For Herman (was Re: Rank 3 temperaments?)

Herman wrote:

> Somewhere I have a document with data for other rank 3 temperaments,
> which have names like Baldur, Odin, and Portent. It's been so long since
> these came up on tuning-math that I don't even know where to start
> searching.

Dear Herman,

thanks an awful lot for this. I've put the three names into Google together with a few additional words and I found something like this: www.robertinventor.com/tuning-math/s___7/msg_6250-6274.html#6251
I've run through the list briefly and I realized some things were not very clear to me. I hope you don't mind if I ask you some questions about that; I'm slowly begining to think you're the only person (and maybe also Graham) who can help me. To make the following text more understandable, I'll use "P" for the pure fifth of 3/2 and "S" for the smaller fifth of 40/27.

Considering untempered planar tunings, I understand that a simple 12-tone one can be made with a chain of fifths like "PPPSPPPSPPP" (reduced into a single octave range) because this is the same as making 5/1 twice and then adding three Ps to each of the three tones, which means there are actually two generators (P and 5/1). My question is this: If I change the chain of fifths to "SPPPSPPPSPP", is it still a planar tuning? The problem is that although the 5/1 generator is still there, the first three fifths in each "cycle of four fifths" are not all the same size. If the answer is no, then the scale I've made recently isn't either.

I was surprised how many of the temperaments on Gene's list used generators approximating 3/1 and 5/1. Maybe I'm doing something wrong but none of my recent planar temperament attempts had generators like this. I also noted that I couldn't find any of them in Gene's list so maybe I'm using a different approach which makes different scales. I'll at least show you one in order you had an idea of what I'm talking about; whether it is a planar temperament or not, I'm still not sure. For example, when I wanted to temper out the "ragisma" of 4375/4374, I took its prime exponents ("-1 -7 4 1") and broke the number into seven 1/3s, four 5/1s, and one 7/2. First, I made a "sequence" of intervals in the order of "ABBABBABBAB", where A was 1/3 and B was 5/1, both of which were one 11th of the "ragisma" lower. This made me arrive at exactly 2/7. IIRC, the "generators" then came up as a tempered 5/9 and a tempered 1/3. Finally, I reduced the intervals to a single octave and I got a 12-tone scale. -- Well, I hope my words aren't tiring you. I'm just wondering if I'm really making something like planar temperaments.

Petr

PS: In case you were interested, this is the sequence of the raising and falling intervals which was the result of breaking the fraction and applying the tempering:

2786.27774
884.28676
-1017.70422
1768.57351
-133.41746
-2035.40844
750.86929
-1151.12169
-3053.11266
-266.83493
2/7

Petr Pa��zek wrote:
> Herman wrote:
> >> Somewhere I have a document with data for other rank 3 temperaments,
>> which have names like Baldur, Odin, and Portent. It's been so long since
>> these came up on tuning-math that I don't even know where to start
>> searching.
> > Dear Herman,
> > thanks an awful lot for this. I've put the three names into Google together > with a few additional words and I found something like this: > www.robertinventor.com/tuning-math/s___7/msg_6250-6274.html#6251
> I've run through the list briefly and I realized some things were not very > clear to me. I hope you don't mind if I ask you some questions about that; > I'm slowly begining to think you're the only person (and maybe also Graham) > who can help me. To make the following text more understandable, I'll use > "P" for the pure fifth of 3/2 and "S" for the smaller fifth of 40/27.

Yes, maybe me as well! I don't like these "dear so-and-so" posts and reply to whatever I want to reply to ;-)

> Considering untempered planar tunings, I understand that a simple 12-tone > one can be made with a chain of fifths like "PPPSPPPSPPP" (reduced into a > single octave range) because this is the same as making 5/1 twice and then > adding three Ps to each of the three tones, which means there are actually > two generators (P and 5/1). My question is this: If I change the chain of > fifths to "SPPPSPPPSPP", is it still a planar tuning? The problem is that > although the 5/1 generator is still there, the first three fifths in each > "cycle of four fifths" are not all the same size. If the answer is no, then > the scale I've made recently isn't either.

That's the formula for a circulating temperament. It may happen to be rank 3 like it may happen to be rank 2. To know the actual rank you need to write it in terms of orthogonal generators. For a linear temperament that'd always be

PPP....PS

where P is the generator and S is some residue that rounds you up to an MOS. Where the period and generator don't match you start this chain at each period.

A rank 3 temperament, then, has two different generators, P and Q

.
.
.
Q
Q
Q
XPPPPP...

You fill in the whole grid with different combinations of P and Q.

> I was surprised how many of the temperaments on Gene's list used generators > approximating 3/1 and 5/1. Maybe I'm doing something wrong but none of my > recent planar temperament attempts had generators like this. I also noted > that I couldn't find any of them in Gene's list so maybe I'm using a > different approach which makes different scales. I'll at least show you one > in order you had an idea of what I'm talking about; whether it is a planar > temperament or not, I'm still not sure. For example, when I wanted to temper > out the "ragisma" of 4375/4374, I took its prime exponents ("-1 -7 4 1") > and broke the number into seven 1/3s, four 5/1s, and one 7/2. First, I made > a "sequence" of intervals in the order of "ABBABBABBAB", where A was 1/3 and > B was 5/1, both of which were one 11th of the "ragisma" lower. This made me > arrive at exactly 2/7. IIRC, the "generators" then came up as a tempered 5/9 > and a tempered 1/3. Finally, I reduced the intervals to a single octave and > I got a 12-tone scale. -- Well, I hope my words aren't tiring you. I'm just > wondering if I'm really making something like planar temperaments.

It looks like Gene gave Hermite-reduced bases. That is, he looks for generators that approximate 3/1 and 5/1 where possible. One thing which makes rank 3 temperaments difficult to identify is that there are different ways of choosing the generators. So, if you can, find a way of reducing the basis or calculating the wedgie.

For the 7-limit that doesn't matter because you're only looking at one comma and that uniquely defines the temperament class. For the ragisma, then, I get this as an optimal tuning map:

<1200.00181343 1901.98688957 2786.27460651 3368.81161439]

That tells you how to tune the prime intervals 2:1, 3:1, 5:1, and 7:1. To get an interesting scale you can try starting with you're favourite chunk of 7-limit JI and see how it comes out. You're AB business gives you, on a 5-limit lattice,

; X
; X X
; X
; X X
; X
; X X
; X
;X X

which doesn't leap out as being an obvious place to start. Although I think it does give you an approximate 7:4 so you probably are thinking along the right lines. Also, I note that 12-equal doesn't temper out the ragisma so a 12 note scale isn't an obvious starting point either. You could try 8, 19, 26, 27, 34, 46, 53, 65, or 72. And probably you need more than 12 notes for such a complicated temperament to be useful.

Graham

Petr Pa��zek wrote:

> Considering untempered planar tunings, I understand that a simple 12-tone > one can be made with a chain of fifths like "PPPSPPPSPPP" (reduced into a > single octave range) because this is the same as making 5/1 twice and then > adding three Ps to each of the three tones, which means there are actually > two generators (P and 5/1). My question is this: If I change the chain of > fifths to "SPPPSPPPSPP", is it still a planar tuning? The problem is that > although the 5/1 generator is still there, the first three fifths in each > "cycle of four fifths" are not all the same size. If the answer is no, then > the scale I've made recently isn't either.

Every note in the scale is still reached with only the two generators (plus the octave). So that makes it a planar (or rank 3) tuning. The difference between these two scales is the selection of different notes from the same tuning system.

\ P +0 +1 +2 +3
5/1
+0 * * * *
+1 * * * *
+2 * * * *

\ P -3 -2 -1 -0
5/1
+0 *
+1 * * * *
+2 * * * *
+3 * * *

> I was surprised how many of the temperaments on Gene's list used generators > approximating 3/1 and 5/1. Maybe I'm doing something wrong but none of my > recent planar temperament attempts had generators like this.

All temperaments with more than one generator can have different choices of generators, but this is especially noticeable starting with rank 3 because of the convention of using the octave as one of the generators. In rank 2 temperaments, if one generator is an octave or equal division of the octave, there are two possible sizes for the other generator that are smaller than the first one. E.g. meantone with a fourth or a fifth as a generator. With a rank 3 temperament, you have more options for generator sizes.

> I also noted
> that I couldn't find any of them in Gene's list so maybe I'm using a > different approach which makes different scales. I'll at least show you one > in order you had an idea of what I'm talking about; whether it is a planar > temperament or not, I'm still not sure. For example, when I wanted to temper > out the "ragisma" of 4375/4374, I took its prime exponents ("-1 -7 4 1") > and broke the number into seven 1/3s, four 5/1s, and one 7/2. First, I made > a "sequence" of intervals in the order of "ABBABBABBAB", where A was 1/3 and > B was 5/1, both of which were one 11th of the "ragisma" lower. This made me > arrive at exactly 2/7. IIRC, the "generators" then came up as a tempered 5/9 > and a tempered 1/3. Finally, I reduced the intervals to a single octave and > I got a 12-tone scale. -- Well, I hope my words aren't tiring you. I'm just > wondering if I'm really making something like planar temperaments.
> > Petr

Yes, this looks like a planar temperament. The reason you can't find it on Gene's list is that it's a list of 11-limit temperaments. 7-limit rank 3 temperaments are most easily identified by the comma which is tempered out, since there are so many possibilities for generator mappings. So you could call this the [-1, -7, 4, 1> temperament.

You could describe this temperament with a generator mapping like this:

[<1, 0, 0, 1], <0, -1, -2, 1], <0, 0, 1, -4]>

although then your generator sizes would be negative cents, so it might make things easier to use 3/1 and 9/5 as generators.

[<1, 0, 0, 1], <0, 1, 2, -1], <0, 0, -1, 4]>

In other words, 7/1 is approximated as 2/1 * 3/1^-1 * 9/5^4.

It's fairly straightforward to find optimal tunings (TOP-MAX or TOP-RMS) from this. The details of how I found the generator mapping from your description, and how to find the optimal tunings, are probably best for the tuning-math list, since you need some basic linear algebra.

Graham Breed wrote:

> It looks like Gene gave Hermite-reduced bases. That is, he > looks for generators that approximate 3/1 and 5/1 where > possible. One thing which makes rank 3 temperaments > difficult to identify is that there are different ways of > choosing the generators. So, if you can, find a way of > reducing the basis or calculating the wedgie.

There's an easy way to get the wedgie of a 7-limit rank 3 temperament: reverse the order of the numbers in the unison vector and flip a couple of signs.

e.g. for starling
unison vector: [1, 2, -3, 1>
wedgie: <<<1, 3, 2, -1]]]

ragisma: [-1, -7, 4, 1>
wedgie: <<<1, -4, -7, 1]]]

The hard way is to calculate the wedge product from a generator mapping (which gives the same result).

Note that the wedgies in Gene's Jan. 2003 post on 11-limit planar temperaments appear to have incorrect signs; this may have been posted before we standardized on the representation of wedgies. E.g. Odin, given as

[6, -12, 18, 12, -24, 12, -8, 17, -10, 2]

should be:

<<<6, 12, 18, 12, 24, 12, 8, 17, 10, 2]]]

> For the 7-limit that doesn't matter because you're only > looking at one comma and that uniquely defines the > temperament class. For the ragisma, then, I get this as an > optimal tuning map:
> > <1200.00181343 1901.98688957 2786.27460651 3368.81161439]

That agrees with what I get for TOP-RMS. The TOP-MAX is:

<1200.016360, 1901.980932, 2786.275726, 3368.779977]

Graham wrote:

> Also, I note
> that 12-equal doesn't temper out the ragisma so a 12 note
> scale isn't an obvious starting point either.

The number 12 came up as a result of 7 (which is the number of used 1/3s) + 4 (the number of used 5/1s) + 1 (the added 2/1). It's a bit like if you make a linear temperament by breaking the fraction into individual prime multiples, which often doesn't make MOS. For example, the prime coordinates for the comma of 78732/78125 are "-2 -9 7" which can be easily split into "-9 -9 0" and "7 0 7". This means that you can also make semisixth by stacking nine 1/6s and seven 10/1s and tempering each of these by 1/16 of the comma, which eventually makes you arrive at 1/1 again. What you get then is actually 16 equal divisions of 60/1 but with only 15 repetitions instead of 16. When you reduce this to a single octave range, you get exactly the scale I used in my recent piece -- 16-tone semisixth chain, which obviously doesn't have MOS but allows me to show the "comma pump", which is what I needed.

Petr

Herman wrote:

> All temperaments with more than one generator can have different choices
> of generators, but this is especially noticeable starting with rank 3
> because of the convention of using the octave as one of the generators.
> In rank 2 temperaments, if one generator is an octave or equal division
> of the octave, there are two possible sizes for the other generator that
> are smaller than the first one. E.g. meantone with a fourth or a fifth
> as a generator. With a rank 3 temperament, you have more options for
> generator sizes.

Okay, so if I decide to temper out 126/125 in such a way that the larger generator is a tempered 4/5 and the smaller one is a tempered 6/5 and that LSS makes 8/7, is it still starling even though the generators now are completely different?

> Yes, this looks like a planar temperament. The reason you can't find it
> on Gene's list is that it's a list of 11-limit temperaments. 7-limit
> rank 3 temperaments are most easily identified by the comma which is
> tempered out, since there are so many possibilities for generator
> mappings. So you could call this the [-1, -7, 4, 1> temperament.

Aha, that's interesting. I always thought that, for example, marvel was the one which tempers out 225/224. But this seems as if marvel was to temper out this one together with 385/384. So I'm not sure if this is an 11-limit version of marvel or if the 7-limit temperament does not count as "marvel". If the latter is true, then some of the temperaments I've played in don't have names as these are only 7-limit ones, some of which could offer good musical possibilities. For example, the one I've been privately calling "anti-orwell" has generators of a tempered 5/4 and a tempered 35/32 and tempers out 6144/6125 in such a way that SSL makes 3/2.

Petr

Petr Pa��zek wrote:

> Okay, so if I decide to temper out 126/125 in such a way that the larger > generator is a tempered 4/5 and the smaller one is a tempered 6/5 and that > LSS makes 8/7, is it still starling even though the generators now are > completely different?

The short answer is yes. A longer answer is that I can take those generators and the 126/125, do some linear algebra on them, and come up with the sequence of numbers 1, 3, 2, -1. This matches the unique "wedgie" <<<1, 3, 2, -1]]] which identifies it as starling temperament. Another way to look at this is that since the generators of a rank 3 temperament are linearly independent, you can add or subtract one from the other to find a new set of generators that define the same temperament.

>> Yes, this looks like a planar temperament. The reason you can't find it
>> on Gene's list is that it's a list of 11-limit temperaments. 7-limit
>> rank 3 temperaments are most easily identified by the comma which is
>> tempered out, since there are so many possibilities for generator
>> mappings. So you could call this the [-1, -7, 4, 1> temperament.
> > Aha, that's interesting. I always thought that, for example, marvel was the > one which tempers out 225/224. But this seems as if marvel was to temper out > this one together with 385/384. So I'm not sure if this is an 11-limit > version of marvel or if the 7-limit temperament does not count as "marvel". > If the latter is true, then some of the temperaments I've played in don't > have names as these are only 7-limit ones, some of which could offer good > musical possibilities. For example, the one I've been privately calling > "anti-orwell" has generators of a tempered 5/4 and a tempered 35/32 and > tempers out 6144/6125 in such a way that SSL makes 3/2.
> > Petr

I'm not sure of the history of "marvel", whether it was first used for a 7-limit or an 11-limit temperament. I did a search on my archived email for 6144/6125 and found "A 7-limit planar temperaments list" from 7/23/2002, but none of these have names. My guess is that few if any of the temperaments on the list have been evaluated for musical uses, due to the practical issues of working with temperaments having more than 2 generators.

If you do the algebra with 2/1, 5/4, and 35/32 as generators, and 6144/6125 as a unison vector, the wedgie comes out as <<<2, -3, -1, 11]]]. And you get this for a generator mapping:

[<1, 1, 2, 3], <0, 1, 1, -1], <0, 2, 0, 1]>
TOP-RMS: 1199.710244, 387.319274, 157.429655

Herman wrote:
> I'm not sure of the history of "marvel", whether it was first
> used for a 7-limit or an 11-limit temperament.

It was originally used to mean the 225/224 7-limit planar
temperament. -Carl