one from the archives

Anybody have anything to say about this tuning? How's
the connectivity, Gene?

-C.

!
"Stairs", 7-limit tuning, 1998 Carl Lumma.
12
!
36/35
8/7
6/5
5/4
48/35
10/7
3/2
5/3
12/7
9/5
40/21
2/1
!

It's been a few days, so I'm joining the resending parade; I hope Carl
still remembers what these are about!

[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]

edge connectivity = 3

characteristic polynomial =
x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*x^4-8*x^3+304*x^2
+96*x

This tells us there are 29 consonant intervals and 22 consonant triads

The scale is not epimorphic, but can be extended in various ways, for
instance to the
h15-epimorphic scale

[1, 36/35, 10/9, 8/7, 6/5, 5/4, 4/3, 48/35, 10/7, 3/2, 8/5, 5/3, 12/7,
9/5, 40/21]

This also has connectivity 3 and a characteristic polynomial

x^15-41*x^13-70*x^12+450*x^11+1316*x^10-899*x^9-6406*x^8-3948*x^7+9034*x^
6+
12050*x^5-298*x^4-7243*x^3-3682*x^2-445*x+24

It therefore has 41 consonant intervals and 35 consonant triads.

I've attached two files which show the graph for each of these scales.
Now I'm wondering what the story is--where does this scale arise from?

>It's been a few days, so I'm joining the resending parade; I hope
>Carl still remembers what these are about!
>
>[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]
>
>edge connectivity = 3

I remember connectivity; 3 isn't so hot for a 12-tone scale in
the 7-limit, eh?

>characteristic polynomial =
>x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*
>x^4-8*x^3+304*x^2+96*x

Never did the homework to understand these.

>This tells us there are 29 consonant intervals and 22 consonant
>triads

That doesn't seem like much to write home about either. What
are some good scales here in your experience, Gene? The three
winners on my list were:

1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8

1/1 16/15 28/25 7/6 5/4 4/3 7/5 112/75 8/5 5/3 7/4 28/15

1/1 21/20 7/6 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 9/5

I ended up using the first for melodic reasons.

>The scale is not epimorphic, but can be extended in various ways,
>for instance to the h15-epimorphic scale
/.../
>I've attached two files which show the graph for each of these
>scales.

Thanks!

>Now I'm wondering what the story is--where does this scale
>arise from?

I came up with it when drawing lattices on paper some time ago,
when looking for good 12-tone 7-limit tunings for my piano.

-Carl

I wrote...

> The three winners on my list were:
>
> 1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8

"lester"

> 1/1 16/15 28/25 7/6 5/4 4/3 7/5 112/75 8/5 5/3 7/4 28/15

"prism"

> 1/1 21/20 7/6 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 9/5

"stelhex"

And this one by Wilson/Hahn:

1/1 21/20 35/32 6/5 5/4 21/16 7/5 3/2 25/16 42/25 7/4 15/8

"class"

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >characteristic polynomial =
> >x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*
> >x^4-8*x^3+304*x^2+96*x
>
> Never did the homework to understand these.

The adjacency matrix of a graph is a square matrix labeled by verticies; it has a "1" if they are connected, and a "0" if not (counting the vertex to itself as a "0".) The characteristic polynomial of this is as above, and is a graph invariant. The
n-2 term gives the number of edges, and the n-3 term twice the number of triads.

> >This tells us there are 29 consonant intervals and 22 consonant
> >triads
>
> That doesn't seem like much to write home about either. What
> are some good scales here in your experience, Gene?

Not that great.

>>>characteristic polynomial =
>>>x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*
>>>x^4-8*x^3+304*x^2+96*x
>>
>>Never did the homework to understand these.
>
>The adjacency matrix of a graph is a square matrix labeled by
>verticies; it has a "1" if they are connected, and a "0" if not
>(counting the vertex to itself as a "0".) The characteristic
>polynomial of this is as above, and is a graph invariant. The
>n-2 term gives the number of edges, and the n-3 term twice the
>number of triads.

That's crazy.

What happened to the n-1 term, above?

Does it show tetrads?

>>>This tells us there are 29 consonant intervals and 22 consonant
>>>triads
>>
>>That doesn't seem like much to write home about either. What
>>are some good scales here in your experience, Gene?
>
>Not that great.

I'd be interested in the stats on the "class" and "stelhex" scales,
for comparison.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> >The adjacency matrix of a graph is a square matrix labeled by
> >verticies; it has a "1" if they are connected, and a "0" if not
> >(counting the vertex to itself as a "0".) The characteristic
> >polynomial of this is as above, and is a graph invariant. The
> >n-2 term gives the number of edges, and the n-3 term twice the
> >number of triads.
>
> That's crazy.

Why?

> What happened to the n-1 term, above?

Since the diagonal terms are all 0, the trace is 0 and so the trace term is 0.

> Does it show tetrads?

It doesn't show anything. To show tetrads, we would need to count principal minors which were all 1s except along the diagonal, which would be 0. I don't know if that can be done using the coefficients of the characteristic polynomial.

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]
> >
> >edge connectivity = 3
> >characteristic polynomial =
> >x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*
> >x^4-8*x^3+304*x^2+96*x

Let's compare this to some other possibilities:

[1, 21/20, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

This is a sentimental favorite--my first scale.

x^12-27*x^10-38*x^9+168*x^8+366*x^7-206*x^6-950*x^5-474*x^4+400*x^3+437*x^2+130*x+12

connectivity = 2

Not quite up to your numbers, but a good extension of JI diatonic.

[1, 15/14, 9/8, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 7/4, 15/8]
x^12-26*x^10-36*x^9+156*x^8+334*x^7-176*x^6-768*x^5-266*x^4+370*x^3+190*x^2-36*x-15
2

Another verion of the above.

[1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15]

This is Euclidean reduced, vertex centered.

x^12-27*x^10-38*x^9+168*x^8+368*x^7-172*x^6-792*x^5-232*x^4+368*x^3+96*x^2-64*x
2

[1, 15/14, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

This is Euclidean reduced, tetrad (shallow hole) centered.

x^12-28*x^10-42*x^9+175*x^8+430*x^7-70*x^6-812*x^5-396*x^4+374*x^3+302*x^2+
32*x-3
2

[1, 15/14, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 7/4, 15/8]

This is Euclidean reduced, hexany (deep hole) centered. Finally something to beat your numbers; I think your scale is actually
quite good. This has 30 intervals and 25 triads.

x^12-30*x^10-50*x^9+189*x^8+506*x^7-103*x^6-1118*x^5-487*x^4+772*x^3+508*x^2-128*x-96
3

[1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

A modified version of the shallow hole reduction.

x^12-30*x^10-48*x^9+193*x^8+498*x^7-96*x^6-1096*x^5-735*x^4+238*x^3+278*x^2-8*x-27
2

[1, 21/20, 35/32, 7/6, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

Another modified shallow hole reduction.

x^12-29*x^10-44*x^9+183*x^8+432*x^7-148*x^6-924*x^5-444*x^4+276*x^3+177*x^2-22*x-15
2

>>>The adjacency matrix of a graph is a square matrix labeled by
>>>verticies; it has a "1" if they are connected, and a "0" if not
>>>(counting the vertex to itself as a "0".) The characteristic
>>>polynomial of this is as above, and is a graph invariant. The
>>>n-2 term gives the number of edges, and the n-3 term twice the
>>>number of triads.
>>
>>That's crazy.
>
>Why?

What makes anything crazy? That you don't understand it? That's
a pretty good def., I guess.

>> Does it show tetrads?
>
>It doesn't show anything. To show tetrads, we would need to count
>principal minors which were all 1s except along the diagonal, which
>would be 0. I don't know if that can be done using the coefficients
>of the characteristic polynomial.

Noted.

-Carl