Tuning-Math Digests messages 9575 - 9599

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 10

Previous Next

9000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950

9550 - 9575 -



top of page bottom of page down


Message: 9575

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,


top of page bottom of page up down


Message: 9576

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 *
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 . . .


top of page bottom of page up down


Message: 9577

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]


top of page bottom of page up down


Message: 9580

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

Subject: Re: Crunch algorithm

From: Paul Erlich

Please see http://www.anaphoria.com/viggo3.PDF - Ok * . . .

--- 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]


top of page bottom of page up down


Message: 9582

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

Subject: Re: kleismic v. hanson

From: Carl Lumma

>> 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?

AFAIK, mainly the 53-note version.

-Carl


top of page bottom of page up down


Message: 9583

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!


top of page bottom of page up down


Message: 9584

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.


top of page bottom of page up down


Message: 9587

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 . . .


top of page bottom of page up down


Message: 9589

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?


top of page bottom of page up down


Message: 9591

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


top of page bottom of page up down


Message: 9593

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!


top of page bottom of page up down


Message: 9594

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!


top of page bottom of page up down


Message: 9595

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


top of page bottom of page up down


Message: 9596

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.


top of page bottom of page up down


Message: 9597

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


top of page bottom of page up down


Message: 9598

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 - Ok *
> >
> >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 *

Kyewl!


top of page bottom of page up

Previous Next

9000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950

9550 - 9575 -

top of page