Re: Third-based MOS

Dave Keenan wrote:

> Reduced to responding to my own posts. Pathetic. :-)

Well, two can play at that game! I wrote:

> 3/2: 2 5 7 10 12 14 17 19 22 24 27 29 31 34 36 39 41 43 46 48 51 43
> 5/4: 3 6 9 12 16 19 22 25 28 31 34 37 40 43 47 50 53
> 6/5: 4 8 11 15 19 23 27 30 34 38 42 46 49 53
>
> By this method, the lowest EDIs with good representations of
> all 5-limit
> intervals are 12, 19, 34 and 53. 65 is the next.

You'll notice 12 isn't in the 6/5 list, so it's only 19,34,53,65,...

[Graham Breed TD234.4, as corrected TD235.1]
>You can generate "good" ETs approximating a given interval by forming an
>equivalence interval from the "ideal" interval and a diesis, then finding
>the best matches to the ideal ratio. Choosing each integer as the number of
>steps in the small interval, and calculating the large interval, gives the
>best approximations. This misses a lot of scales however, so taking the
>large interval and calculating the small is how these lists were produced:
>
>3/2: 2 5 7 10 12 14 17 19 22 24 27 29 31 34 36 39 41 43 46 48 51 43
>5/4: 3 6 9 12 16 19 22 25 28 31 34 37 40 43 47 50 53
>6/5: 4 8 11 15 19 23 27 30 34 38 42 46 49 53
>
>By this method, the lowest EDIs with good representations of all 5-limit
>intervals are 19, 34 and 53. 65 is the next.

This seems a bit restrictive. But if it means "good - for their size" then
that is useful. Is that what it's doing?

[Dave Keenan]
>> I find it interesting that the diatonic scale falls out of
>> this formula for
>> ET's (EDO's) with reasonable fifths, N = 5T + 2s where T >= 0
>> and T/9 <= s <= 6T/7.

[Graham Breed]
>Er, hang on, where do these numbers come from?

"T" for "tone", "s" for "semitone". By restricting "s" in this way I limit
the error in the fifths to something like +-13c (specifically the best and
second best fifths in 47-tET) and eliminate the ETs that are multiples of 5
or 7 as being not good enough. But this was irrelevant to my point, which
was just what I think you said elsewhere, that the diatonic 5T+2s is simply
the smallest fifth-based MOS. In future I'll use W and h, after Dan
Stearns, instead of T and s.

>I don't think these [major-third based] scales would work
>melodically as d is so much smaller than T for good major thirds.

Agreed. The minor-thirds aren't so bad at only 4 to 1. But still pretty
bad. As Paul Erlich said, you can just keep going up the sequence (down the
tree as you drew them) adding the appropriate number of the smaller
interval each time, until you get a proper scale. But that's often too many
notes. And we have seen that the number of consonances can get quite large
enough long before propriety is achieved. We'd need 15 notes for propriety
in the minor-thirds MOS and 16 for major-thirds. But we get sufficient
harmonic resources with 11 and 10 notes respectively. It sure is hard to
beat 6 triads from a proper 7 notes (diatonic).

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote:

> >3/2: 2 5 7 10 12 14 17 19 22 24 27 29 31 34 36 39 41 43 46 48 51 43
> >5/4: 3 6 9 12 16 19 22 25 28 31 34 37 40 43 47 50 53
> >6/5: 4 8 11 15 19 23 27 30 34 38 42 46 49 53
> >
> >By this method, the lowest EDIs with good representations of
> all 5-limit
> >intervals are 19, 34 and 53. 65 is the next.
>
> This seems a bit restrictive. But if it means "good - for
> their size" then
> that is useful. Is that what it's doing?

For each interval, it means you can't get a better approximation by using a
different smaller-step-size with the same larger-step-size. Taking each
interval alone, it has something to do with economy. Taking all intervals
together it's fairly meaningless. There are better methods for showing
which ETs have good approximations to a range of intervals. And they show
12 as extremely economical. The advantage of this method is that it can
help you find a two-step generalisation, and also that all approximations
must be consistent.

However, I may as well point out that that this method gives no all-round
7-limit ET, but 31= is good for all of what I think Partch called "intervals
of 7" that is 7/4, 7/5, 7/6 and equivalents.

> the diatonic 5T+2s is simply the smallest fifth-based MOS.

No, 2V+T is the smallest (V=fourth), then the classic pentatonic scale
2U+3T.

> >I don't think these [major-third based] scales would work
> >melodically as d is so much smaller than T for good major thirds.
>
> Agreed. The minor-thirds aren't so bad at only 4 to 1. But
> still pretty
> bad. As Paul Erlich said, you can just keep going up the
> sequence (down the
> tree as you drew them) adding the appropriate number of the smaller
> interval each time, until you get a proper scale. But that's
> often too many
> notes. And we have seen that the number of consonances can
> get quite large
> enough long before propriety is achieved. We'd need 15 notes
> for propriety
> in the minor-thirds MOS and 16 for major-thirds. But we get sufficient
> harmonic resources with 11 and 10 notes respectively. It sure
> is hard to
> beat 6 triads from a proper 7 notes (diatonic).

I think there's more to it than propriety, as my favourite "Blues Scale" as
outlined on my website was designed to work melodically, but is improper.
In that case, it sounds like either a 5-note scale with "quartertone"
inflections, or an uneven 7-note scale, or an even more uneven 8-note scale
if you have good pitch discrimination. I haven't tried the 5/4 scales -- I
suppose I should -- but they would logically either sound like a 3-note
scale with inflections, which is boring, or a complete mess of small
intervals with big jumps.

The 6/5 scales look a bit more promising, and I may well try them out. The
4+7 scale has the largest atomic intervals as 3 steps from 19, which should
be small enough, and it can fit into one octave of a normal keyboard.

Dave Keenan wrote,

>> the diatonic 5T+2s is simply the smallest fifth-based MOS.

Graham Breed wrote,

>No, 2V+T is the smallest (V=fourth), then the classic pentatonic scale
>2U+3T.

Thanks, Graham, for not letting that slip. 2V+T is called a "bare
tetrachordal framework" in my paper. Coincidentally, I was trying to use it
in a jam session last night; it sounds very Coltranesque. It's particularly
useful when playing in 5-tET.

Would the two-note scale V+T count as MOS?