Yahoo Tuning Groups Ultimate Backup tuning Fwd: Re: a 24 tone non-edo scale

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti" <giordanobruno76@y...> wrote:
--- In tuning@yahoogroups.com, "stephenszpak" <stephen_szpak@h...>
wrote:
...
> A scale came to me a number of months ago. (I call it the Szpak
> Scale since I don't know who came up with it first.) If anyone
> here can get some use out of it that's fine with me. It is very
> closely related to 24 EDO. It includes all notes of the 12 EDO
> scale.
>
> Some advantages (of the Szpak Scale):
>
> Using the tonics of 12 EDO:
>
> subminor 3rd is less than 6 cents off ideal
> neutral 3rd is less than 10 cents off ideal
> sub 4th (470.781 cents) is less than 10 cents off ideal
> 11th harmonic is less than 10 cents off ideal
> 7th harmonic is less than 8 cents off ideal
>
> The scale goes like this: (in cents)
>
> 0-60.88-100-160.88-200-260.88 etc. until 1200 cents

So you have a scale with two stepsizes, L=100 s=60.88

And it's built sLsLsLs...

IF YOU SAY IT'S BUILT sLsLsL I'll TAKE YOUR WORD FOR IT.

Just out of curiosity, which are your ideal references, Stephen?

For instance, your approximation for neutral third is 360.88 cents,
which is more than 10 cents away from

347.41 cents 11/9
350 cents (24-eq neutral third)
348.39 cents (31-eq neutral third)

REFERENCES:

NEUTRAL 3RD = 350.9775

ASSUMING I DID THE MATH CORRECTLY, I TOOK THE JI MAJOR 3RD AND MINOR 3RD
AND SPLIT THE DIFFERENCE.

I understand your sub-diminished 4th's reference is septimal fourth,
that is, 21/16. And your ideal subminor 3rd must be 7/6. So may be
it's 49/40 what you're talking about? 7-limit?

REFERENCES:

SUBMINOR 3RD 7/6 @ 266.871
SUB 4TH 21/16 @ 470.781

49/40 IS 186.340 THIS SCALE DOESN'T HAVE THIS VALUE.(?)

I DON'T KNOW MUCH ABOUT TUNING. 7-LIMIT DOESN'T MEAN ANYTHING TO ME. I
KNOW THESE TERMS ARE AT MONZ'S SITE. I COULD READ THEM BUT SOMETIMES
IT'S JUST WORDS TO ME.

Max.
--- End forwarded message ---

Stephen Szpak

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🔗Maximiliano G. Miranda Zanetti <giordanobruno76@yahoo.com.ar>

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> REFERENCES:
>
> NEUTRAL 3RD = 350.9775
>
> ASSUMING I DID THE MATH CORRECTLY, I TOOK THE JI MAJOR 3RD AND
MINOR 3RD
> AND SPLIT THE DIFFERENCE.

Maths are pretty correct. Notice you have taken the geometric mean of
5/4 and 6/5 (or the arithmetic mean of those ratios in cents).

Is that a useful reference? In doing this, you destroy the
rationality of both ratios to get square root of 3/2. It's the same
with 1/1 and 2/1 (fundamental and the replicated octave). The
geometric mean yields the 12-eq tritone.

What's wrong with sqrt(3/2) or sqrt(2)? Nothing intrinsecally, they
have their own rationale. One thing is sure, they have not much in
common with JI's visions. Shall I belittle your 350.9775 and
recommend 11/9 (347.4079) instead? That would be unwise. You'll find
for yourself a useful answer for your purposes.

>> I understand your sub-diminished 4th's reference is septimal
>>fourth,
>> that is, 21/16. And your ideal subminor 3rd must be 7/6. So may be
>> it's 49/40 what you're talking about? 7-limit?
>>
> REFERENCES:
>
> SUBMINOR 3RD 7/6 @ 266.871
> SUB 4TH 21/16 @ 470.781
>
> 49/40 IS 186.340 THIS SCALE DOESN'T HAVE THIS VALUE.(?)

I suggest that you try the maths again...

49/40 equiv. 351.3381 cents

>
> I DON'T KNOW MUCH ABOUT TUNING. 7-LIMIT DOESN'T MEAN ANYTHING TO
ME. I
> KNOW THESE TERMS ARE AT MONZ'S SITE. I COULD READ THEM BUT
SOMETIMES
> IT'S JUST WORDS TO ME.
>

Stephen:

You should check some previous posts on the topic.
Usually, two senses are devised for the term n-limit:

a) ODD
b) PRIME

The first meaning (n-odd-limit) applies to ratios not containing a
numerator and derivator larger than n, once discarded the factors of
2 (octave transpositions).

In this sense, 11/9 is a 13-odd-limit ratio, as well as 18/11. Both
are also 15-odd-limit, 17-odd-limit, etc.
49/40 is not a 13-odd-limit ratio. It's 49-odd-limit.

The second sense involves decomposition into prime factors.
E.g., 75=3*5^2 77=7*11 etc.
So, n-prime limit implies that neither num. nor denom. contain prime
factors larger than n.

For instance, 11/9 is not a 7-prime-limit ratio, since the numerator
contains the prime factor 11.

49/40, on the other hand, is a 7-prime-limit ratio. [And 11-prime-
limit, 13, 17 and so...]

I was talking about the fact that you apparently took a 7-prime-limit
reference (which is of course OK) as 49/40, rather than the more
traditional 11/9 (11-prime-limit neutral third).

Max.

🔗Stephen Szpak <stephen_szpak@hotmail.com>

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti" <giordanobruno76@y...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

>REFERENCES:
>
> NEUTRAL 3RD = 350.9775
>
> ASSUMING I DID THE MATH CORRECTLY, I TOOK THE JI MAJOR 3RD AND
MINOR 3RD
> AND SPLIT THE DIFFERENCE.

Maths are pretty correct. Notice you have taken the geometric mean of
5/4 and 6/5 (or the arithmetic mean of those ratios in cents).

>>I understand your sub-diminished 4th's reference is septimal fourth,
>>that is, 21/16. And your ideal subminor 3rd must be 7/6. So may be
>>it's 49/40 what you're talking about? 7-limit?
>>
>REFERENCES:
>
> SUBMINOR 3RD 7/6 @ 266.871
> SUB 4TH 21/16 @ 470.781
>
> 49/40 IS 186.340 THIS SCALE DOESN'T HAVE THIS VALUE.(?)

I suggest that you try the maths again...

49/40 equiv. 351.3381 cents

>
> I DON'T KNOW MUCH ABOUT TUNING. 7-LIMIT DOESN'T MEAN ANYTHING TO
ME. I
> KNOW THESE TERMS ARE AT MONZ'S SITE. I COULD READ THEM BUT
SOMETIMES
> IT'S JUST WORDS TO ME.
>

Stephen:

You should check some previous posts on the topic.
Usually, two senses are devised for the term n-limit:

a) ODD
b) PRIME

The first meaning (n-odd-limit) applies to ratios not containing a
numerator and derivator larger than n, once discarded the factors of
2 (octave transpositions).

In this sense, 11/9 is a 13-odd-limit ratio, as well as 18/11. Both
are also 15-odd-limit, 17-odd-limit, etc.
49/40 is not a 13-odd-limit ratio. It's 49-odd-limit.

The second sense involves decomposition into prime factors.
E.g., 75=3*5^2 77=7*11 etc.
So, n-prime limit implies that neither num. nor denom. contain prime
factors larger than n.

For instance, 11/9 is not a 7-prime-limit ratio, since the numerator
contains the prime factor 11.

49/40, on the other hand, is a 7-prime-limit ratio. [And 11-prime-
limit, 13, 17 and so...]

I was talking about the fact that you apparently took a 7-prime-limit
reference (which is of course OK) as 49/40, rather than the more
traditional 11/9 (11-prime-limit neutral third).

Max.
--- End forwarded message ---
Stephen Szpak writes:

First, the 49/40 =186.340 stuff I got from

http://home.earthlink.net/~kgann/Octave.html (Anatomy of an Octave)

If this is wrong there is an error at that site.

Having just joined this group I'll probably be leaving soon. I have no idea what
you just wrote to me. The same is true for most other responses I have recieved. There is no
sense wasting my time, not to mention the time of others. There are just certain things beyond
my grade, so to speak.

Stephen Szpak

_________________________________________________________________
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🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Maximiliano G. Miranda Zanetti <giordanobruno76@yahoo.com.ar>

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
> >
>

Max had written

>> Stephen:
>>
>> You should check some previous posts on the topic.
> Usually, two senses are devised for the term n-limit:
>
> a) ODD
> b) PRIME
>
> The first meaning (n-odd-limit) applies to ratios not containing a
> numerator and derivator larger than n, once discarded the factors of
> 2 (octave transpositions).
>
> In this sense, 11/9 is a 13-odd-limit ratio, as well as 18/11. Both
> are also 15-odd-limit, 17-odd-limit, etc.
> 49/40 is not a 13-odd-limit ratio. It's 49-odd-limit.
>
> The second sense involves decomposition into prime factors.
> E.g., 75=3*5^2 77=7*11 etc.
> So, n-prime limit implies that neither num. nor denom. contain prime
> factors larger than n.
>
> For instance, 11/9 is not a 7-prime-limit ratio, since the numerator
> contains the prime factor 11.
>
> 49/40, on the other hand, is a 7-prime-limit ratio. [And 11-prime-
> limit, 13, 17 and so...]
>
> I was talking about the fact that you apparently took a 7-prime-
limit
> reference (which is of course OK) as 49/40, rather than the more
> traditional 11/9 (11-prime-limit neutral third).
>
> Max.

And Stephen:

Stephen Szpak writes:
>
> First, the 49/40 =186.340 stuff I got from
>
> http://home.earthlink.net/~kgann/Octave.html (Anatomy of
an Octave)
>
> If this is wrong there is an error at that site.
>
> Having just joined this group I'll probably be leaving soon. I
have no
> idea what
> you just wrote to me. The same is true for most other responses
I have
> recieved. There is no
> sense wasting my time, not to mention the time of others. There
are just
> certain things beyond
> my grade, so to speak.
>
> Stephen Szpak

Me again:

UPS! There's indeed a mistake on the page. If you scroll down up to
351.351, you'll find the correct ratio 49/40. I don't know which
fraction they wanted to link to 186.340.

I'm so sorry if my comments have been unhelpful.They were intended to
help you, rather than disscourage you. It's my fault, in any case.
Don't feel frustraded if I have made my message unclear for you.

Let me try to be plainer, and focus on prime-limit, rather than odd-
limit:

You may think of the ratio 5/4 as "going up in the ratio of a fifth
harmonic, and descending two octaves".

You could think of the ratio 9/8 in the same fashion: "ninth harm.
up, 3 octaves down". But as you recognize that 4 and 8 are related to
the 2nd harmonic (the octave), you could also recognize that going up
a ninth-harmonic ratio up is equivalent to going up two third-
harmonic ratios. So, in a "prime" sense, you just take into account
the prime factors:2,3,5,7,11, etc.

Decomposed in such way, 49/40 is 7*7/2*2*2*5. In words, "going up two
seventh harmonics, going down a fifth harmonic, then going down three
octaves.

11/9 is 11/3*3 decomposed.

We say that 49/40 is 7-prime-limit, because when decomposed it
contains no factor greater than 7.

11/9 is 11-prime-limit, but not a 7-prime-limit ratio, because it has
11 as a factor, which is greater than 7.

These are labels that help us to understand some of the
characteristics of ratios.

You may hear: Though 12-eq properly represents some 5-(prime)-limit
ratios, it can't approximate many 7-limit fractions. That is because
it does a great job with ratios like 3/2, 4/3, 5/4 (hum?), 9/8, etc.
but cannot represent accurately the seventh harmonic (7/4).

I hope this stands as a better explanation than the previous post,
but please do not hesitate to ask for info, or inquire about
anything. I'm pretty sure there are people here that can help you
with better, clearer comments, but in any case do not let yourself
down. There's much info on tuning I can't understand at the moment
(maybe I'll never do), but that doesn't make this group less
interesting.

🔗Stephen Szpak <stephen_szpak@hotmail.com>

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti" <giordanobruno76@y...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
>--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti" ><giordanobruno76@y...> wrote:
> >
>

Max had written

>>Stephen:
>>
>>You should check some previous posts on the topic.
>Usually, two senses are devised for the term n-limit:
>
>a) ODD
>b) PRIME
>
>The first meaning (n-odd-limit) applies to ratios not containing a
>numerator and derivator larger than n, once discarded the factors of
>2 (octave transpositions).
>
>In this sense, 11/9 is a 13-odd-limit ratio, as well as 18/11. Both
>are also 15-odd-limit, 17-odd-limit, etc.
>49/40 is not a 13-odd-limit ratio. It's 49-odd-limit.
>
>The second sense involves decomposition into prime factors.
>E.g., 75=3*5^2 77=7*11 etc.
>So, n-prime limit implies that neither num. nor denom. contain prime
>factors larger than n.
>
>For instance, 11/9 is not a 7-prime-limit ratio, since the numerator
>contains the prime factor 11.
>
>49/40, on the other hand, is a 7-prime-limit ratio. [And 11-prime-
>limit, 13, 17 and so...]
>
>I was talking about the fact that you apparently took a 7-prime-
limit
>reference (which is of course OK) as 49/40, rather than the more
>traditional 11/9 (11-prime-limit neutral third).
>
>Max.

And Stephen:

Stephen Szpak writes:
>
> First, the 49/40 =186.340 stuff I got from
>
> http://home.earthlink.net/~kgann/Octave.html (Anatomy of
an Octave)
>
> If this is wrong there is an error at that site.
>
> Having just joined this group I'll probably be leaving soon. I
have no
>idea what
> you just wrote to me. The same is true for most other responses
I have
>recieved. There is no
> sense wasting my time, not to mention the time of others. There
are just
>certain things beyond
> my grade, so to speak.
>
> Stephen Szpak

Me again:

UPS! There's indeed a mistake on the page. If you scroll down up to
351.351, you'll find the correct ratio 49/40. I don't know which
fraction they wanted to link to 186.340.

Let me try to be plainer, and focus on prime-limit, rather than odd-
limit:

You may think of the ratio 5/4 as "going up in the ratio of a fifth
harmonic, and descending two octaves".

Decomposed in such way, 49/40 is 7*7/2*2*2*5. In words, "going up two
seventh harmonics, going down a fifth harmonic, then going down three
octaves.

11/9 is 11/3*3 decomposed.

We say that 49/40 is 7-prime-limit, because when decomposed it
contains no factor greater than 7.

11/9 is 11-prime-limit, but not a 7-prime-limit ratio, because it has
11 as a factor, which is greater than 7.

These are labels that help us to understand some of the
characteristics of ratios.

You did fine. There is nothing to apologize about. Thanks for your comments
and insight.

Stephen Szpak

_________________________________________________________________
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🔗Stephen Szpak <stephen_szpak@hotmail.com>

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti" <giordanobruno76@y...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

>REFERENCES:
>
> NEUTRAL 3RD = 350.9775
>
> ASSUMING I DID THE MATH CORRECTLY, I TOOK THE JI MAJOR 3RD AND
MINOR 3RD
> AND SPLIT THE DIFFERENCE.

Maths are pretty correct. Notice you have taken the geometric mean of
5/4 and 6/5 (or the arithmetic mean of those ratios in cents).

I suggest that you try the maths again...

49/40 equiv. 351.3381 cents

>
> I DON'T KNOW MUCH ABOUT TUNING. 7-LIMIT DOESN'T MEAN ANYTHING TO
ME. I
> KNOW THESE TERMS ARE AT MONZ'S SITE. I COULD READ THEM BUT
SOMETIMES
> IT'S JUST WORDS TO ME.
>

Stephen:

You should check some previous posts on the topic.
Usually, two senses are devised for the term n-limit:

a) ODD
b) PRIME

The first meaning (n-odd-limit) applies to ratios not containing a
numerator and derivator larger than n, once discarded the factors of
2 (octave transpositions).

In this sense, 11/9 is a 13-odd-limit ratio, as well as 18/11. Both
are also 15-odd-limit, 17-odd-limit, etc.
49/40 is not a 13-odd-limit ratio. It's 49-odd-limit.

The second sense involves decomposition into prime factors.
E.g., 75=3*5^2 77=7*11 etc.
So, n-prime limit implies that neither num. nor denom. contain prime
factors larger than n.

For instance, 11/9 is not a 7-prime-limit ratio, since the numerator
contains the prime factor 11.

49/40, on the other hand, is a 7-prime-limit ratio. [And 11-prime-
limit, 13, 17 and so...]

I was talking about the fact that you apparently took a 7-prime-limit
reference (which is of course OK) as 49/40, rather than the more
traditional 11/9 (11-prime-limit neutral third).

Max.
--- End forwarded message ---

Stephen Szpak writes:

(Please note this is, I believe your first reply to my original message a few days ago.)

I think we're thinking on 2 separate plains here. You are analyzing the Szpak Scale
with "odd-limit" "prime-limit" and other terms, and I am looking at the scale regarding
the harmonics it contains.

If one uses the tonics of the 12 EDO system (ignoring the richness of the other half of
the Szpak Scale) you have the ability to reach, access, use all of the first 12 harmonics.
The 12 EDO scale does NOT have the 7th and 11th harmonics.

12 EDO 7th harmonic not available
12 EDO 11th harmonic not available

24 EDO 7th harmonic not available
24 EDO 11 harmonic available in 24 tonics

Szpak Scale 7th harmonic available in 12 tonics (the 12 western tonic notes)
Szpak Scale 11th harmonic available in 12 tonics (the 12 western tonic notes)

The reason that the neutral 3rd may be slightly off (I guess it is off by more than 10 cents)
if because of this. The 24 EDO scale, which is what the Szpak Scale is based on, is really
2 12 EDO scales that have been interlaced. If the first scale is untouched the 12 familiar
notes of western music are intact. The second scale has to be shifted however, if the
harmonic 7th is to be included.
But here's the important point. If the second scale is shifted too much the 11 th harmonic
is lost. The Szpak Scale includes the 7th and 11 harmonics BOTH at less than 10 cents off
their ideal locations at the expense of losing other ratios. In fact the shift of the second
scale can only exist (you can check the math if you really want to ) within a 2.49 cent
window. Otherwise the 7th harmonic is off by 10 cents or more OR the 11 th harmonic
is off by 10 cents or more. I realize that the "10 cent" business is arbitrary on my part, but
the line has to be drawn somewhere.

0 58.827 100 158.827 200 258.827 etc. until 1200

0 61.317 100 161.317 200 261.317 etc until 1200

0 60.88 100 160.88 200 260.88 etc until 1200 (Szpak Scale)

I suppose you might have realized all this already. Feel free to get back to me either way.

Stephen Szpak

_________________________________________________________________
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🔗Maximiliano G. Miranda Zanetti <giordanobruno76@yahoo.com.ar>

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
> --- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
> wrote:
>
> Max.
> --- End forwarded message ---
>
> Stephen Szpak writes:
>
> (Please note this is, I believe your first reply to my original
message a
> few days ago.)
>
> I think we're thinking on 2 separate plains here. You are
analyzing
> the Szpak Scale
> with "odd-limit" "prime-limit" and other terms, and I am looking
at the
> scale regarding
> the harmonics it contains.
>

Well, Stephen, indeed it all started because I just wanted to know
which was your neutral third's reference, because it was not clear to
me.

I didn't mean to analyse deeply your proposal or turn it down in
terms of n-prime-limit approximation.

Please notice that prime-limit stuff and harmonics' representation
capabilities are not opposite but related concepts.

If I say this scale represents quite well 7-prime-limit ratios, I may
well think it does a good job with the 7/4 ratio (seventh harmonic).

> If one uses the tonics of the 12 EDO system (ignoring the
richness of the
> other half of
> the Szpak Scale) you have the ability to reach, access, use all
of the
> first 12 harmonics.
> The 12 EDO scale does NOT have the 7th and 11th harmonics.
>
> 12 EDO 7th harmonic not available
> 12 EDO 11th harmonic not available
>
> 24 EDO 7th harmonic not available
> 24 EDO 11 harmonic available in 24 tonics
>
> Szpak Scale 7th harmonic available in 12 tonics (the 12
western
> tonic notes)
> Szpak Scale 11th harmonic available in 12 tonics (the 12
western
> tonic notes)
>

Yes, that's a good point on behalf of your scale.

> The reason that the neutral 3rd may be slightly off (I guess it
is off
> by more than 10 cents)
> if because of this. The 24 EDO scale, which is what the Szpak
Scale is
> based on, is really
> 2 12 EDO scales that have been interlaced. If the first scale
is
> untouched the 12 familiar
> notes of western music are intact. The second scale has to be
shifted
> however, if the
> harmonic 7th is to be included.
> But here's the important point. If the second scale is
shifted too
> much the 11 th harmonic
> is lost. The Szpak Scale includes the 7th and 11 harmonics BOTH
at less
> than 10 cents off
> their ideal locations at the expense of losing other ratios. In
fact the
> shift of the second
> scale can only exist (you can check the math if you really want
to )
> within a 2.49 cent
> window. Otherwise the 7th harmonic is off by 10 cents or more
OR the
> 11 th harmonic
> is off by 10 cents or more. I realize that the "10 cent"
business is
> arbitrary on my part, but
> the line has to be drawn somewhere.
>

That's alright. If I were in your shoes, I wouldn't be too worried
about neutral third; 7th and 11th harmonic are more important to
approximate.

As long as the neutral third is neither too minor nor too major, it
will work, in my humble opinion.

As you have read, there's no thing such as an unanimous reference for
the neutral 3rd. But 7/4 is just 7/4.

One last apologetic word: in one of my posts i had used the word
derivator (yeah, derivator ¿!) instead of denominator.
PARDON MY ENGLISH.

> 0 58.827 100 158.827 200 258.827 etc. until 1200
>
> 0 61.317 100 161.317 200 261.317 etc until 1200
>
> 0 60.88 100 160.88 200 260.88 etc until
1200
> (Szpak Scale)
>
> I suppose you might have realized all this already. Feel free
to get
> back to me either way.
>
> Stephen Szpak
>

Max.

🔗Paul Erlich <paul@stretch-music.com>

🔗Maximiliano G. Miranda Zanetti <giordanobruno76@yahoo.com.ar>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
>
> > If I say this scale represents quite well 7-prime-limit ratios, I
> may
> > well think it does a good job with the 7/4 ratio (seventh
harmonic).
>
> Max,
>
> Wouldn't 7-odd-limit be more relevant and more to the point than 7-
> prime-limit here?
>
> -Paul

Dammit, the best option may be my refraining from posting unclear
examples.

I could have said 5-odd vs 7-odd (3/2, 5/4 vs 7/4). You're right on
that point.

Or, else I could have said "some" 5-prime vs "some" 7-prime ratios
(3/2,5/4,15/8 vs 7/4).

I think sometimes it's useful the interaction of both concepts. For
instance, in regards to a tritriadic 5-limit JI scale, it's good to
think of the intersection of 5-prime and 15-odd subsets of ratios.

In this way, one gets 5/4, 15/8, etc. but discards 7/4. (Well, forget
about 16/15!)

Of course, I may have missed some convention on this.

Max.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> I think sometimes it's useful the interaction of both concepts. For
> instance, in regards to a tritriadic 5-limit JI scale, it's good to
> think of the intersection of 5-prime and 15-odd subsets of ratios.
>
> In this way, one gets 5/4, 15/8, etc. but discards 7/4. (Well,
forget
> about 16/15!)
>
> Of course, I may have missed some convention on this.
>
> Max.

Well, one thing to keep in mind is that there are pitch-ratios (which
are usually written as 5/4, etc.) and there are interval-ratios
(which are usually written as 5:4), etc.

Scales are typically given as a list of pitch-ratios, frequency
ratios with respect to a (possibly arbitrary) tonic. It's extremely
rare to use the odd-limit concept when looking at these. The only
exception I know of is Partch's Tonality Diamond concept.

In general, a scale with N notes will have N*(N-1)/2 possible
intervals. Some of these may be duplicates of one another. And of
course their prime limit of their ratios will be the same as that of
the original scale's pitch-ratios.

When you speak of 5-limit harmony, though, the idea is that
*intervals* within the 5-*odd*-limit will be considered consonant;
chords which contain such intervals and only such intervals will also
be considered consonant; and all other intervals and chords will be
considered dissonant. This is where *triads* come from in the first
place -- they are the largest possible *consonant* chords in 5-limit
harmony; that is, in the 5-odd-limit.

When combining these 5-odd-limit triads into larger Just Intonation
scales, you will of course never get any prime numbers higher than 5
entering the ratios, so you will have a 5-prime-limit scale. But I
think the basic generating idea behind these scales is the 5-odd-
limit, if you follow my drift.

Hope I'm not belaboring the obvious,
Paul

🔗Paul Erlich <paul@stretch-music.com>

🔗Maximiliano G. Miranda Zanetti <giordanobruno76@yahoo.com.ar>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Well, one thing to keep in mind is that there are pitch-ratios
(which
> are usually written as 5/4, etc.) and there are interval-ratios
> (which are usually written as 5:4), etc.
>
> Scales are typically given as a list of pitch-ratios, frequency
> ratios with respect to a (possibly arbitrary) tonic. It's extremely
> rare to use the odd-limit concept when looking at these. The only
> exception I know of is Partch's Tonality Diamond concept.
>
> In general, a scale with N notes will have N*(N-1)/2 possible
> intervals. Some of these may be duplicates of one another. And of
> course their prime limit of their ratios will be the same as that
of
> the original scale's pitch-ratios.
>
> When you speak of 5-limit harmony, though, the idea is that
> *intervals* within the 5-*odd*-limit will be considered consonant;
> chords which contain such intervals and only such intervals will
also
> be considered consonant; and all other intervals and chords will be
> considered dissonant. This is where *triads* come from in the first
> place -- they are the largest possible *consonant* chords in 5-
limit
> harmony; that is, in the 5-odd-limit.
>
> When combining these 5-odd-limit triads into larger Just Intonation
> scales, you will of course never get any prime numbers higher than
5
> entering the ratios, so you will have a 5-prime-limit scale. But I
> think the basic generating idea behind these scales is the 5-odd-
> limit, if you follow my drift.
>
> Hope I'm not belaboring the obvious,
> Paul

It's alright. A "Clear as can be" elaboration like this is always to
be welcome by a somewhat troubled and confused intellect, like mine
yesterday.

Max.