I'm not getting it... [Ligon scale]

I'm not following how Jacky gets the "second octave" of his scales:

http://tma.asgarddesign.net/TMA_vol_3_pg_06.htm

Ratio Cents Steps
1/1 0 0
11/10 165.004 165.004
11/9 347.408 182.404
11/8 551.318 203.910
11/7 782.492 231.174 Appended Otonal Ratios
11/6 1049.363 266.871 1/1
2/1 1200.000 150.637 12/11
7/3 1466.871 266.871 14/11
8/3 1698.045 231.174 16/11
3/1 1901.955 203.910 18/11
10/3 2084.359 182.404 20/11
11/3 2249.363 165.004 2/1

How is the 2/1, 7/3, 8/3, 3/1, 10/3, 11/3 derived??

I could keep puzzling over this but I *know* somebody here can save
me some time... I tried.

JP

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

/tuning/topicId_unknown.html#32640
>
> Joseph,
>
> When you impose a 2/1 onto the end of the 11 Limit Utonal Series
> (Subharmonics):
>
> Ratio Cents Steps
> 11/10 165.004 165.004
> 11/9 347.408 182.404
> 11/8 551.318 203.910
> 11/7 782.492 231.174
> 11/6 1049.363 266.871
> 2/1 1200.000 150.637
>
> You can see that the interval between 11/6 and 2/1 is 12/11.
>
> 12/11 = 150.637 cents
>
> If you treat the 11/6 as the "1/1" for the inversions of the above
> ratios, and add them to the series, you get two overlapping
octaves,

Hi Jacky...

Thanks for the help, but I'm still not exactly seeing how you "add
them to the series" above. Could you please write out the math for
me...

I told people I shouldn't be over on "Tuning Math..." :)

JP

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

/tuning/topicId_unknown.html#32644

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> >
> > /tuning/topicId_unknown.html#32640
> > >
> > > Joseph,
> > >
> > > When you impose a 2/1 onto the end of the 11 Limit Utonal
Series
> > > (Subharmonics):
> > >
> > > Ratio Cents Steps
> > > 11/10 165.004 165.004
> > > 11/9 347.408 182.404
> > > 11/8 551.318 203.910
> > > 11/7 782.492 231.174
> > > 11/6 1049.363 266.871
> > > 2/1 1200.000 150.637
> > >
> > > You can see that the interval between 11/6 and 2/1 is 12/11.
> > >
> > > 12/11 = 150.637 cents
> > >
> > > If you treat the 11/6 as the "1/1" for the inversions of the
> above
> > > ratios, and add them to the series, you get two overlapping
> > octaves,
> >
> > Hi Jacky...
> >
> > Thanks for the help, but I'm still not exactly seeing how
you "add
> > them to the series" above. Could you please write out the math
for
> > me...
> >
> > I told people I shouldn't be over on "Tuning Math..." :)
> >
> > JP
>
> Gladly Dear Friend!
>
> It's very simple really. What you do to add ratios is to multiply
the
> numerators and denominators, then divide the results.
>
> So here, if you add 12/11 to 11/6 you would multiply:
>
> 11*12=132
>
> 6*11=66
>
> 132/66=2/1 reduced for 1200 cents
>
>
> Next, to add 14/11 to 11/6, you would:
>
> 11*14=154
>
> 6*11=66
>
> 154/66=7/3 reduced for 1466.871 cents.
>
>

Thanks so much Jacky, for the help! This explains everything. My
confusion was that somehow I thought you were "adding" (which I
*did,* fortunately, understand) the Otonal set of ratios to the
*existing* utonal ratios, and they weren't coming out right... rather
than "adding" each to the "root" of 11/6.

I don't know if *anybody* else could have possibly been confused by
that, or if I was just being a "dunce" but that's what happened.

Congrats on all the wonderful new resources and the great music that
has resulted!!!

best,

Joseph

----- Original Message -----
From: jacky_ligon <jacky_ligon@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, January 13, 2002 8:16 AM
Subject: [tuning] Re: I'm not getting it... [Ligon scale]

> It's very simple really. What you do to add ratios is to multiply the
> numerators and denominators, then divide the results.
>
> So here, if you add 12/11 to 11/6 you would multiply:
>
> 11*12=132
>
> 6*11=66
>
> 132/66=2/1 reduced for 1200 cents

It's really much easier to understand if you show how to
do it by the old-fashioned way I learned in school to
multiply fractions:

(of course, Yahoo's deletion of leading spaces is going
to completely mess up the way this looks on the web inerface)

12 11
-- * --
11 6

It's obvious doing it this way that the 11's cancel each
other, and the 6 divides 12 twice, so your answer is 2/1.

>
>
> Next, to add 14/11 to 11/6, you would:
>
> 11*14=154
>
> 6*11=66
>
> 154/66=7/3 reduced for 1466.871 cents.

Same thing again here:

11 14
-- * --
6 11

The 11's cancel, you divide 14 and 6 in half,
you get 7/3 for your answer.

And so on.

-monz

_________________________________________________________
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> From: jacky_ligon <jacky_ligon@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 13, 2002 8:49 AM
> Subject: [tuning] Re: I'm not getting it... [Ligon scale]
>
>
> Master Monz,
>
> Would you agree that the math for adding and subtracting ratios
> should be a part of any FAQ that might come from this group?

Yes, absolutely. You're speaking here specifically of
multiplying and dividing fractions.

> Do you have this on your web??

Nope, not this method ... quite a glaring omission.

I show how to do it with prime-factor notation

http://www.ixpres.com/interval/monzo/article/article.htm#calculate

using vector addition, which is more complicated for simple
examples like this, but is even easier for more complex
ratios.

To add 12/11 to 11/6 :

2 3 5 7 11

11/6 = [-1 -1 0 0 1]
* 12/11 = + [ 2 1 0 0 -1]
------------------
[ 1 0 0 0 0] = 2/1

Your next example: to add 14/11 to 11/6 :

2 3 5 7 11

11/6 = [-1 -1 0 0 1]
* 14/11 = + [ 1 0 0 1 -1]
------------------
[ 0 -1 0 1 0] = 7/3

Etc.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

> It's very simple really. What you do to add ratios is to multiply the
> numerators and denominators, then divide the results.
>
> So here, if you add 12/11 to 11/6 you would multiply:
>
> 11*12=132
>
> 6*11=66
>
> 132/66=2/1 reduced for 1200 cents

Actually, this is the rule for multiplying fractions. To add fractions you represent both numbers by a fractions with a common denominator, add the numerators, and divide by the common denominator. If you want to make that into a rule of the sort you gave, cross-multiply each of the numerators by the denominator of the other fraction, and add the results, and then divide by the product of the two denominators:

12/11 + 11/6 = (12*6 + 11*11)/(11*6) = 193/66

We can now reduce by factoring--noting, for instance, that 193 is a prime; or that it is not divisible by either 2, 3, or 11--or else by means of the Euclidean algorithm which is related to continued fractions.

The language in tuning has always referred to placing one ratio above the other as adding a
ratio. As as far as i can see (although i would be interested) the other "correct" formula lacks a
musical application and thus i see no reason to change the language that already is easily
understood by musicians universally. and that is what we are talking about.

I also have to voice an objection to the idea of placing such artifacts under the
interpretation of ET's or ratios and such unless there is reason to believe that this is the basis
of construction. I am not trying to be insulting but i can't help thinking of the act of placing
those myths around the world and comparing them to stories found in the Bible. Pardon my lack of
imagination in coming up with such a harsh metaphor. It is even under our own western disciplines
extremely unscientific. What we need to do is really look at these things and figure out what is
going on instead of attempting to put them as square pegs into round holes.

The fault with this lies with our method of description and we should always be on guard to
watch how our description influences our analysis. For instance since i do not know the actual
range in which this instrument is I chose 440 as a base pitch to look to see if looking at in it
terms of Vibration per sec. might actually tells us something about what might be really going on.
(regardless of the base pitch, v.p.s.. relations will hold) i got.....

1. 440
2. 476.78
3. 517.24
4. 570.61
5. 584. 96
6. 661.93
7. 733.18

I find it interesting that pitches 1,3,5,6 are all separated by about 77 vps
with 2, 4,6 between 94-101 but that is stretching it
there are larger relationships also.
Now if we knew the base pitch we might find that that these difference tones might fall within
the range we find the voices chanting. easily proven or disproved.
Anyway pardon if i put forth this example of an attempt to interpret a musical instrument
within the context of its own culture or at least within the confines of the very limited
knowledge we all have as such. It is my own observation in listening to the vast array of world
music that others hear in way surprisingly different than ourselves. more is to be gain by
comprehending their musical "technologies".

jacky_ligon wrote:

>
>
> I think in my effort to make this as simple as possible I might have
> chosen words that didn't have the desired effect.
>
> What I was trying to convey was to "append" one ratio to another,
> which is multiplication. I use the word append in my article, but
> likely should have clarified on this.
>
> J:L
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

> The language in tuning has always referred to placing one ratio above the other as adding a
> ratio.

I was groggy from lack of morning coffee when I read Jacky's post, and didn't understand what she meant at first. She said she was adding
two fractions, and it would be a really bad idea for musicians to adopt another definition than that of the grade school students of the world for the meaning of "adding fractions", it seems to me. Why not at least say adding intervals instead?

As as far as i can see (although i would be interested) the other "correct" formula lacks a
> musical application and thus i see no reason to change the language that already is easily
> understood by musicians universally. and that is what we are talking about.

I think if you asked musicians to add 12/11 and 11/6 you would *not* normally get 2 for an answer--musicians went to school like the rest of us.

> I also have to voice an objection to the idea of placing such artifacts under the
> interpretation of ET's or ratios and such unless there is reason to believe that this is the basis
> of construction.

I don't know what this means.

Jacky!
Not knowing what the beginning pitch of the Gyaling., i choose an arbitrary 440. it would be
good to know what the bottom pitch really is/was if possible.
it was an attempt to show how the same instrument data written out differently shows different
results or aspects of the instrument.

jacky_ligon wrote:

>
>
> J:L:
>
> All good points and all well taken and understood in my quarter.
>
> May I inquire as to what instrument you speak of above?
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Jacky!
great i'll remap the scale!

jacky_ligon wrote:

> Kraig,
>
> I measure the 1/1 to be approximately 369.994 cps. An F#.
>
> Thanks,
>
> J:L
>
> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > Jacky!
> > Not knowing what the beginning pitch of the Gyaling., i choose
> an arbitrary 440. it would be
> > good to know what the bottom pitch really is/was if possible.
> > it was an attempt to show how the same instrument data written
> out differently shows different
> > results or aspects of the instrument.
> >
> > jacky_ligon wrote:
> >
> > >
> > >
> > > J:L:
> > >
> > > All good points and all well taken and understood in my quarter.
> > >
> > > May I inquire as to what instrument you speak of above?
> > >
> > >
> >
> > -- Kraig Grady
> > North American Embassy of Anaphoria island
> > http://www.anaphoria.com
> >
> > The Wandering Medicine Show
> > Wed. 8-9 KXLU 88.9 fm
>
>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Hi Jacky and Kraig,

> From: jacky_ligon <jacky_ligon@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 13, 2002 4:29 PM
> Subject: [tuning] Re: I'm not getting it... [Ligon scale]
>
>
> KG:
> > I also have to voice an objection to the idea of placing such
> artifacts under the interpretation of ET's or ratios and such unless
> there is reason to believe that this is the basis of construction. I
> am not trying to be insulting but i can't help thinking of the act of
> placing those myths around the world and comparing them to stories
> found in the Bible. Pardon my lack of imagination in coming up with
> such a harsh metaphor. It is even under our own western disciplines
> > extremely unscientific. What we need to do is really look at these
> things and figure out what is going on instead of attempting to put
> them as square pegs into round holes.
>
> J:L:
> I completely agree. Although there may be some value in the
> comparative mathematical tools we use, when we are studying the
> tunings of other cultures we must take them as what they are, rather
> than try to impose another frames of reference onto them as if this
> is what they may be adhereing to. They should stand on their on
> merits.

I fully understand what both of you are writing about here.
Part of the reason I work so hard with my own dissemination
of tuning history and theory is because I've seen first the
understanding of African music and then the music itself
become distorted because of analysis of it in terms of 12-EDO.
These days, nearly all African music one hears is being played
on western instruments, tuned to 12-EDO.

But I have to dissent with you both at least to some degree.
I know the goals of many tuning theorists, myself included,
lean towards finding out the most objective mathematical
"truth" about historic tunings that may be found. A very
big part of my endeavor in this is to study the history of
tuning itself.

Certainly, it's difficult to impose objectivity upon a study
of something so ephemeral as music. But if one tries to take
into account as many factors as possible, and *includes them
in his mathematical models*, then surely one will be coming
as close as possible to the "truth".

A goal of modern tuning theory is therefore not to "impose
another frame of reference", but rather, to find the proper
frame of reference *from which* one may make a proper study
of any given musical system. It's often discovered that
a certain individual or culture was doing something systematic
that he/she/it never even knew about. Humans thrive on
communication and pattern recognition, and certainly these
two things are present in all music and can be modeled
mathematically.

-monz

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Hi there,

Actually, addition does have a meaning in music.

Harmonic sereis - continually adding 1
1 2 3 4 5

Now if you start at 1/1 and add, say 1/5
1+1/5 + 1/5 +...

you get
5/5 6/5 7/5 8/5...

i.e. the harmonic series starting from the 5th harmonic.

Adding say 6/5 gives
1 + 6/5 + 6/5 + ...

gives
5/5 11/5 17/5 ...

i.e. every sixth harmonic from the fifth upwards.

Starting from say 3/2 instead of 1/1

3/2 + 6/5 + 6/5 + ...

then the easiest thing is to multiply by 2/3 to make the first
term 1/1:

1/1 + 4/15 + 4/15 + ...

i.e. every fourth harmonic starting from the 15th upwards.

So if you start with a ratio and repeatedly add another ratio
you always go up a harmonic series by a fixed number
of terms at a time.

I added this to FTS some time back - ratio-wise additive scale steps, thinking
it might give something completely new (and you never know, possibly
musically interesting).

However, it only does so if the thing you are adding is irrational:

1 + g + g + ... (g = golden ratio)

isn't going to be anything in the harmonic series.

Robert

Correction, sorry:

Starting from say 3/2 instead of 1/1

3/2 + 6/5 + 6/5 + ...

then the easiest thing is to multiply by 2/3 to make the first
term 1/1:

1/1 + 4/5 + 4/5 + ...

i.e. every fourth harmonic starting from the 5th upwards.

Robert