! middle-path 7-limit tetradic scales for kalle

pajara-10 (can be approximated in 22-equal):
0
108.81
217.63
382.37
491.19
600.00
708.81
817.63
982.37
1091.19
steps:
108.81
108.81
164.74
108.81
108.81
108.81
108.81
164.74
108.81
108.81

injera-14 (can be approximated in 26-equal):
0
93.65
187.30
280.95
319.05
412.70
506.35
600.00
693.65
787.30
880.95
919.05
1012.70
1106.35
steps
93.65
93.65
93.65
38.09
93.65
93.65
93.65
93.65
93.65
93.65
38.09
93.65
93.65
93.65

blackwood-10 (can be approximated in 15-equal, 25-equal, 40-equal):
0
149.39
240.00
389.39
480.00
629.39
720.00
869.39
960.00
1109.39
steps
149.39
90.61
149.39
90.61
149.39
90.61
149.39
90.61
149.39
90.61

new ones:
10 notes, 6 tetrads (can be approximated in 19-equal):

0
125.47
250.94
376.41
501.88
572.66
698.12
823.59
949.06
1074.53
steps
125.47
125.47
125.47
125.47
70.78
125.47
125.47
125.47
125.47
125.47

14 notes, 12 tetrads (can be approximated in 19-equal):

0
63.71
189.03
252.74
316.45
441.77
505.48
630.80
694.52
758.23
883.55
947.26
1010.97
1136.29
steps
63.71
125.32
63.71
63.71
125.32
63.71
125.32
63.71
63.71
125.32
63.71
63.71
125.32
63.71

14 notes, 12 tetrads (can be approximated in 23-equal):
0
103.53
157.18
260.71
364.23
417.88
521.41
575.06
678.59
782.12
835.77
939.29
1042.82
1096.47
steps
103.53
53.65
103.53
103.53
53.65
103.53
53.65
103.53
103.53
53.65
103.53
103.53
53.65
103.53

10 notes, 4 tetrads (can be approximated in 26-equal):
0
137.40
231.30
368.70
462.60
600.00
737.40
831.30
968.70
1062.60
steps
137.40
93.89
137.40
93.89
137.40
137.40
93.89
137.40
93.89
137.40

10 note scale, 8 tetrads (can be approximated in 28-equal):
0
128.51
257.02
342.98
471.49
600.00
728.51
857.02
942.98
1071.49
steps
128.51
128.51
85.95
128.51
128.51
128.51
128.51
85.95
128.51
128.51
note that this uses 28-equal's second-best fifth . . .

> 10 notes, 6 tetrads (can be approximated in 19-equal):
>
> 0
> 125.47
> 250.94
> 376.41
> 501.88
> 572.66
> 698.12
> 823.59
> 949.06
> 1074.53
> steps
> 125.47
> 125.47
> 125.47
> 125.47
> 70.78
> 125.47
> 125.47
> 125.47
> 125.47
> 125.47

Strictly proper.
This is like the 10-tone MOS of quadrafourths.

What's the new name for quadrafourths?

> 10 notes, 4 tetrads (can be approximated in 26-equal):
> 0
> 137.40
> 231.30
> 368.70
> 462.60
> 600.00
> 737.40
> 831.30
> 968.70
> 1062.60
> steps
> 137.40
> 93.89
> 137.40
> 93.89
> 137.40
> 137.40
> 93.89
> 137.40
> 93.89
> 137.40

Strictly proper.
A little too homogeneous, and thus only 4 tetrads.

> 10 note scale, 8 tetrads (can be approximated in 28-equal):
> 0
> 128.51
> 257.02
> 342.98
> 471.49
> 600.00
> 728.51
> 857.02
> 942.98
> 1071.49
> steps
> 128.51
> 128.51
> 85.95
> 128.51
> 128.51
> 128.51
> 128.51
> 85.95
> 128.51
> 128.51
> note that this uses 28-equal's second-best fifth . . .

Indeed; it isn't very good.

-Carl

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

> What's the new name for quadrafourths?

Negri.

--- In tuning@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> > 10 notes, 6 tetrads (can be approximated in 19-equal):
> >
> > 0
> > 125.47
> > 250.94
> > 376.41
> > 501.88
> > 572.66
> > 698.12
> > 823.59
> > 949.06
> > 1074.53
> > steps
> > 125.47
> > 125.47
> > 125.47
> > 125.47
> > 70.78
> > 125.47
> > 125.47
> > 125.47
> > 125.47
> > 125.47
>
> Strictly proper.
> This is like the 10-tone MOS of quadrafourths.
>
> What's the new name for quadrafourths?

like gene said, negri -- and in fact negri's design for a 19-tone
keyboard has this 10-tone scale on the white keys; the other 9
are black.

>
> > 10 notes, 4 tetrads (can be approximated in 26-equal):
> > 0
> > 137.40
> > 231.30
> > 368.70
> > 462.60
> > 600.00
> > 737.40
> > 831.30
> > 968.70
> > 1062.60
> > steps
> > 137.40
> > 93.89
> > 137.40
> > 93.89
> > 137.40
> > 137.40
> > 93.89
> > 137.40
> > 93.89
> > 137.40
>
> Strictly proper.
> A little too homogeneous, and thus only 4 tetrads.
>
> > 10 note scale, 8 tetrads (can be approximated in 28-equal):
> > 0
> > 128.51
> > 257.02
> > 342.98
> > 471.49
> > 600.00
> > 728.51
> > 857.02
> > 942.98
> > 1071.49
> > steps
> > 128.51
> > 128.51
> > 85.95
> > 128.51
> > 128.51
> > 128.51
> > 128.51
> > 85.95
> > 128.51
> > 128.51
> > note that this uses 28-equal's second-best fifth . . .
>
> Indeed; it isn't very good.
>
> -Carl

and the other two new ones?

>and the other two new ones?

Too many notes!

But! This technique is what I've been dreaming of
for 5 years -- a shortcut to brute force. Is your
list complete for 4:5:6:7 and 1/1:6/5:3/2:12/7?
It doesn't catch 4:5:6:7 and 7/(7:6:5:4), does it?

If the former is to be answered yes, then perhaps
it's time to plug in other desirable chord pairs,
such as higher/lower harmonic clusters, ASSs, etc.
We'll loose the o/utonal doubling feature of the
unbroken chain, but. . .

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >and the other two new ones?
>
> Too many notes!

sorry you feel 14 notes per octave is too many.

> But! This technique is what I've been dreaming of
> for 5 years -- a shortcut to brute force.

yes. kudos to kalle for bringing it to the surface.

> Is your
> list complete for 4:5:6:7 and 1/1:6/5:3/2:12/7?

it's complete for rms error below 20 cents and geometric badness
below some reasonable limit . . . see tuning-math.

> It doesn't catch 4:5:6:7 and 7/(7:6:5:4), does it?

umm . . . 4:6 = 5:7?? you won't acheive that with rms below 20 cents,
anyhow . . .

>
> If the former is to be answered yes, then perhaps
> it's time to plug in other desirable chord pairs,
> such as higher/lower harmonic clusters, ASSs, etc.

sure, why not?

>>Is your list complete for 4:5:6:7 and 1/1:6/5:3/2:12/7?
>
>it's complete for rms error below 20 cents and geometric badness
>below some reasonable limit

Cool.

>. . . see tuning-math.

I read tuning-math.

>>It doesn't catch 4:5:6:7 and 7/(7:6:5:4), does it?
>
>umm . . . 4:6 = 5:7? you won't acheive that with rms below 20
>cents, anyhow . . .

Isn't it really 21:20 = 15:14?

>>If the former is to be answered yes, then perhaps
>>it's time to plug in other desirable chord pairs,
>>such as higher/lower harmonic clusters, ASSs, etc.
>
>sure, why not?

Which chord pairs do you think would make good
candidates?

-Carl

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

> >>It doesn't catch 4:5:6:7 and 7/(7:6:5:4), does it?
> >
> >umm . . . 4:6 = 5:7? you won't acheive that with rms below 20
> >cents, anyhow . . .
>
> Isn't it really 21:20 = 15:14?

4:6 "=" 5:7, but that could be a chromatic unison vector, not
necessarily a commatic one. thus there may very well be some viable
possibilities. excuse my error, i was sick when i wrote that (and i'm
still sick) . . .

i'll take a look at such possibilities, and others, when i
recover . . .

>>>>It doesn't catch 4:5:6:7 and 7/(7:6:5:4), does it?
>>>
>>>umm . . . 4:6 = 5:7? you won't acheive that with rms below 20
>>>cents, anyhow . . .
>>
>>Isn't it really 21:20 = 15:14?
>
>4:6 "=" 5:7, but that could be a chromatic unison vector, not
>necessarily a commatic one. thus there may very well be some viable
>possibilities. excuse my error, i was sick when i wrote that (and
>i'm still sick) . . .

Of course. What I meant there was, since in an lt there is only
one chromatic uv, mustn't it serve as both 21:20 and 15:14 in the
above example? If so, it wouldn't seem to push us out of the 20
cents RMS cutoff (which I think is reasonable).

>i'll take a look at such possibilities, and others, when i
>recover . . .

Hope you're feeling better! By the way, I really appreciate your
willingness to provide solutions, if you'll excuse the project-
management speak. I understood all but one or two steps in your
procedure...

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>>>It doesn't catch 4:5:6:7 and 7/(7:6:5:4), does it?
> >>>
> >>>umm . . . 4:6 = 5:7? you won't acheive that with rms below 20
> >>>cents, anyhow . . .
> >>
> >>Isn't it really 21:20 = 15:14?
> >
> >4:6 "=" 5:7, but that could be a chromatic unison vector, not
> >necessarily a commatic one. thus there may very well be some
viable
> >possibilities. excuse my error, i was sick when i wrote that (and
> >i'm still sick) . . .
>
> Of course. What I meant there was, since in an lt there is only
> one chromatic uv, mustn't it serve as both 21:20 and 15:14 in the
> above example?

i can't see where you're getting 21:20 from. then again, i'm still
sick, and i have three gigs coming up in the next week (thank
goodness for DayQuil) . . . rumor has it Trey's drummer will be
sitting in on Tuesday . . .

>>Of course. What I meant there was, since in an lt there is only
>>one chromatic uv, mustn't it serve as both 21:20 and 15:14 in the
>>above example?
>
>I can't see where you're getting 21:20 from.

Whoops, sorry! 3/2-7/5 and 5/4-7/6 are both 15:14.

>then again, i'm still sick, and i have three gigs coming up in the
>next week (thank goodness for DayQuil) . . . rumor has it Trey's
>drummer will be sitting in on Tuesday . . .

Good luck!

-Carl