Modes of a Bar

Thanks to those who replied to my query. I suppose ths next step is to
look for interesting scales to fit the timbre and spectra of my metal
bars.

Best Wishes

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> Thanks to those who replied to my query. I suppose ths next step is
to
> look for interesting scales to fit the timbre and spectra of my
metal
> bars.
>
> Best Wishes

Kraig Grady would probably advise you to ignore the partials and tune
the bars to a JI system or something like it.

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> Thanks to those who replied to my query. I suppose ths next step is
to
> look for interesting scales to fit the timbre and spectra of my
metal
> bars.

Maybe Graham Breed could run his linear-temperament-finding program
for the intervals in the metal bar spectrum, instead of JI intervals.
Then again, I'm not sure what would substitute for the "seeding" of
the "Breeding", which was consistent ETs in the harmonic-timbre case.
But I'm sure it can be done.

In-Reply-To: <9oqtc7+6808@eGroups.com>
Paul wrote:

> Maybe Graham Breed could run his linear-temperament-finding program
> for the intervals in the metal bar spectrum, instead of JI intervals.
> Then again, I'm not sure what would substitute for the "seeding" of
> the "Breeding", which was consistent ETs in the harmonic-timbre case.
> But I'm sure it can be done.

Oh, alright then. I'm using the usual odd-limit matrices, although they
aren't quite right here. Results using the octave as the equivalence
interval are at

<http://x31eq.com/limit5.tubulong>
<http://x31eq.com/limit7.tubulong>
<http://x31eq.com/limit9.tubulong>
<http://x31eq.com/limit11.tubulong>
<http://x31eq.com/limit13.tubulong>
<http://x31eq.com/limit15.tubulong>

and taking the second partial as the equivalence interval

<http://x31eq.com/limit5.tubulong.nonoctave>
<http://x31eq.com/limit7.tubulong.nonoctave>
<http://x31eq.com/limit9.tubulong.nonoctave>
<http://x31eq.com/limit11.tubulong.nonoctave>
<http://x31eq.com/limit13.tubulong.nonoctave>
<http://x31eq.com/limit15.tubulong.nonoctave>

The first usable scales seem to be in the "11 limit" results in both
cases. For a 2:1 octave:

11/24, 274.6 cent generator

basis:
(0.5, 0.22886330498902493)

mapping by period and generator:
[(2, 0), (3, 0), (9, -9), (9, -6), (6, 3)]

mapping by steps:
[(26, 22), (39, 33), (63, 54), (81, 69), (96, 81)]

unison vectors:
[[-3, 2, 0, 0, 0], [-6, 1, -2, 3, 0], [12, 1, -1, 0, -3]]

highest interval width: 12
complexity measure: 24 (26 for smallest MOS)
highest error: 0.006922 (8.306 cents)

and the nonoctave generator:

44/135, 390.9 cent generator

basis:
(1.0, 0.32574817436635989)

mapping by period and generator:
[(1, 0), (0, 5), (6, -12), (8, -17), (7, -13)]

mapping by steps:
[(89, 46), (145, 75), (186, 96), (219, 113), (246, 127)]

unison vectors:
[[2, -1, 1, -1, 0], [4, -2, -3, 0, 2], [13, -5, -1, 0, -1]]

highest interval width: 27
complexity measure: 27 (28 for smallest MOS)
highest error: 0.005267 (6.321 cents)

Note that wherever it says "cents" for nonoctave generators, these aren't
real cents. The equivalence interval happens to be exactly 1.5 octaves.
so the real generator here is 390.9*1.5 = 586.35 cents. And it's really
9.5 cents from the ideal ratios.

Although these both use the 11-limit matrix, the nonoctave scale does buy
you one more partial. These "7-limit" nonoctave scales also look good:

7/31, 274.6 cent generator [really 411.95 cents]

basis:
(1.0, 0.22886330498902491)

mapping by period and generator:
[(1, 0), (3, -6), (3, -4), (2, 2)]

mapping by steps:
[(22, 9), (36, 15), (46, 19), (54, 22)]

unison vectors:
[[2, -1, 1, -1], [-3, -2, 3, 0]]

highest interval width: 8
complexity measure: 8 (9 for smallest MOS)
highest error: 0.004614 (5.537 cents) [really 8.3 cents]

11/35, 377.4 cent generator [really 566.1 cents]

basis:
(1.0, 0.31449601452617937)

mapping by period and generator:
[(1, 0), (1, 2), (-2, 13), (-1, 11)]

mapping by steps:
[(19, 16), (31, 26), (40, 33), (47, 39)]

unison vectors:
[[2, -1, 1, -1], [13, -11, 0, 2]]

highest interval width: 13
complexity measure: 13 (16 for smallest MOS)
highest error: 0.002885 (3.462 cents) [really 5.2 cents]

I happen to get slightly different values to McLaren for the frequency
ratios. Here are mine:

2.82843
5.42326
8.77058
12.86626
17.70875
23.29741

If you're planning to use this method to tune real bars, I reckon
measuring their spectra would be a good idea instead of relying on the
ideal theory.

Graham

--- In tuning@y..., graham@m... wrote:

> Although these both use the 11-limit matrix, the nonoctave scale
does buy
> you one more partial. These "7-limit" nonoctave scales also look
good:
>
>
> 7/31, 274.6 cent generator [really 411.95 cents]
>
> basis:
> (1.0, 0.22886330498902491)
>
> mapping by period and generator:
> [(1, 0), (3, -6), (3, -4), (2, 2)]
>
> mapping by steps:
> [(22, 9), (36, 15), (46, 19), (54, 22)]
>
> unison vectors:
> [[2, -1, 1, -1], [-3, -2, 3, 0]]
>
> highest interval width: 8
> complexity measure: 8 (9 for smallest MOS)
> highest error: 0.004614 (5.537 cents) [really 8.3 cents]

I vote for this one. I think the others have too high a complexity
measure.
>
> If you're planning to use this method to tune real bars, I reckon
> measuring their spectra would be a good idea instead of relying on
the
> ideal theory.

I completely agree.