Yahoo Tuning Groups Ultimate Backup tuning Understanding notation for regular mappings

When reading Graham Breeds text "The Regular Mapping Paradigm" (http://x31eq.com/paradigm.html). I am citing his text.

"In the regular mapping paradigm, miracle temperament is a rank 2 temperament with the mapping

[< 1, 1, 3, 3, 2],
< 0, 6, -7, -2, 15]>

...

You can also write the mapping for the meantone (and so typical do re mi) according to it's period (the octave) and generator (here a perfect fourth)

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

Unfortunately, I was unable to understand this notation from the text. But I assume one of you guys here could help -- or simply point me to some other text to read on this matter :)

Thank you!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

🔗Graham Breed <gbreed@...>

2008/8/2 Torsten Anders <torsten.anders@...>:
> Hi
>
> When reading Graham Breeds text "The Regular Mapping
> Paradigm" (http://x31eq.com/paradigm.html). I am citing his text.
>
> "In the regular mapping paradigm, miracle temperament is a rank 2
> temperament with the mapping
>
> [< 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]>

I can see I spent a lot of time explaining the rank 1 mappings and
then threw in a rank 2 mapping with no explanation. But it's two rank
1 mappings. The first one describes the period and the other one
describes the generator.

> You can also write the mapping for the meantone (and so typical do re
> mi) according to it's period (the octave) and generator (here a
> perfect fourth)
>
> [<1, 2, 4],
> <0,-1,-4]>"
>
> Unfortunately, I was unable to understand this notation from the
> text. But I assume one of you guys here could help -- or simply point
> me to some other text to read on this matter :)

I think this is still the standard introductory text, but if you can
work it out you can write something better.

How many fourths and octaves approximate a 3:1 in meantone? And a 5:1?

Graham

🔗Torsten Anders <torsten.anders@...>

Dear Graham,

Thanks for your reply.

On Aug 2, 2008, at 9:51 PM, Graham Breed wrote:

> 2008/8/2 Torsten Anders <torsten.anders@...>:
> > Hi
> >
> > When reading Graham Breeds text "The Regular Mapping
> > Paradigm" (http://x31eq.com/paradigm.html). I am citing his text.
> >
> > "In the regular mapping paradigm, miracle temperament is a rank 2
> > temperament with the mapping
> >
> > [< 1, 1, 3, 3, 2],
> > < 0, 6, -7, -2, 15]>
>
> I can see I spent a lot of time explaining the rank 1 mappings and
> then threw in a rank 2 mapping with no explanation. But it's two rank
> 1 mappings. The first one describes the period and the other one
> describes the generator.
>
So, each line is an interval specification (one stating the period and the other the generator)? How do I read them? For example, how does [<1, 2, 4] indicate and octave and <0,-1,-4] indicate a fourth?

Thank you!

Best
Torsten

> > You can also write the mapping for the meantone (and so typical > do re
> > mi) according to it's period (the octave) and generator (here a
> > perfect fourth)
> >
> > [<1, 2, 4],
> > <0,-1,-4]>"
> >
> > Unfortunately, I was unable to understand this notation from the
> > text. But I assume one of you guys here could help -- or simply > point
> > me to some other text to read on this matter :)
>
> I think this is still the standard introductory text, but if you can
> work it out you can write something better.
>
> How many fourths and octaves approximate a 3:1 in meantone? And a 5:1?
>
> Graham
>

🔗Herman Miller <hmiller@...>

Torsten Anders wrote:

> So, each line is an interval specification (one stating the period > and the other the generator)? How do I read them? For example, how > does [<1, 2, 4] indicate and octave and <0,-1,-4] indicate a fourth?

<1, 2, 4] indicates the number of octaves to reach an approximation of the prime intervals 2/1, 3/1, and 5/1 -- when combined with the number of fourths specified by <0, -1, -4].

2/1 is +1 octave, +0 fourths.
3/1 is +2 octaves, -1 fourth.
5/1 is +4 octaves, -4 fourths.

The first column is <1, 2, 4] (octaves) and the second column is <0, -1, -4] (fourths).

🔗Petr Parízek <p.parizek@...>

Herman wrote:

> The first column is <1, 2, 4] (octaves) and the second column is <0, -1,
> -4] (fourths).

Interestingly, at the times I made my first experiments with 2D tempering, which was long before I knew anything about 2D temperament webpages, somewhat intuitively I was writing all my mappings 90° off what Herman describes -- i.e. meantone is:
1 0
2 -1
4 -4

A few months later, I realized that the only person who seemed to be doing it the same way was Graham, namely his unison vector temperament finder: http://x31eq.com/temper/vectors.html

Petr

🔗Torsten Anders <torsten.anders@...>

On Aug 3, 2008, at 2:04 AM, Herman Miller wrote:
> Torsten Anders wrote:
> > So, each line is an interval specification (one stating the period
> > and the other the generator)? How do I read them? For example, how
> > does [<1, 2, 4] indicate an octave and <0,-1,-4] indicate a fourth?
>
> <1, 2, 4] indicates the number of octaves to reach an approximation of
> the prime intervals 2/1, 3/1, and 5/1 -- when combined with the number
> of fourths specified by <0, -1, -4].
>
> 2/1 is +1 octave, +0 fourths.
> 3/1 is +2 octaves, -1 fourth.
> 5/1 is +4 octaves, -4 fourths.
>
> The first column is <1, 2, 4] (octaves) and the second column is > <0, -1,
> -4] (fourths).
>
Ah, thanks for that explanation! So, these two interval representations depend on each other. Now the notion of two dimensions makes more sense for me.

Best
Torsten