nonoctave equal temperament: 13th root of 7:5

Hi Jacky,

I have been timid about broadcasting my
little nonoctave scales for all to see.

So here is the first of the ones I've found
that I like the most.

The first scale I call "Just Tritone/13".

"An excellent scale with jazzy feel.
Great harmonies and expressive melodies.

Adjectives:
Opening, Thoughtful, Reflective,
Healing, Invigorating, Exciting,
Bright, Thrilling, Breathless,
Free

Good timbres:
Piano, strings, brass, or anything."

You can start it at middle C and repeat it in
both directions. It is jazzy and has pretty melodic
possibilities. I think it is a wonderful tuning
and encourage people to try it and let me know what they
think -- especially if you get any vibes or have any
interesting adjectives you find.

Here it is. The scale is:

7:5 divided up by [4,1,1,1,4,2]

Your basic pattern:

ratio cents
(7:5)^(0/13) 0.000000
(7:5)^(4/13) 179.234521
(7:5)^(5/13) 224.043151
(7:5)^(6/13) 268.851781
(7:5)^(7/13) 313.660411
(7:5)^(11/13) 492.894932
(7:5)^(13/13) 582.512193

Here it is in terms of 13 odd-limit where
you can really see some of its great
resources:

00 1/1
01 10/9 - 3.169191 cents
02 8/7 - 7.130943 cents
03 7/6 + 1.980876 cents
04 6/5 - 1.980876 cents
05 4/3 - 5.150067 cents
06 7/5
07 14/9 - 3.169191 cents
08 8/5 - 7.130943 cents
09 18/11 - 1.228086 cents
10 5/3 + 11.813891 cents
11 13/7 + 3.705370 cents
12 2/1 - 34.975615 cents
13 13/6 + 5.686245 cents
14 20/9 + 6.663824 cents
15 16/7 + 2.702073 cents
16 7/3 + 11.813891 cents
17 13/5 + 3.705370 cents
18 11/4 - 3.781365 cents
19 3/1 + 24.816098 cents
...

------
What I was thinking/my method:

I liked 7:5 and thought it would make a better foregone
conclusion than 2:1 -- make 7:5 my octave (what I call the
'Repeat Ratio'). Also, I liked the subminor third 7:6 from
working with it in 88cET. So I wanted to see if I could
have both.

I looked at all the equal divisions of 7:5.

I found that in decreasing amount of error:

* 7:6 is the 4th step in 9th root of 7:5
(7.977 cents deviation)

* 7:6 is the 5th step in 11th root of 7:5
(2.093 cents deviation)

* 7:6 is the 6th step in 13th root of 7:5
(1.981 cents deviation)

So I tried working with all three of these (listening) and
decided that I liked 13th root of 7:5 the best. That gave
me a chromatic equal scale with a step size of 44.809
cents.

But there were too many notes to make it manageable on the
keyboard so I wanted to find a nice subset to work with.
This was probably a matter of taste I suppose but I
haven't found many other subsets that I like nearly as
much as the one I'm showing you here, though maybe someone
else will find something interesting. I don't remember now
but it was probably more than a coincidence that a decided
to subset it to have 6 notes so that the tritone ended up
in the usual place on the standard keyboard.

--

Other notes:

* I've tried the 9th & 11th root more and have found at
least one other scale I like. I'll dig it up later.

* I've tried working with 10:7 but I just can't get it
going with 10:7 like I can with 7:5.

* A couple years after finding this scale I was reading
some stuff by Brian and he mentioned some study in which
listeners found that 7:5 was the very most consonant
interval they heard. I don't know the conditions of the
study but I believe it. 7:5 is more consonant than just
about anything else. It's just a great interval and one of
the few small order just intervals I really enjoy tuned
absolutely pure.

---------

I was going to write more but I was up way too late last
night because I couldn't sleep cos I was tossing and
turning and obsessing over the number 23 and could not
get any rest and now I am so tired I am going to bed now
early.

Another scale tommorow when I am rested if people
are interested. In the meantime, have fun playing
with it!

- Jeff

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:
> The first scale I call "Just Tritone/13".
>
> Adjectives:
>
> You can start it at middle C and repeat it in
> both directions. It is jazzy and has pretty melodic
> possibilities. I think it is a wonderful tuning
> and encourage people to try it and let me know what they
> think -- especially if you get any vibes or have any
> interesting adjectives you find.

Jeff,

I'll start here:

> Adjectives:

Joyous, Expansive, Interstellar, Meditative

>
> Here it is. The scale is:
>
> 7:5 divided up by [4,1,1,1,4,2]
>
> Your basic pattern:
>
> ratio cents
> (7:5)^(0/13) 0.000000
> (7:5)^(4/13) 179.234521
> (7:5)^(5/13) 224.043151
> (7:5)^(6/13) 268.851781
> (7:5)^(7/13) 313.660411
> (7:5)^(11/13) 492.894932
> (7:5)^(13/13) 582.512193
>

Thanks for sharing your scale here. I really enjoyed it. I played it
last night for about an hour, on a variety of timbres, and it sounds
great!
Was nice on Strings, Metallophones, a number of pads sounds, and my
favorite timbre for this was my "Liquid Glass", a hand made FM Timbre
seasoned with age, which sounds kind of like a Glass Harmonica with a
slow attack and long sustain - inharmonic in nature. The scale was
rich with interesting consonances, and beating intervals, and when I
found the pattern, I was able to get some lovely melodies happening,
but it also had wonderful chordal properties too.

What is really interesting to me, is that your scale directly
anticipated what I was going to post next about Tritone non-octave
scales. It's really fascinating to realize that we share this common
feeling about 7/5, which I also find to be a consonant interval. It
may sound perverse, but I like the chord 1/1-7/5-10/7, which I find
particularly lovely on strings.

> ------
> What I was thinking/my method:
>
> I liked 7:5 and thought it would make a better foregone
> conclusion than 2:1 -- make 7:5 my octave (what I call the
> 'Repeat Ratio'). Also, I liked the subminor third 7:6 from
> working with it in 88cET. So I wanted to see if I could
> have both.

I like this term. Mind if I adopt it? Of course with a scale of
constant structure, all of the intervals are "Repeat Ratios", but I
know what you mean - it is the interval from which the entire scale
is constructed.

>
> I looked at all the equal divisions of 7:5.
>
> I found that in decreasing amount of error:
>
> * 7:6 is the 4th step in 9th root of 7:5
> (7.977 cents deviation)
>
> * 7:6 is the 5th step in 11th root of 7:5
> (2.093 cents deviation)
>
> * 7:6 is the 6th step in 13th root of 7:5
> (1.981 cents deviation)
>
> So I tried working with all three of these (listening) and
> decided that I liked 13th root of 7:5 the best. That gave
> me a chromatic equal scale with a step size of 44.809
> cents.

I could hear that this was a good choice too when I played it.

>
> But there were too many notes to make it manageable on the
> keyboard so I wanted to find a nice subset to work with.
> This was probably a matter of taste I suppose but I
> haven't found many other subsets that I like nearly as
> much as the one I'm showing you here, though maybe someone
> else will find something interesting. I don't remember now
> but it was probably more than a coincidence that a decided
> to subset it to have 6 notes so that the tritone ended up
> in the usual place on the standard keyboard.

Very cool.

>
> --
>
> Other notes:
>
> * I've tried the 9th & 11th root more and have found at
> least one other scale I like. I'll dig it up later.
>
> * I've tried working with 10:7 but I just can't get it
> going with 10:7 like I can with 7:5.

I like 10/7 in two contexts:

1. JI Scales with inversional symmetry, where 10/7 is present
along with it's compliment 7/5.
2. A totally Utonal JI Scale, where 10/7 is the tritone.

>
> * A couple years after finding this scale I was reading
> some stuff by Brian and he mentioned some study in which
> listeners found that 7:5 was the very most consonant
> interval they heard. I don't know the conditions of the
> study but I believe it. 7:5 is more consonant than just
> about anything else. It's just a great interval and one of
> the few small order just intervals I really enjoy tuned
> absolutely pure.

I would like to read this article, I will try to see if I can get it
somehow. I wonder if this has anything to do with the findings about
the Tritone Paradox.

>
> ---------
>
> I was going to write more but I was up way too late last
> night because I couldn't sleep cos I was tossing and
> turning and obsessing over the number 23 and could not
> get any rest and now I am so tired I am going to bed now
> early.

I have this same issue sometimes. I'll get excited about some new
possibility, and it becomes hard to sleep, and when I do drop off I
feel I'm still dreaming about it. The work goes on in the still of
the night!

>
> Another scale tomorrow when I am rested if people
> are interested. In the meantime, have fun playing
> with it!

Yes! Please - more!

Jeff - I want to thank you for this great presentation. It always
helps me to understand other's work when they take time to explain
things as well as you have here. You spoke about the theory, timbres,
moods and musical context, making this a wonderful read. I had allot
of fun with this scale - which was entirely new to me. I haven't
worked with these "Root Tunings", but I'm sold on how good they
sound. I'll have to experiment with it some.

This makes me very eager to learn more about your approach to non-
octave structures - a topic of deep interest here.

Thanks - and bring out the Scales(!),

Jacky Ligon

Jeff Scott wrote:

> I liked 7:5 and thought it would make a better foregone
> conclusion than 2:1 -- make 7:5 my octave (what I call the
> 'Repeat Ratio').

Around here it's usually called "interval of equivalence," or IoE for short.

Equally dividing 7:5 - neat idea! Of course, it might be difficult for the typical listener to
identify the 7:5 as the IoE unless you construct your compositions in such a way to as to make it
clear. 44.81cents*27 is about 1210 cents - you probably should avoid having that interval in your
scales or most ears will inevitably assume standard octave equivalence.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

Hi David!

>> (what I call the 'Repeat Ratio').

> Around here it's usually called "interval of equivalence,"
> or IoE for short.

Yes, I'd gathered that some people were occasionally using
the term recently. What do you think? Doesn't its use
presume a mechanism of perceptual octave/other-ratio class
equivalence? I don't find any evidence of that myself.
What I believe is that a not-too-long repeating _pattern_
of intervals is *definately* what makes scales cohesive.
Thus the word 'Repeat' is key for me, and definately not
'Equivalence', which I think presumes a model I find
fallacious.

Also,

12 3456 78 90 1
Re peat Ra ti o

Int er val of e quiv a lence
123 45 678 90 1 2345 6 78901

11 characters and 5 syllables
vs
21 characters and 8 syllables

And 'Repeat Ratio' is also easier to say, has
alliteration, is more poetic, and more self-explanatory
(well IMHO at least).

Is IoE really in wide usage? What says the list? I guess I
hadn't given it much though becouse it did not seem to me
that it had caught on but the more I think about it, the
more it seems like a weird term to me. Seems 'clunky'.

> Equally dividing 7:5 - neat idea! Of course, it might be
> difficult for the typical listener to identify the 7:5 as
> the IoE unless you construct your compositions in such a
> way to as to make it clear. 44.81cents*27 is about 1210
> cents - you probably should avoid having that interval in
> your scales or most ears will inevitably assume standard
> octave equivalence.

Thanks! But, well the 27 chromatic step span of 1210 cents
does not appear regularly in the scale... which is a nice
feature of course.

Now, the 26 chromatic step span does occur regularly, it
being the square of the 7:5 Repeat Ratio. Here's the cents
values of the scale taken out a little further.

0.000000, 179.234521, 224.043151, 268.851781, 313.660411,
492.894932, 582.512193, 761.746713, 806.555344,
851.363974, 896.172604, 1075.407125, 1165.024385,
1344.258906, 1389.067536, 1433.876166, 1478.684797,
1657.919317, 1747.536578

Now the 1165.024385 cent interval -- which is of course
(7:5)^2 and which is mapped out every 12 notes of the um
'diatonic' scale (so-to-speak) -- I think is more likely
to pass for the octave since it is also a Repeat Ratio of
its own and 1165 cents can sound like an octave
melodically, say in an arpeggio figure. Though usually to
make that pass (in my experience) you need to have a fifth
like interval in there. The 716.938 cents interval (16
chromatic steps) would work ok except it's not regular in
the diatonic/subset scale. The way the subset is spaced
out makes it so that you can have an octave like interval
melodically if you want to force it but it's just not easy
to make any near-1200 interval come out sounding like the
base of the scale.

> http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--------

Hey, I read this by you in a previous post:

> ...music that reflects the order and complexity (unity and
> diversity) of creation in a way that is recognizable and
> digestible to the listener is likely to last a long time. To
> put it another way, lasting music helps put the listener in a
> frame of mind that raises or enhances the ability to see, or at
> least sense, the beauty and/or profundity in creation.

...and really agreed with it -- I think it says much
better and more correctly what I would liked to have said
when I just superficially glanced past the whole idea by
saying:

> 3) Model things on patterns found in nature:
> a) Use the golden ratio.

So anyway, It inspired me to take a look at your web site.
Very nice, Mr. INTP!

Liked the color design, illustrations & navigation a lot
too. By the way, I was especially impressed in that I
think your website is the only one I have been to where I
found the use of frames sensible rather than offensive
and/or annoying.

- Jeff

Jeff Scott wrote:

> Hi David!
>
> >> (what I call the 'Repeat Ratio').
>
> > Around here it's usually called "interval of equivalence,"
> > or IoE for short.
>
> Yes, I'd gathered that some people were occasionally using
> the term recently. What do you think? Doesn't its use
> presume a mechanism of perceptual octave/other-ratio class
> equivalence? I don't find any evidence of that myself.
> What I believe is that a not-too-long repeating _pattern_
> of intervals is *definately* what makes scales cohesive.
> Thus the word 'Repeat' is key for me, and definately not
> 'Equivalence', which I think presumes a model I find
> fallacious.

You're right. I hadn't thought about that. If your compositional vision doesn't include equivalency of
the scale at the point where it repeats, then the term IoE would carry unwanted baggage.

> 12 3456 78 90 1
> Re peat Ra ti o
>
> Int er val of e quiv a lence
> 123 45 678 90 1 2345 6 78901
>
> 11 characters and 5 syllables
> vs
> 21 characters and 8 syllables
>
> And 'Repeat Ratio' is also easier to say, has
> alliteration, is more poetic, and more self-explanatory
> (well IMHO at least).

Well, you're certainly right about the articulacy of it. It's clunky, like an academic term should be.
>: - ) (Maybe it would sound better in Latin?) But what if the repeated interval is not a rational?
What if it's arrived at by some other means? I mean, of course we'd have to burn the heretic at the
stake, but other than that? :-)

The only thing is that "Repeated Ratio" doesn't imply that said ratio is being further divided. A
previous suggestion was x-Equal Division of the Non-octave followed by the interval being divided. IOW,
"13-EDN 7:5", in this case. But that implies something special about the 2:1 ratio, an idea you don't
seem to hold with.

> Is IoE really in wide usage?

No, but it's rarity is only due to the fact that 99% of the scales and tunings discussed here are octave
equivalent.

Thanks for the compliment on the web site!

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--