Yahoo Tuning Groups Ultimate Backup tuning Layout for 32-tones on the Terpstra (Bosanquet)?

Hi Bogdan,

I never played on an actual Terpstra keyboard, so I can't speak from experience, but this is what I think would be a good approach:

If you're just looking for a good pattern for 32-tone tunings, where the specific tuning doesn't matter, then I would start off with a 32-note MOS and use that as a basis.

The way the Terpstra keyboard layout you're refering to is tilted means that only patterns based on 5L2s or 2L5s divisions will "work fine" octave-wise, otherwise notes an octave apart won't be at the same level. However, this is just a design aspect that facilitates working with patterns related to the diatonic (or antidiatonic) scale. If octave-equivalent notes aren't at the same level, simply rotate the keyboard until they are, and everything is fine. ;)

But let's say you want to keep the keyboard's orientation, and you're therefore only interested in an MOS that is, or contains an MOS with a 5L2s or 2L5s pattern (or repetitions thereof). Then there are two options:

1) A 32-note MOS with a generator between ~711.1 and 720 cents (16\27 and 3\5), e.g. Superpyth[32]
Possible layout: http://abload.de/img/terpstralayout-superpi7d2u.png http://abload.de/img/terpstralayout-superpi7d2u.png
Both blocks in the diagram cover a whole octave range (well, at least after moving the -5 and -4, or the 29, 30 and 31 tiles up or down an octave). The darker tiles mark the "diatonic" MOS that is contained in the larger 32-note "chromatic" MOS. Step information in the green block refers to the small (s) and large (L) steps of the "diatonic" MOS. The numbers in the blue block indicate the step distance to a possible root note "0".
The chroma for the 32-note MOS is |6s - L|, which also gives you the amount of steps after which a block almost identically repeats, only transposed by said chroma - though you can choose to make those exact repetitions. ;)

2) A 32-note MOS with a generator between ~257.1 and ~266.7cents (3\14 and 2\9), and a half-octave period, e.g. Bamity[32]
Possible layout: http://abload.de/img/terpstralayout-bamity1jeha.png http://abload.de/img/terpstralayout-bamity1jeha.png
Both blocks cover a half-octave range, and are apart by that interval. The dark tiles from both blocks concatenated form some kind of "double-antidiatonic scale" sLsssLssLsssLs. The numbers refer to the lower half-octave; for the higher half-octave, just add 16. The chroma for the 32-note MOS is |3s - 2L|.

There are many more possible layouts if we drop the above requirement, and allow for arbitrary MOS patterns as a basis. And there are even more options, like embedding the tuning in a larger MOS, e.g. using a 32-note subset of a 34- or 41-note MOS, and ignoring the surplus keys that aren't part of the tuning. Which layout is best depends on the tuning, how the pattern should be spread horizontally / vertically, and whether octave-equivalent notes should be at the same level without having to rotate the keyboard.

- Gedankenwelt

🔗baros_ilogic@...

Hi Gedankenwelt,

No one played the Terpstra - except maybe Garnet who has one in his studio, and Mike who played the other one once to shoot a video. All of us backers are waiting for the Second Generation Terpstras to be built; in the meantime we prepare for when they come. Since my curiosity towards layout mappings.

Thanks for the thorough explanation, it makes a lot of sense! The first option is a nice arrangement and I've figured out that, through the design of the keybed, the only way to have "octaves" at the same level is to go 5 steps pseudo-right and 2 steps pseudo-down.

From this the complete layout can be generated. Every key is surrounded by 6 others, so there are 3 different step-directions and their "inverses". But once 2 of them are known, the 3rd is automatically generated. So all the work consists in finding a solution of integers to the equation 5*x + 2*y = N-tone tuning.

As I was playing around with these figures, trying to come up with a simple way to calculate all possibilities (in a spreadsheet maybe) I realized there has to be one extra condition needing to be fulfilled, in order for the pattern to work: x and y must never be both even or odd.

For my 32-tone example, the best layout having "octaves" at the same level seems to be your first example: 5*6 + 2*1. The next one doesn't work, because it only maps even tones (5*4 + 2*6). The last one that works is 5*2 + 2*11, but this pattern is rather strange as consecutive tones are 3 pseudo-rows apart, and fingerings are inconvenient.

Now I'm looking for other layouts that don't have to satisfy the "octave" requirement - unless there is more to it than the above. I'm thinking about studying the many layouts possible by changing the tuning but not the number of tones: regular and arbitrary MOSes, ETs, harmonic series, etc.

All the best,
Bogdan

---In TUNING@yahoogroups.com, <gedankenwelt94@...> wrote :

Hi Bogdan,

I never played on an actual Terpstra keyboard, so I can't speak from experience, but this is what I think would be a good approach:

If you're just looking for a good pattern for 32-tone tunings, where the specific tuning doesn't matter, then I would start off with a 32-note MOS and use that as a basis.

- Gedankenwelt

🔗gedankenwelt94@...

Hi,

Bogdan wrote:
> From this the complete layout can be generated. Every key is surrounded by 6 others,
> so there are 3 different step-directions and their "inverses". But once 2 of them are known,
> the 3rd is automatically generated. So all the work consists in finding a solution of integers
> to the equation 5*x + 2*y = N-tone tuning.

> As I was playing around with these figures, trying to come up with a simple way to calculate
> all possibilities (in a spreadsheet maybe) I realized there has to be one extra condition
> needing to be fulfilled, in order for the pattern to work: x and y must never be both even or odd.

Interesting, I haven't thought of this simple approach. You're correct that both x and y being
even is an important special case. If x and y are both odd, you won't find solutions in this case,
but it's not generally a problem; a counter example is 5*5 + 2*3 = 31, which describes the
meantone diatonic scale as a subscale of either a 31-note meantone MOS, or 31-EDO.

The important thing is that d = gcd(x, y) divides N. So if d > 1, the whole equation 5*x + 2*y = N
can be divided by d, which leads to solutions for N/d-note tunings, which isn't desired,
since we're interested in solutions for an N-note tuning. We can still "fix" this and get
a usable solution by dividing x and y by d, and chaining the pattern d times together,
meaning we get an MOS which has d periods per octave.

For example, the solution 5*4 + 2*6 = 32 can be viewed as 2*(5*2 + 2*3) = 32, which means the
resulting pattern consists of two concatenated antidiatonic patterns: 2x sLsssLs = sLsssLssLsssLs,
with s = 2 and L = 3.
This is exactly my previous second example, here again the picture:
http://abload.de/img/terpstralayout-bamity1jeha.png http://abload.de/img/terpstralayout-bamity1jeha.png
http://abload.de/img/terpstralayout-bamity1jeha.png
http://abload.de/img/terpstralayout-bamity1jeha... http://abload.de/img/terpstralayout-bamity1jeha.png

View on abload.de http://abload.de/img/terpstralayout-bamity1jeha.png
Preview by Yahoo

It should be clearer if you imagine the numbers 0 to 15 were in the green block, and increase the numbers
in the blue block by 16, s.th. they range from 16 to 31, and both blocks together cover an octave.

> The last one that works is 5*2 + 2*11, but this pattern is rather strange as consecutive tones are
> 3 pseudo-rows apart, and fingerings are inconvenient.
Hm, seems I overlooked that one. Yeah, that's a rather extreme MOS, with a tritone generator of ~15\32.

If you're interested, there's another method to find the MOSes. Take a look at the following section of the
Stern-Brocot tree (aka Wilson's Scale Tree):
http://abload.de/img/sternbrocotuis1n.png http://abload.de/img/sternbrocotuis1n.png
http://abload.de/img/sternbrocotuis1n.png
http://abload.de/img/sternbrocotuis1n.png http://abload.de/img/sternbrocotuis1n.png

View on abload.de http://abload.de/img/sternbrocotuis1n.png
Preview by Yahoo

If we start at the 3/7 node (purple rectangle), we can go downwards to find generators for a diatonic scale
LsLLLsL, or upwards for generators of an antidiatonic scale sLsssLs, if we interpret the ratios as octave fractions.
All we need to do is to look there for generators with a denominator of 32, which will generate a diatonic or
antidiatonic scale, and 32-EDO. At the bottom there's 13/32, and at the top there's 15/32 (both green) - or,
13\32 and 15\32, if we write them as octave fractions. 13\32 corresponds to my first example, and generates
a diatonic scale, while 15\32 corresponds to the third example which I overlooked, and generates an antidiatonic
scale. If we slightly modify those generators, the 32-note scales won't be EDOs, but MOSes.

To find the MOSes with more than one period per octave, we also have to consider the divisors of 32 which are
greater than 7, i.e. we have to look for nodes with a denominator of 16 or 8.
The only one is 7\16, which generates an antidiatonic scale. But since we're interested in a 32-note MOS,
we'll have to make that 7\32, and apply a period of 1\2 (= a half octave), which corresponds to my second example.

If you're looking for all possible layouts based on MOSes, you'll have to locate all ratios starting from the 1/3 node
which have a denominator of 32, 16, 8 and 4, each of which will have an odd numerator. Then, each node on the
path represents a smaller MOS, and therefore an own layout (yep, that's a lot).

Example:
If we have a generator of ~13\32, the nodes on the path are 1/2, 1/3, 2/5, 3/7, 5/12, 7/17, 9/22, and 11/27.
The denominator tells us the number of notes the associated MOS has, and the predecessor's denominator
tells us the number of either the large or small steps it has.
E.g. the MOS associated with 5/12 has 12 notes. Since its predecessor 3/7 has a denominator of 7, the
12-note MOS is either a 7L5s or a 5L7s MOS (here, it's the latter).

Do you have specific criteria for what a good layout would be? For example, maybe the number of notes and the
L/s ratio for the smaller MOS shouldn't be too small, or too large.
Also, ideally the larger MOS would correspond to a scale that is a tempered version of the original tuning, meaning
no interval size in the original tuning corresponds to different intervals in the MOS. If an MOS doesn't work,
it's possible to use a MODMOS instead.

Best
- Gedankenwelt

🔗baros_ilogic@...

Hi Gedankenwelt,

I suspected that 5*4 + 2*6 = 32 which can be viewed as 2*(5*2 + 2*3) = 32 could be mapped as in your example of "two concatenated antidiatonic patterns". And it seems like there's no fixed rule as for x and y being both odd or even - unless I'm missing out on something.

As another similar example, Siemen Terpstra's design specifications state that 24-et has two "circles of fifths", and mapping it splits the keyboard into two sections - something I never quite understood. Is this done just like in your above example?

What do you mean by "interpreting the ratios as octave fractions"; has this anything to do with "interval of equivalence"?

> If you're looking for all possible layouts based on MOSes, you'll have to locate all ratios starting from the 1/3 node
Why the 1/3 node?

And generally about MOSes: I never encountered any account of ratios from the 2/1 half. Why is just the 1/2 half of the Scale Tree used as ratio generators?

Everything else is clear (except maybe the MODMOS part), though it took me a while to get it! Mapping MOSes is a lot of fun once you are familiar with the concept. One of the options left unexplored for 32 is harmonic series, and I'm not sure whether there can be any MOS through which to map such a layout, once specific requirements are in place.

For example, it would be convenient to have comfortable finger positions for chords on small number ratios like primarily every 4th tone, and secondarily every 8th, leaving all others to self-adjust around these. Of course to achieve that, "octave" on the same level would no longer be a requirement.

All the best,
Bogdan

PS How did you generate that version of the Scale Tree?

🔗gedankenwelt94@...

Hi Bogdan,

I wanted to write a detailed answer, but didn't manage to find the time for that (I'm midway through moving), so here's at least a "short" answer:

> As another similar example, Siemen Terpstra's design specifications state that 24-et has two "circles of fifths", and mapping it splits the keyboard into two sections - something I never quite understood. Is this done just like in your above example?

A way to look at this would be mohajira temperament, which has a neutral third generator that splits the fifth in two. If you stack 24 generators, you get two cycles of fifths which are a neutral third apart.

Another way would be godzilla temperament, which has a generator that splits the fourth in two "semifourths" of approximately 250 cents. Here again, stacking 24 generators gives two cycles of fifths (or fourths), this time a "semifourth" apart.

However, if two chromatic scales that are a quartertone interval apart are mapped to the terpstra keyboard via standard layout, the coordinate system isn't consistent, and moving in a specific axis direction can correspond to different step sizes. This isn't necessarily bad, but it's different from my examples, which are all based on a consistent coordinate system.

A third way to look at it may be catler temperament, but it's clearly based on 12-EDO, not on a chromatic scale generated by stacking fifths, so this may be a stretch.

> What do you mean by "interpreting the ratios as octave fractions"; has this anything to do with "interval of equivalence"?

Yes. In the case of MOSes, there's always an interval of equivalence (IoE), and I assumed that it's the octave.
(though it's no problem to use something else, like the tritave)
Often, period and IoE are the same for an MOS. However, sometimes the period divides the IoE, in which case we'd interpret the ratios as fractions of the period (which is a fraction of the IoE), instead.

> Why the 1/3 node?
> And generally about MOSes: I never encountered any account of ratios from the 2/1 half. Why is just the 1/2 half of the Scale Tree used as ratio generators?

MOSes are strongly related to equal temperaments, and equal tempered intervals can be expressed as octave fractions, like 7\12 for the perfect fifth in 12-EDO.
When applying a period (which we associate with 1/1), we can reduce the generators s.th. they fall between 0/1 and 1/1. We can always take the smaller of the two generators, which falls between 0/1 and 1/2. This is the range that we get when we start at the 1/3 node.

Finding an intuitive layout for a subset of the harmonic series may be difficult, since it is kind of an opposite to MOSes: All steps are different in size, and they are sorted by size, instead of distributed evenly.
Some good EDOs to start may be 46- or 87-EDO (good in 13-limit), 72-EDO (good in no-13s 19-limit), or 94-EDO (good as a 23-limit tuning), though that's still far from the 32-note 61-limit scale between the 32nd and 63rd harmonic, if that's what you're looking for.

> PS How did you generate that version of the Scale Tree?
That's a scale tree implemented as a foldable "mind map", that can be opened with FreeMind or similar programs, which offers a very useful search function. I generated it with a C++ program. I'll see that I upload it in the tuning-math file section, when I find the time.

Best
- Gedankenwelt