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) . . .