Mohajira sequences

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:

> > (Jacques) :
> > But one thing I should say about 13 and a "Mohajira sequence" :
> > 13, it's true, is perhaps the prime that's approached the worse way, in a "Mohajira temperament" (by -1 generator, but as you said, with some approximation). But on the other hand, it is a necessary basic ingredient to start a differentially-coherent sequence such as :
> > [18 22 26 32 39 48 59 72 88 >>...
> > where x^5 - x^4 = 1/2, that will much later on converge to closer and >closer semififths.

> I don't know how to calculate this sequence further- what are the figures when taken out to 17 tones?
>
> -Cameron Bobro

If you write it x^5 = x^4 + 1/2 it gives the next term of the series (x^5) ;
so next one will be : 88 + 39/2 = 215/2, etc., and because of "1/2",
in order to keep it in whole numbers you need some 2^n provisions.

Also because this series repeats 2 octaves higher the same terms 18, 22,
(=72, 88), which is cool because it adds more -c properties, it is like if the series was starting from 13 (hence my answer to Gene) !
You can certainly have a 17 tones scale, but 17 semififths here will give 6459, 52 cents under 13*2^9 = 6656.
24 semififths will cycle better with 52951, 9.6833 cents under 13*2^12 = 53248

This series ("Mogol"), octaves reduced is :
13
1
39
3
59
9
11
215
263
161 = 7*23
197
241
1179
721
441 = 3^2 * 7^2
1079
165 = 3*5*11
6459
7901
9665
11823
14663
35385
21643

Another series ("Osir") would start from 9, to start with a 11-limit scale :
9
11
27
33
81
99
121
37
181
443
271
663
811
31
2427
2969
227
4443
5435
13297
8133
9949
24341
1861

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

Thanks Gene and Jacques for the answers, very clear.

I take it semififths refer only to divisions of pure or smaller (meantone) fifths? Because for example a chain of 38/31s or 27/22s will give you MOSs near to 17 equal but that does not count as a Mohajira temperament in either interpretation, does it?

-Cameron Bobro

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
>
> > > (Jacques) :
> > > But one thing I should say about 13 and a "Mohajira sequence" :
> > > 13, it's true, is perhaps the prime that's approached the worse
> way, in a "Mohajira temperament" (by -1 generator, but as you said,
> with some approximation). But on the other hand, it is a necessary
> basic ingredient to start a differentially-coherent sequence such as :
> > > [18 22 26 32 39 48 59 72 88 >>...
> > > where x^5 - x^4 = 1/2, that will much later on converge to
> closer and >closer semififths.
>
> > I don't know how to calculate this sequence further- what are the
> figures when taken out to 17 tones?
> >
> > -Cameron Bobro
>
> If you write it x^5 = x^4 + 1/2 it gives the next term of the series
> (x^5) ;
> so next one will be : 88 + 39/2 = 215/2, etc., and because of "1/2",
> in order to keep it in whole numbers you need some 2^n provisions.
>
> Also because this series repeats 2 octaves higher the same terms 18, 22,
> (=72, 88), which is cool because it adds more -c properties, it is
> like if the series was starting from 13 (hence my answer to Gene) !
> You can certainly have a 17 tones scale, but 17 semififths here will
> give 6459, 52 cents under 13*2^9 = 6656.
> 24 semififths will cycle better with 52951, 9.6833 cents under
> 13*2^12 = 53248
>
> This series ("Mogol"), octaves reduced is :
> 13
> 1
> 39
> 3
> 59
> 9
> 11
> 215
> 263
> 161 = 7*23
> 197
> 241
> 1179
> 721
> 441 = 3^2 * 7^2
> 1079
> 165 = 3*5*11
> 6459
> 7901
> 9665
> 11823
> 14663
> 35385
> 21643
>
> Another series ("Osir") would start from 9, to start with a 11-limit
> scale :
> 9
> 11
> 27
> 33
> 81
> 99
> 121
> 37
> 181
> 443
> 271
> 663
> 811
> 31
> 2427
> 2969
> 227
> 4443
> 5435
> 13297
> 8133
> 9949
> 24341
> 1861
>
> - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
> Jacques
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Thanks Gene and Jacques for the answers, very clear.
>
> I take it semififths refer only to divisions of pure or smaller (meantone) fifths? Because for example a chain of 38/31s or 27/22s will give you MOSs near to 17 equal but that does not count as a Mohajira temperament in either interpretation, does it?
>
> -Cameron Bobro

I would think that after a good performance with the 11th harmonic it would gain credibility with 13 but would loose it with 5, and 7, to find it back progressively with 7 again at -4 generators : that would be another temperament. Only for linear temperaments Graham mentions "Hemififths", then "Beatles". But I'm not sure if it comes from this good old Georges Harrison, I don't know the story here.

Considering recurrent sequences, in the meantime sqrt of 3/2 would be a VERY strong attractor after Mohajira ;
Then, Semi-Soria and other Hemi-superpyths would have some chances.

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Thanks Gene and Jacques for the answers, very clear.
> >
> > I take it semififths refer only to divisions of pure or smaller (meantone) fifths? Because for example a chain of 38/31s or 27/22s will give you MOSs near to 17 equal but that does not count as a Mohajira temperament in either interpretation, does it?
> >
> > -Cameron Bobro
>
> I would think that after a good performance with the 11th harmonic it would gain credibility with 13 but would loose it with 5, and 7, to find it back progressively with 7 again at -4 generators : that would be another temperament. Only for linear temperaments Graham mentions "Hemififths", then "Beatles". But I'm not sure if it comes from this good old Georges Harrison, I don't know the story here.
>
> Considering recurrent sequences, in the meantime sqrt of 3/2 would be a VERY strong attractor after Mohajira ;
> Then, Semi-Soria and other Hemi-superpyths would have some chances.
>

The MOS tuning from 17 27/22s sounds quite good, but functions quite lopsided, more than it should considering how close it is to an equal division, and I guess that might be due to irregular emphases in the spectrum.

I'll be trying out these sequences, the one from 13 looks great.

-Cameron Bobro

On 18 May 2010 15:03, jacques.dudon <fotosonix@...> wrote:

> I would think that after a good performance with the 11th harmonic
> it would gain credibility with 13 but would loose it with 5, and 7, to
> find it back progressively with 7 again at -4 generators : that would
> be another temperament. Only for linear temperaments Graham
> mentions "Hemififths", then "Beatles". But I'm not sure if it comes
> from this good old Georges Harrison, I don't know the story here.

I have these names in the software, it doesn't mean I'm familiar with
them. But, yes, Beatles (17&10) is named because it has a 19/64
generator, which reminds us that John, Paul, George, and Ringo became
famous with Love Me Do in 1963. Until this thread I didn't realize
that it covers the approximate 7:6 that rounds off the 7 note neutral
thirds scale in 17-equal. And it (Beatles) is quite a good
2.3.7.11-limit temperament class. So I'll look at back-porting some
names to the 2.3.7-limit. Beatles, Pajara (22&12), and a hemifourths
type thing are all contorted versions of Dominant (12&5 or something,
8:7 equivalent to 9:8) in 2.3.7.

The whole neutral-thirds family is here as "mosh":

http://xenharmonic.wikispaces.com/MOSNamingScheme

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> But, yes, Beatles (17&10) is named because it has a 19/64
> generator, which reminds us that John, Paul, George, and Ringo became
> famous with Love Me Do in 1963.

No, it commemorates the fact that Beatles mania descended on the US with sudden and overwhelming force after their first appearance on the Ed Sullivan Show on February 9, 1964.

http://www.time.com/time/80days/640209.html

Here's a list of MOS in the range 11-29 notes to the octave I compiled, along with the name of a rank 2 temperament whose tunings fall into the range defined by the MOS.

Keemun[11] [4L 7s]
Sensi[11] [8L 3s]

Meantone[12] [7L 5s]
Superpyth[12] [5L 7s]
Pajara[12] [10L 2s]
Injera/Doublewide[12] [2L 10s]
Augene[12] [3L 9s]

Godzilla[14] [5L 9s]

Porcupine[15] [7L 8s]
Myna[15] [4L 11s]
Superkleismic[15] [11L 4s]
Valentine[15] [1L 14s]

Mothra[16] [5L 11s]
Wizard[16] [6L 10s]

Garibaldi[17] [12L 5s]
Mohajira[17] [7L 10s]
Beatles[17] [10L 7s]
Squares[17] [14L 3s]

Meantone[19] [12L 7s]
Flattone[19] [7L 12s]
Magic[19] [3L 16s]
Muggles[19] [16L 3s]
Myna[19] [4L 15s]
Sensi[19] [8L 11s]
Negri[19] [10L 9s]
Tritonic[19] [2L 17s]
Hemiwuerschmidt[19] [6L 13s]

Roman[20] [3L 17s]

Miracle[21] [10L 11s]
Tritonic[21] [2L 19s]
Rodan[21] [5L 16s]

Magic[22] [19L 3s]
Orwell[22] [9L 13s]
Wizard[22] [6L 16s]
Diaschismic[22] [12L 10s]
Shrutar[22] [2L 20s]
Coendou[22] [7L 15s]

Hemikleismic[23] [15L 8s]
Unidec[23] [13L 10s]
Roman[23] [3L 20s]

Mohajira[24] [7L 17s]

Hemithirds[25] [6L 19s]

Ennealimmal[27] [18L 9s]
Octacot[27] [14L 13s]

Leapday[29] [17L 12s]
Garibaldi[29] [12L 17s]

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Injera/Doublewide[12] [2L 10s]

Sorry, should be

Injera[12] [2L 10s]

Injera[14] [12L 2s]
Doublewide[14] [4L 10s]