's ranking

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


>> You rescale the monzos by multiplying, but it all comes out the 
> same 

>> up to a constant factor,
> 

> I don't think so! The element that you were formerly multiplying by 
> the largest factor, you're now dividing by the largest factor, and 
> vice versa!


If you mulitply by p5p7 and then divide by p2p3p5p7, you get 1/p2p3, 
and so forth.

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