CPS tunings (for dummies)

Hello list,

For a while I've been interested in using CPS tunings in composition. This goal hasn't been realised yet because I haven't made sense of the available writing on the topic. What little I do understand hasn't yet allowed me to use these pitch structures with the tools I have available. I'm hoping to get a discussion going about CPS tunings so that I can get some idea of how to apply them (and to ask some specific questions while at it).

Members of this list who have caught any of my previous posts might remember that I have very limited understanding of tuning theory and maths. Be aware that this is still the case. :)

--Working out the pitches--

I've been able to create hexanies, dekanies and eikosanies using the Scala microtuning software. The result, however, is a set of pitches in ascending order - this reveals nothing about the structure of the set and which pitches are harmonious with which. Personally I don't understand how I will figure this out.

To gain some understanding I recently (started with an easy one) created a hexany in Scala and tried to map the pitches to a diagram of an octahedron. I copied the shape from diagrams I found online (e.g. this from wikipedia http://en.wikipedia.org/wiki/File:Hexanyfacets.gif and some found at the Anaphorian Embassy) and tried to match the pitches from Scala with the diagram. I'm unsure if I did this correctly because the diagrams I have found describe the vertices like "3x7" and "1x5" for example.

What is the method by which "3x5" and the rest are converted into cents or ratios? I need to convert to either cents or ratios to more easily understand the geometry of the pitches (by drawing diagrams which I can be certain are *correct*) and to retune my synths using Scala.

--Scala Options--

When creating a new CPS in Scala there are (at least) two options whose implications I don't fully understand. Those are "delete 1/1 from result scale" and "normalize result by 2/1".

Is it absolutely necessary to delete the 1/1? How about if one simply avoids using 1/1, rather than delete it? My issue is that when you compare a scale where the 1/1 has been deleted with a scale where it hasn't, the other pitches change dramatically. Even the intervals between pitches are changing. What does it mean and why is this so?

Also, when you delete the 1/1 the resultant scale also has a 1/1, which is very confusing.

Is it absolutely necessary to normalise the result by 2/1? Personally I would like to map each pitch of the CPS to my keyboard once, with no octave equivalents. That is to say, for a hexany there will be only 6 pitches mapped, for an eikosany only 20. (My keyboard is an AXiS-49 from C-Thru Music, one of those hexagonal lattice MIDI controllers). In composition I would probably have every instrument use the same mapping but with certain instruments orchestrated to lower octaves (e.g. the bass 2 octaves lower). Am I right in thinking that normalising the result by 2/1 would be unhelpful in this case?

--Mapping to AXiS keyboard--

Last time I played around with the hexany, trying to understand it, I mapped the 6 notes to 6 keys on my keyboard. First I chose one hexagon-button, then I imagined the 6 hexagon-buttons around it were the vertices of the hexany's octahedron. I think this worked quite well (playing it for the first time was fun for a while), though I'm still unsure if each pitch was mapped to the correct vertex.

Are there any projections of the larger dekany and eikosany shapes which will map well to the AXiS-49 (or similar hexagon-keyboard)?

--Existing music--

What are your favourite musical works composed with CPS tunings?

Thanks folks, I'm interested to see how much fog can be removed from this topic! :)

Sean

--- In tuning@yahoogroups.com, "sevishmusic" <sevish@...> wrote:
> I've been able to create hexanies, dekanies and eikosanies using the Scala microtuning software. The result, however, is a set of pitches in ascending order - this reveals nothing about the structure of the set and which pitches are harmonious with which. Personally I don't understand how I will figure this out.

One simple way is just to play the scale and figure out - if it sounds harmonious, it is harmonious.

> What is the method by which "3x5" and the rest are converted into cents or ratios? I need to convert to either cents or ratios to more easily understand the geometry of the pitches (by drawing diagrams which I can be certain are *correct*) and to retune my synths using Scala.

3x5 = 15

15 itself is a ratio (same as 15/1).

If you want to octave-reduce it, you keep dividing by 2 until the result is smaller than 2. 15/2 > 2, 15/4 > 2... so 15/8 is the octave-reduced version.

As for cents, the ratio 15/1 in cents is log(15/1)/log(2) * 1200 = 4688.26871 cents

> When creating a new CPS in Scala there are (at least) two options whose implications I don't fully understand. Those are "delete 1/1 from result scale" and "normalize result by 2/1".
>
> Is it absolutely necessary to delete the 1/1? How about if one simply avoids using 1/1, rather than delete it? My issue is that when you compare a scale where the 1/1 has been deleted with a scale where it hasn't, the other pitches change dramatically. Even the intervals between pitches are changing. What does it mean and why is this so?

Of course it's not absolutely necessary to do anything. You can use any scale you want. It's just not really what most people mean when they say "hexany", "dekany", etc.

The intervals between the other pitches do not change when you delete 1/1. If you think they are, you're misunderstanding something.

> Also, when you delete the 1/1 the resultant scale also has a 1/1, which is very confusing.

It's because Scala automatically transposes everything.

Take a simple scale like {1/1, 5/4, 3/2, 7/4}.

If you delete 1/1 it becomes just {5/4, 3/2, 7/4}. But Scala can't represent it that way - it has to transpose it so that one of the remaining pitches becomes 1/1. So for example let's transpose it down by 5/4 so that the 5/4 becomes 1/1.

Then you have {1/1, 6/5, 7/5}. This is the same scale as {5/4, 3/2, 7/4}, just transposed. All the intervals between the three notes are still the same.

> Is it absolutely necessary to normalise the result by 2/1? Personally I would like to map each pitch of the CPS to my keyboard once, with no octave equivalents. That is to say, for a hexany there will be only 6 pitches mapped, for an eikosany only 20. (My keyboard is an AXiS-49 from C-Thru Music, one of those hexagonal lattice MIDI controllers). In composition I would probably have every instrument use the same mapping but with certain instruments orchestrated to lower octaves (e.g. the bass 2 octaves lower). Am I right in thinking that normalising the result by 2/1 would be unhelpful in this case?

I thought (maybe I'm wrong?) that CPSs were usually used as any other scales in an octave-equivalent context. But certainly you don't have to do that.

However, if you don't assume octave periodicity, then don't forget 2,4,etc. as possible factors for your CPSs! For example, you have the {2,3,4,5} hexany:

{2*3,2*4,2*5,3*4,3*5,4*5}
{6/1,8/1,10/1,12/1,15/1,20/1}
or transposed down by 6/1:
{1/1,4/3,5/3,2/1,5/2,20/3}

or the {3,4,5,6,7} dekany:

{3*4,3*5,3*6,3*7,4*5,4*6,4*7,5*6,5*7,6*7}
{12/1,15/1,18/1,21/1,20/1,24/1,28/1,30/1,35/1,42/1}
or transposing down by 12/1 and also swapping the two out-of-order pitches:
{1/1,5/4,3/2,5/3,7/4,2/1,7/3,5/2,35/12,7/2}

Keenan

BTW, I believe I've heard your music on last.fm. Sounded pretty sweet!

Keenan

Keenan, thanks for the ideas.

> 3x5 = 15
>
> 15 itself is a ratio (same as 15/1).

Thanks! I know that was so simple, but this point has never been made clear to me before now.

> > Is it absolutely necessary to delete the 1/1? How about if one simply avoids using 1/1, rather than delete it? My issue is that when you compare a scale where the 1/1 has been deleted with a scale where it hasn't, the other pitches change dramatically. Even the intervals between pitches are changing. What does it mean and why is this so?
>
> Of course it's not absolutely necessary to do anything. You can use any scale you want. It's just not really what most people mean when they say "hexany", "dekany", etc.
>
> The intervals between the other pitches do not change when you delete 1/1. If you think they are, you're misunderstanding something.

I certainly was misunderstanding something... As you rightly said, Scala automatically transposes everything and this hurt my little mind while trying to find patterns in the madness.

> > Is it absolutely necessary to normalise the result by 2/1? Personally I would like to map each pitch of the CPS to my keyboard once, with no octave equivalents. That is to say, for a hexany there will be only 6 pitches mapped, for an eikosany only 20. (My keyboard is an AXiS-49 from C-Thru Music, one of those hexagonal lattice MIDI controllers). In composition I would probably have every instrument use the same mapping but with certain instruments orchestrated to lower octaves (e.g. the bass 2 octaves lower). Am I right in thinking that normalising the result by 2/1 would be unhelpful in this case?
>
> I thought (maybe I'm wrong?) that CPSs were usually used as any other scales in an octave-equivalent context. But certainly you don't have to do that.

My first concern was not being able to fit more than one octave of a CPS onto my keyboard at any one time. The hexany at least is small enough that I can fit three of them on the AXiS-49 while keeping their ring shape. I've decided to octave-reduce after all, but this wasn't an obvious good idea at first.

> However, if you don't assume octave periodicity, then don't forget 2,4,etc. as possible factors for your CPSs! For example, you have the {2,3,4,5} hexany:
>
> {2*3,2*4,2*5,3*4,3*5,4*5}
> {6/1,8/1,10/1,12/1,15/1,20/1}
> or transposed down by 6/1:
> {1/1,4/3,5/3,2/1,5/2,20/3}
>
> or the {3,4,5,6,7} dekany:
>
> {3*4,3*5,3*6,3*7,4*5,4*6,4*7,5*6,5*7,6*7}
> {12/1,15/1,18/1,21/1,20/1,24/1,28/1,30/1,35/1,42/1}
> or transposing down by 12/1 and also swapping the two out-of-order pitches:
> {1/1,5/4,3/2,5/3,7/4,2/1,7/3,5/2,35/12,7/2}
>
> Keenan

Thanks again for all the tips Keenan! The first breakthrough was understanding how to turn say 3x5 into a ratio (and to know to remove the 1/1) - yes I know it was simple, but since then I was able to draw a bunch of this little diagrams for myself and work out the pitches without Scala. 7-9-11-15 is sounding the slickest of all. My project this week is to compose a tune with it. After that I'll dip my toes into a dekany!

Should any other AXiS owners want to try my mapping I've posted it below. Each dark blue (G#) key has 6 pitches around it. Try with any hexany.

Sean

! put 6 notes around dark blue keys on axis 49 or 64
! Size of map:
12
! First MIDI note number to retune:
0
! Last MIDI note number to retune:
127
! Middle note where scale degree 0 is mapped to:
59
! Reference note for which frequency is given:
60
! Frequency to tune the above note to (floating point e.g. 440.0):
256.000000
! Scale degree to consider as formal octave:
6
! Mapping.
0
2
4
x
1
3
5
x
x
x
x
x