A new scale -- tiny, though nice

Hi everyone.

I'm not sure if this has been ever mentioned on this list or in any tuning literature. Yesterday, I was just examining various sets of intervals and trying to make something out of it. What came out had surprised me quite a lot, much more than I could imagine at that time. The resulting scale, although it has good possibilities in the usual octave-periodic field, is not octave-periodic. Instead, the period is half an octave. And, more interestingly, it is actually a linear temperament with a generator being the 5th root of 8/5 (something like a neutral or very narrow major second). When you stack three of these, you're only about 112 cents below the period of 600 cents -- and this smaller interval can work okay as a nice minor second. - Even better, it approximates (more or less acceptably) many harmonic intervals, some of which are even as high as in the 11-limit region.
Okay, so the actual intervals from the bottom tone are:
111.788228
274.525486
437.262743
600.0 cents

I was so struck by the nice sound of the scale that I immediately made a smal improvized recording without any sequencer or multitracking. There are a few bad notes I played but I don't think it makes any harm to the general character of the tuning. Here it is: http://download.yousendit.com/2FA9A0BD2C629F20

Petr

That recording sounds great, Petr. It's a really neat tuning. However,
I'm hardly the foremost tuning theorist, and I couldn't really
understand how this scale is derived. I understand that the 5th root
of 8/5 is the 162.7 interval, but how did you decide on the half
octave as the period? It does seem to hit some interesting intervals.
I guess I'm interested in how you stumble upon it.
Thanks,
Cody

On Dec 9, 2007 2:43 PM, Petr Pařízek <p.parizek@chello.cz> wrote:
> Hi everyone.
>
> I'm not sure if this has been ever mentioned on this list or in any tuning
> literature. Yesterday, I was just examining various sets of intervals and
> trying to make something out of it. What came out had surprised me quite a
> lot, much more than I could imagine at that time. The resulting scale,
> although it has good possibilities in the usual octave-periodic field, is
> not octave-periodic. Instead, the period is half an octave. And, more
> interestingly, it is actually a linear temperament with a generator being
> the 5th root of 8/5 (something like a neutral or very narrow major second).
> When you stack three of these, you're only about 112 cents below the period
> of 600 cents -- and this smaller interval can work okay as a nice minor
> second. - Even better, it approximates (more or less acceptably) many
> harmonic intervals, some of which are even as high as in the 11-limit
> region.
> Okay, so the actual intervals from the bottom tone are:
> 111.788228
> 274.525486
> 437.262743
> 600.0 cents
>
> I was so struck by the nice sound of the scale that I immediately made a
> smal improvized recording without any sequencer or multitracking. There are
> a few bad notes I played but I don't think it makes any harm to the general
> character of the tuning. Here it is:
> http://download.yousendit.com/2FA9A0BD2C629F20
>
> Petr
>
>
>
>
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Some parts sounds maqamish Petr.

Oz.

----- Original Message -----
From: "Petr Pa��zek" <p.parizek@chello.cz>
To: <tuning@yahoogroups.com>
Sent: 10 Aral�k 2007 Pazartesi 0:43
Subject: [tuning] A new scale -- tiny, though nice

> Hi everyone.
>
> I'm not sure if this has been ever mentioned on this list or in any tuning
> literature. Yesterday, I was just examining various sets of intervals and
> trying to make something out of it. What came out had surprised me quite a
> lot, much more than I could imagine at that time. The resulting scale,
> although it has good possibilities in the usual octave-periodic field, is
> not octave-periodic. Instead, the period is half an octave. And, more
> interestingly, it is actually a linear temperament with a generator being
> the 5th root of 8/5 (something like a neutral or very narrow major
second).
> When you stack three of these, you're only about 112 cents below the
period
> of 600 cents -- and this smaller interval can work okay as a nice minor
> second. - Even better, it approximates (more or less acceptably) many
> harmonic intervals, some of which are even as high as in the 11-limit
> region.
> Okay, so the actual intervals from the bottom tone are:
> 111.788228
> 274.525486
> 437.262743
> 600.0 cents
>
> I was so struck by the nice sound of the scale that I immediately made a
> smal improvized recording without any sequencer or multitracking. There
are
> a few bad notes I played but I don't think it makes any harm to the
general
> character of the tuning. Here it is:
> http://download.yousendit.com/2FA9A0BD2C629F20
>
> Petr
>
>

Petr Pa��zek wrote:
> Hi everyone.
> > I'm not sure if this has been ever mentioned on this list or in any tuning > literature. Yesterday, I was just examining various sets of intervals and > trying to make something out of it. What came out had surprised me quite a > lot, much more than I could imagine at that time. The resulting scale, > although it has good possibilities in the usual octave-periodic field, is > not octave-periodic. Instead, the period is half an octave. And, more > interestingly, it is actually a linear temperament with a generator being > the 5th root of 8/5 (something like a neutral or very narrow major second).

Nice improvisation!

Okay, so with a period of half an octave, 2/1 is represented as (+2, +0) (2 periods up). Then, 5/4 is (+2, -5) (5 generators down from the octave), so 5/1 is (+6, -5). This in itself isn't enough to identify the temperament, but there are really only two possibilities that make sense: hedgehog and kwasy. And if you're using less than 9 notes in each half-octave period, that would rule out kwasy. 14 notes per octave is a good size for hedgehog.

I don't know if an 11-limit mapping of hedgehog has been proposed, but here's one that would work with this scale:

[<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]>
TOP-MAX: P = 599.090477, G = 163.146764
TOP-RMS: P = 600.129682, G = 164.649752

Cody Hallenbeck wrote:

> That recording sounds great, Petr. It's a really neat tuning. However,
> I'm hardly the foremost tuning theorist, and I couldn't really
> understand how this scale is derived. I understand that the 5th root
> of 8/5 is the 162.7 interval, but how did you decide on the half
> octave as the period? It does seem to hit some interesting intervals.
> I guess I'm interested in how you stumble upon it.
> Thanks,
> Cody

Glad you like it. :-)
The first step was that I took the harmonic chord of 3:5:6:7:9 and started subtracting the intervals in various ways until I found two commas I could temper out, one of them being 245/243 and the other one being 2430/2401. So I actually ended up with four new interval sizes that I could also use to describe the chord - 49/45, 15/14, and the two commas. After tempering the commas out, I found out that a single octave is made of 6 neutral seconds and two minor seconds in this system. So it was pretty clear to me that I can also view it as three neutral seconds and one minor second and stack two of these results, which will eventually make the octave.

Maybe I'll write more, now I'm leaving for school again.

Petr

Petr Pa��zek wrote:
> Cody Hallenbeck wrote:
> >> That recording sounds great, Petr. It's a really neat tuning. However,
>> I'm hardly the foremost tuning theorist, and I couldn't really
>> understand how this scale is derived. I understand that the 5th root
>> of 8/5 is the 162.7 interval, but how did you decide on the half
>> octave as the period? It does seem to hit some interesting intervals.
>> I guess I'm interested in how you stumble upon it.
>> Thanks,
>> Cody
> > Glad you like it. :-)
> The first step was that I took the harmonic chord of 3:5:6:7:9 and started > subtracting the intervals in various ways until I found two commas I could > temper out, one of them being 245/243 and the other one being 2430/2401.

Definitely hedgehog temperament, then. Paul Erlich lists a few of the commas tempered out: 50/49, 245/243, 250/253, and 2430/2401.

Hedgehog is one of the more unusual temperaments. Most of the well known regular temperaments can be defined by combining ET's -- such as meantone, which could be viewed as intermediate between 12-ET and 19-ET. Hedgehog, with its 8- and 14-note MOS, can be defined as an unusual 8-ET mapping <8, 13, 19, 23] combined with an unusual 14-ET mapping <14, 22, 32, 39]. As it's so unusual, it's been pretty much neglected, so it's nice to hear that it has some good musical uses. I'll have to play around with it and see what I can do with it one of these days.

Herman wrote:

> Okay, so with a period of half an octave, 2/1 is represented as (+2, +0)
> (2 periods up). Then, 5/4 is (+2, -5) (5 generators down from the
> octave), so 5/1 is (+6, -5). This in itself isn't enough to identify the
> temperament, but there are really only two possibilities that make
> sense: hedgehog and kwasy. And if you're using less than 9 notes in each
> half-octave period, that would rule out kwasy.

What is kwasy?

> 14 notes per octave is a good size for hedgehog.

If I understand it well, you mean an octave-periodic linear temperament with a generator being the "slightly-more-than-neutral" second, is that it? If so, then I agree that 14 generators is a good choice here. In fact, I've found this out myself too, soon after I sent the piece.

> [<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]>
> TOP-MAX: P = 599.090477, G = 163.146764
> TOP-RMS: P = 600.129682, G = 164.649752

Where can I find some explanation on how the numbers on the first of these lines work and what "TOP-MAX" means?

Petr

Herman wrote:

> Definitely hedgehog temperament, then. Paul Erlich lists a few of the
> commas tempered out: 50/49, 245/243, 250/253, and 2430/2401.

1. Are you sure you meant 250/253? I'm asking just because you made only this one go downwards.

2. What? Tempering out something as large as 35 cents? Hmmm, ... I see, maybe we'll be tempering out minor seconds one day. :-D -- Well, I probably won't go this way.

Petr

Petr Pa��zek wrote:
> Herman wrote:
> >> Definitely hedgehog temperament, then. Paul Erlich lists a few of the
>> commas tempered out: 50/49, 245/243, 250/253, and 2430/2401.
> > 1. Are you sure you meant 250/253? I'm asking just because you made only > this one go downwards.

Sorry, that was a typo for 250/243.

> 2. What? Tempering out something as large as 35 cents? Hmmm, ... I see, > maybe we'll be tempering out minor seconds one day. :-D -- Well, I probably > won't go this way.
> > Petr

It's actually pretty common for scales that repeat at the half octave (or quarter octave) to temper out 50/49. The practical result is that the half octave represents both 7/5 and 10/7.

Besides, if you're tempering out both 245/243 and 2430/2401, then 50/49 (which is just the two multiplied together) will also be tempered out.

Petr Pa��zek wrote:
> Herman wrote:
> >> Okay, so with a period of half an octave, 2/1 is represented as (+2, +0)
>> (2 periods up). Then, 5/4 is (+2, -5) (5 generators down from the
>> octave), so 5/1 is (+6, -5). This in itself isn't enough to identify the
>> temperament, but there are really only two possibilities that make
>> sense: hedgehog and kwasy. And if you're using less than 9 notes in each
>> half-octave period, that would rule out kwasy.
> > What is kwasy?

One of the more obscure temperaments; it probably wouldn't make anyone's top 40 list of interesting temperaments. With the scheme of using ET's to identify regular temperaments, which I find useful, it would be 22&74 (intermediate between 22-ET and 74-ET), or 74&96 (between 74-ET and 96-ET). If you look at a chart of equal temperaments like the one at http://tonalsoft.com/enc/e/equal-temperament.aspx , you can draw a line between two ET's, and that shows you the range of possible tunings.

The generator mapping of kwasy is:

[<2, 1, 6], <0, 8, -5]>

with example tunings for the generators:
P = 600.002686, G = 162.742975 (TOP tuning of the generators)
P = 600.003398, G = 162.743548 (TOP-RMS tuning)

This describes how to take any JI interval and convert it to an interval of the temperament. If you take the first generator to be 600.002686 cents, and the second one as 162.742975 cents (usually we call these the "period" and the "generator"), the first three numbers of the generator mapping describe how many 600.0-cent periods to approximate the primes 2, 3, and 5, when combined with the number of 162.7-cent generators specified by the second group of three numbers. For example:

2/1: 2 * 600.0 + 0 * 162.7 = 1200.0 cents
5/1: 6 * 600.0 - 5 * 162.7 = 2786.3 cents.
3/2: (1-2) * 600.0 + (8-0) * 162.7 = 701.9 cents.

These generator mappings are very useful for identifying and studying the properties of temperaments, although they're not unique (they depend on which intervals are chosen for the generators). These days we typically have a convention where the first generator is an octave or 1/n of an octave, and the second generator is less than half the size of the first one.

>> 14 notes per octave is a good size for hedgehog.
> > If I understand it well, you mean an octave-periodic linear temperament with > a generator being the "slightly-more-than-neutral" second, is that it? If > so, then I agree that 14 generators is a good choice here. In fact, I've > found this out myself too, soon after I sent the piece.
> >> [<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]>
>> TOP-MAX: P = 599.090477, G = 163.146764
>> TOP-RMS: P = 600.129682, G = 164.649752
> > Where can I find some explanation on how the numbers on the first of these > lines work and what "TOP-MAX" means?

TOP-MAX is what we used to refer to simply as TOP. It's an optimized tuning of the generators, which minimizes the maximum error (weighted by the logarithms of the primes, so that e.g. 2/1 is more accurate than 3/1). TOP-RMS is a similar optimization which minimizes the RMS (root mean square) error instead of the maximum error. TOP stands for "Tenney Optimal", as it's based on Tenney's "Harmonic Distance".

I have some explanation of regular temperaments on my web page, although some of the terminology may be out of date, and it doesn't really go into the details of how to use the generator mapping as I've explained above.

http://www.io.com/~hmiller/music/regular-temperaments.html

Also some brief MIDI examples of a handful of these temperaments:

http://www.io.com/~hmiller/music/zireen-music.html

Paul Erlich's paper "A Middle Path" is a good general explanation of this approach to regular temperaments, and includes many useful diagrams.

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

Herman wrote:

> with example tunings for the generators:
> P = 600.002686, G = 162.742975 (TOP tuning of the generators)
> P = 600.003398, G = 162.743548 (TOP-RMS tuning)

If this is the case, then I've actually played the piece in kwasy, didn't I? Why should be 5 generators "ruling out kwasy"?

> This describes how to take any JI interval and convert it to an interval
> of the temperament. If you take the first generator to be 600.002686
> cents, and the second one as 162.742975 cents (usually we call these the
> "period" and the "generator"), the first three numbers of the generator
> mapping describe how many 600.0-cent periods to approximate the primes
> 2, 3, and 5, when combined with the number of 162.7-cent generators
> specified by the second group of three numbers. For example:
>
> 2/1: 2 * 600.0 + 0 * 162.7 = 1200.0 cents
> 5/1: 6 * 600.0 - 5 * 162.7 = 2786.3 cents.
> 3/2: (1-2) * 600.0 + (8-0) * 162.7 = 701.9 cents.

Okay, now I think I understand. The reason why these numbers were so unclear to me was simply that I usually do it, when writing my own linear temperament schemes, always in exactly the opposite way -- one pair of numbers per line, the first of which is the number of periods required and the second one is the number of generators. Don't know if there's any general preference for either way or if it's like fifty-fifty. But for me the method you describe is less clear, even though it's nothing more than essentially turning mine 90 degrees. :-D Maybe it is, indeed, just because I'm not used to it, who knows?

Petr

Petr Pa��zek wrote:
> Herman wrote:
> >> with example tunings for the generators:
>> P = 600.002686, G = 162.742975 (TOP tuning of the generators)
>> P = 600.003398, G = 162.743548 (TOP-RMS tuning)
> > If this is the case, then I've actually played the piece in kwasy, didn't I? > Why should be 5 generators "ruling out kwasy"?

The difference between hedgehog and kwasy is not the exact tuning, but how the notes are used to approximate ratios of small integers.

[<2, 1, 6], <0, 8, -5]>

is the generator mapping for kwasy, which means to get a fifth, you need at least 9 notes per half-octave period (or 18 in all per octave):

3/2 is 1 period down (1 - 2) and 8 generators up (8 - 0). To save space, I usually write this as (-1, +8). A full 4:5:6 triad also has a major third (5/4 approximation), which is (+2, -5), so you'd need a chain of 14 generators for that.

Since you're using intervals that sound like fifths or fourths, but you've only got 12 notes per octave in your scale, you must be using a different approximation of 3/2, probably (+2, -3). Another way of looking at this is that the generator divides a perfect fourth into 3 equal parts. The optimal tuning for 7-limit hedgehog is very similar to kwasy, but the scale uses fewer notes.

[<2, 4, 6, 7], <0, -3, -5, -5]>
P = 598.446711, G = 162.315961 (TOP)
P = 599.619018, G = 164.247626 (TOP-RMS)

But these are just suggested tunings, and the 5th root of 8/5 (162.7372572) is well within the range of possible tunings for either temperament. Note that the tempered value of 8/5 in both of these temperaments is 5 generators up (+0, +5), which agrees with this tuning.