A Property of MOS/DE Scales

Hi!

Could someone explain why MOS/DE scales have the following property:

The difference between larger and smaller interval is the same in
every interval class with two sizes of interval.

I'm not sure about the correctness of my formulation or my English
but hopefully you'll understand what I'm asking. :)

Kalle

--- In tuning@y..., "Kalle Aho" <kalleaho@m...> wrote:
> Hi!
>
> Could someone explain why MOS/DE scales have the following property:
>
> The difference between larger and smaller interval is the same in
> every interval class with two sizes of interval.

you got it!
>
> I'm not sure about the correctness of my formulation or my English
> but hopefully you'll understand what I'm asking. :)
>
> Kalle

one explanation involves periodicity blocks. kind of an indirect
route, but it helps when dealing with harmony-based examples such as
the diatonic scale, blackjack, etc.

if you start with any fokker periodicity block, and temper out all
its defining unison vectors except one, the result is an MOS/DE
(that's my Hypothesis) -- and the difference between the larger and
smaller version of each generic interval (that is, each interval
composed of a given number of steps of the scale) *is* that one
untempered unison vector, now called the "chromatic unison vector".

gene and graham may be able to shed more light on this
mathematically, though they'd probably take a more direct route to
the answer . . .

or, for more of a ji-founded, visual approach, you can check out my
paper "The Forms of Tonality" -- write me offlist.

Kalle asked:
>Could someone explain why MOS/DE scales have the following property:
>The difference between larger and smaller interval is the same in
>every interval class with two sizes of interval.

It's actually quite trivial, I answered a similar question from
Johnny Reinhard on 21-2-2002, who was wondering about the
interval differences which are a multiple of 6 cents in
Werckmeister's temperament:

"For the sake of simplicity, let's take the semitone as
generating interval instead of the fifth. This also visits
all tones in the scale and we don't need to take wrapping
around the octave into account.
Say we have two semitones of X and X+C cents, for example
96 and 96+6=102 cents.
Then each interval consists of some number of semitones of
size X and some of size X+C, say nX + m(X+C). Rewrite this
into (n+m)X + mC. Then n+m is the interval class. Then
you see that for a particular interval class n+m is constant
and m=0,1,2,etc. which is 0x6, 1x6, 2x6, etc. cents,
the difference between the interval sizes is a multiple of
6 cents."

The two sizes of interval for every interval class is in
the definition of distributional evenness, so that answers
half the question. Why the size difference is always the
same, lies in the above, and you can also see that, because of
the evenness, the number of semitones of each size in each interval
never differs by more than one (from that in another interval
of the same class). So the difference in m can only be 0 or 1.

>I'm not sure about the correctness of my formulation or my English
>but hopefully you'll understand what I'm asking. :)

Me likewise :)

Manuel

Thank you, Paul and Manuel for your explanations! You are great!

This property can be exploited in the following way. I'm interested
in 7-limit scales that will produce major and minor tetrads in root
position with the same pattern of steps.

Those tetrads in just intonation are:

1/1 5/4 3/2 7/4
1/1 6/5 3/2 12/7

Let's say I want to search for these scales in equal temperaments.
Let App(x) mean the best approximation of x in an equal temperament.
If we require that

App(5/4)=App(6/5) or
App(7/4)=App(12/7) or
App(5/4)-App(6/5)=App(7/4)-App(12/7)

we will find all equal temperaments which may contain MOS/DE scales
where root position major and minor tetrads are produced with the
same pattern of steps.

I get equal temperaments 1-26, 29, 44 and 48. Those that are 7-limit
consistent are 4, 5, 6, 9, 10, 12, 15, 16, 18, 19, 22, 26 and 29.

Now this is a very manageable number of ETs to do a search upon. I
believe the most interesting systems are found in 15, 22 and 26.

Why I want MOS/DE scales? Well, it's because of Paul's hypothesis.
These MOS/DE scales will be tempered periodicity blocks. And I
suppose reshuffling the steps in MOS/DE scales will give scales that
correspond to other shapes of periodicity blocks with same unison
vectors.

Kalle

--- In tuning@y..., manuel.op.de.coul@e... wrote:

> The two sizes of interval for every interval class is in
> the definition of distributional evenness, so that answers
> half the question. Why the size difference is always the
> same, lies in the above, and you can also see that, because of
> the evenness, the number of semitones of each size in each interval
> never differs by more than one (from that in another interval
> of the same class). So the difference in m can only be 0 or 1.

manuel, this doesn't look right. what do you mean by semitones? if
you mean 12-equal semitones, a counterexample is the augmented scale -
- there the difference is 2 semitones. if you mean general steps of
an ET, blackjack is a counterexample -- there the difference is 3
steps of 72-equal. besides, distributional evenness does not require
a superset or universe scale of any sort, so in general m will not
even be an integer.

maybe you're confusing distributional evenness with maximal evenness?

thanks,
paul

--- In tuning@y..., "Kalle Aho" <kalleaho@m...> wrote:
> Thank you, Paul and Manuel for your explanations! You are great!
>
> This property can be exploited in the following way. I'm interested
> in 7-limit scales that will produce major and minor tetrads in root
> position with the same pattern of steps.
>
> Those tetrads in just intonation are:
>
> 1/1 5/4 3/2 7/4
> 1/1 6/5 3/2 12/7
>
> Let's say I want to search for these scales in equal temperaments.
> Let App(x) mean the best approximation of x in an equal
temperament.
> If we require that
>
> App(5/4)=App(6/5) or
> App(7/4)=App(12/7) or
> App(5/4)-App(6/5)=App(7/4)-App(12/7)
>
> we will find all equal temperaments which may contain MOS/DE scales
> where root position major and minor tetrads are produced with the
> same pattern of steps.
>
> I get equal temperaments 1-26, 29, 44 and 48.

something's wrong here. in most of these ETs, for example 12-equal,
15-equal, 22-equal, and 26-equal, App(5/4)=App(6/5) does not hold. i
understand what you're trying to do, but perhaps you've slipped up in
trying to explain this? could you try to flesh this out more
explicitly?

> Those that are 7-limit
> consistent are 4, 5, 6, 9, 10, 12, 15, 16, 18, 19, 22, 26 and 29.
>
> Now this is a very manageable number of ETs to do a search upon. I
> believe the most interesting systems are found in 15, 22 and 26.
>
> Why I want MOS/DE scales? Well, it's because of Paul's hypothesis.
> These MOS/DE scales will be tempered periodicity blocks. And I
> suppose reshuffling the steps in MOS/DE scales will give scales
that
> correspond to other shapes of periodicity blocks with same unison
> vectors.

this last bit sounds right.

if i understand what you're trying to do above, you'd like to find
MOS/DE scales where the chromatic unison vector is, equivalently,

(5/4)/(6/5) = 25/24

or

(7/4)/(12/7) = 49/48

and a commatic (vanishing; tempered out) unison vector is

(25/24)/(49/48) = 50/49

these requirements will result in MOS/DE scales that will produce
major and minor tetrads in root position with the same pattern of
steps.

i've thought about this before but i'm not sure if i've posted it in
just these terms.

to conduct the survey, one simply has to supply a list of candidates
for the third unison vector, also to be commatic (tempered out) --
since 7-limit scales are defined by three independent unison vectors.

note that there's no need to generate a list of ETs, or to consider
ETs at all, in this process.

perhaps gene could easily post such a survey -- and maybe doing so on
tuning-math might keep the natives happy :)

Paul wrote:

> something's wrong here. in most of these ETs, for example 12-equal,
> 15-equal, 22-equal, and 26-equal, App(5/4)=App(6/5) does not hold.
i
> understand what you're trying to do, but perhaps you've slipped up
in
> trying to explain this? could you try to flesh this out more
> explicitly?

Look more carefully!

> > App(5/4)=App(6/5) or
> > App(7/4)=App(12/7) or
> > App(5/4)-App(6/5)=App(7/4)-App(12/7)

Don't you see the "or"s?

App(7/4)=App(12/7) is actually there because of 15-equal.

Kalle

--- In tuning@y..., "Kalle Aho" <kalleaho@m...> wrote:
> Paul wrote:
>
> > something's wrong here. in most of these ETs, for example 12-
equal,
> > 15-equal, 22-equal, and 26-equal, App(5/4)=App(6/5) does not
hold.
> i
> > understand what you're trying to do, but perhaps you've slipped
up
> in
> > trying to explain this? could you try to flesh this out more
> > explicitly?
>
> Look more carefully!
>
> > > App(5/4)=App(6/5) or
> > > App(7/4)=App(12/7) or
> > > App(5/4)-App(6/5)=App(7/4)-App(12/7)
>
> Don't you see the "or"s?
>
> App(7/4)=App(12/7) is actually there because of 15-equal.
>
> Kalle

oh! i'm so sorry, i didn't see the "or"s!! but it doesn't seem you're
getting the answer to the question you asked this way. if the MOS/DE
scale is going to have the same pattern of steps for both types of
tetrad, then *all* of the above identities must hold for the MOS/DE
(in terms of generic step sizes). meanwhile, the embedding ET (which
is not even necessary at all) would *have* to satisfy, and *only*
have to satisfy,

> > > App(5/4)-App(6/5)=App(7/4)-App(12/7)

since the difference between these two sides is a commatic unison
vector.

am i getting warmer?

Paul wrote:

> oh! i'm so sorry, i didn't see the "or"s!! but it doesn't seem
you're
> getting the answer to the question you asked this way. if the
MOS/DE
> scale is going to have the same pattern of steps for both types of
> tetrad, then *all* of the above identities must hold for the MOS/DE
> (in terms of generic step sizes). meanwhile, the embedding ET
(which
> is not even necessary at all) would *have* to satisfy, and *only*
> have to satisfy,
>
> > > > App(5/4)-App(6/5)=App(7/4)-App(12/7)
>
> since the difference between these two sides is a commatic unison
> vector.
>
> am i getting warmer?

Not quite yet!

I'm not trying to get an exact list of ETs that have the required
property. I'm just trying to narrow my search with these procedures.
If an ET is only required to satisfy

App(5/4)-App(6/5)=App(7/4)-App(12/7)

we get 12, 16, 17, 18, 21, 22, 23, 25, 26, 44 and 48. 7-limit
consistent ones are 12, 16, 18, 22, 26.

(Remember that App(x) means here the best approximation of x in an
ET.)

That leaves out 15 which contains Blackwood's 10-tone symmetric scale
which satisfies the required property. That's why I'm including the
possibilities that

App(5/4)=App(6/5) or App(7/4)=App(12/7).

These conditions narrow my search by giving a small list of ETs:

4, 5, 6, 9, 10, 12, 15, 16, 18, 19, 22, 26, 29.

Now every MOS/DE scale that is embedded in an ET and has the required
property must be among these (if we exclude inconsistent mappings).
This has very little to do with periodicity blocks at this point!

The periodicity block stuff answers why I want to find MOS/DE scales
and not something else.

Kalle

--- In tuning@y..., "Kalle Aho" <kalleaho@m...> wrote:
> Paul wrote:
>
> > oh! i'm so sorry, i didn't see the "or"s!! but it doesn't seem
> you're
> > getting the answer to the question you asked this way. if the
> MOS/DE
> > scale is going to have the same pattern of steps for both types
of
> > tetrad, then *all* of the above identities must hold for the
MOS/DE
> > (in terms of generic step sizes). meanwhile, the embedding ET
> (which
> > is not even necessary at all) would *have* to satisfy, and *only*
> > have to satisfy,
> >
> > > > > App(5/4)-App(6/5)=App(7/4)-App(12/7)
> >
> > since the difference between these two sides is a commatic unison
> > vector.
> >
> > am i getting warmer?
>
> Not quite yet!
>
> I'm not trying to get an exact list of ETs that have the required
> property. I'm just trying to narrow my search with these
procedures.
> If an ET is only required to satisfy
>
> App(5/4)-App(6/5)=App(7/4)-App(12/7)
>
> we get 12, 16, 17, 18, 21, 22, 23, 25, 26, 44 and 48. 7-limit
> consistent ones are 12, 16, 18, 22, 26.
>
> (Remember that App(x) means here the best approximation of x in an
> ET.)
>
> That leaves out 15 which contains Blackwood's 10-tone symmetric
scale
> which satisfies the required property. That's why I'm including the
> possibilities that
>
> App(5/4)=App(6/5) or App(7/4)=App(12/7).
>
> These conditions narrow my search by giving a small list of ETs:
>
> 4, 5, 6, 9, 10, 12, 15, 16, 18, 19, 22, 26, 29.
>
> Now every MOS/DE scale that is embedded in an ET and has the
required
> property must be among these (if we exclude inconsistent mappings).
> This has very little to do with periodicity blocks at this point!
>
> The periodicity block stuff answers why I want to find MOS/DE
scales
> and not something else.
>
> Kalle

ok. i was wrong about the criteria for ETs. i think i see how you got
the list:

App(5/4)=App(6/5) or
App(7/4)=App(12/7) or
App(5/4)-App(6/5)=App(7/4)-App(12/7)

let x be the difference between intervals with the same generic size.
then, if major and minor tetrads have the same pattern of generic
sizes, then the following must be true:

App(5/4)-App(6/5)=0 or App(5/4)-App(6/5)=x
App(7/4)=App(12/7)=0 or App(7/4)=App(12/7)=x

from these two conditions you can derive the tripartite statement you
made.

sorry i got confused on this matter!

but the main thrust of my reply was to suggest to you that we have
more sophisticated tools at our disposal now than i did when i wrote
that paper. we should not be restricted to an ET conception. each
MOS/DE scale can be defined independently in terms of its own
generator, in cents; and period of repetition, in 1/octaves. by
restricting ourselves to ETs and their best approximations to
consonant intervals, we may miss some interesting and wonderful
possibilities.

it might be a good idea to follow up to tuning-math, where
the "searchers" may be more willing to publically help.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> perhaps gene could easily post such a survey -- and maybe doing so on
> tuning-math might keep the natives happy :)

I'll skip the wedgies and post the results:

The only "standard" ets which work are 4,6, and 10. However, we also have [14,22,32,39], [16,26,37,45], [18,28,41,50], [24,38,55,67],
[34,54,78,95]. Use at your own risk.

Paul wrote:
>manuel, this doesn't look right. what do you mean by semitones? if
>you mean 12-equal semitones, a counterexample is the augmented scale -
>- there the difference is 2 semitones. if you mean general steps of
>an ET, blackjack is a counterexample

Oh, my use of the word semitone carried over from that old post
about Werckmeister's temperament. I should have replaced it by
"one step interval", so I meant neither semitone nor ET step.
That's all, thanks for catching that.

>maybe you're confusing distributional evenness with maximal evenness?

No.

Manuel

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > perhaps gene could easily post such a survey -- and maybe doing
so on
> > tuning-math might keep the natives happy :)
>
> I'll skip the wedgies and post the results:
>
> The only "standard" ets which work are 4,6, and 10. However, we
also have [14,22,32,39], [16,26,37,45], [18,28,41,50], [24,38,55,67],
> [34,54,78,95]. Use at your own risk.

this doesn't answer the question. we're looking for linear-tempered
scales, and we already know that pajara-10, blackwood-10, and injera-
14 are possible solutions.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> this doesn't answer the question.

That's because you changed the question.

we're looking for linear-tempered
> scales, and we already know that pajara-10, blackwood-10, and injera-
> 14 are possible solutions.

Why not state the new question in terms of commas?

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > this doesn't answer the question.
>
> That's because you changed the question.

kalle's question has been the same all along.

> > we're looking for linear-tempered
> > scales, and we already know that pajara-10, blackwood-10, and
injera-
> > 14 are possible solutions.
>
> Why not state the new question in terms of commas?

there will be three classes of answer, as kalle pointed out.

1st: 25:24 is commatic, 49:48 is chromatic
2nd: 49:48 is commatic, 25:24 is chromatic (e.g., blackwood-10)
3rd: 50:49 is commatic, 49:48=25:24 is chromatic (paj-10, inj-14)

to review, commatic means it's tempered out; chromatic means it isn't
tempered out.

so for each of the cases, two of the unison vectors of the
periodicity block are fixed; now plug in many many choices for the
third unison vector, temper it out, and present the resulting linear-
tempered scales (preferably with some sort of badness measure) . . .