A well-temperament with constant maj.third 0f 400 and min.sixth of 800

Hi all

With 3 divisions as:

A=105.000
B=98.477
C=98.045

And Making an intervallic pattern as : BABC BABC BABC which has 3 repeating blocks ,
You see a well-temperament with constant maj.thirds of 400 c and min.sixths of 800 c in interval matrix . 3 Fifth sizes are :

701.955
701.5225
695.

And scale is :

0.
98.4775
203.4775
301.955
400.
498.4775
603.4775
701.955
800.
898.4775
1003.4775
1101.955
1200.

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web siteوب سايت شاهين مهاجري <http://240edo.tripod.com/>

My farsi page in Harmonytalk صفحه اختصاصي در هارموني تاك <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia شاهين مهاجري دردائره المعارف ويكي پديا <http://en.wikipedia.org/wiki/Shaahin_mohajeri>

Hi Shahin,

--- In tuning@yahoogroups.com, "Mohajeri Shahin"wrote:
> With 3 divisions as:
>
> A=105.000
> B=98.477
> C=98.045
>
> And Making an intervallic pattern as : BABC BABC BABC which has 3
repeating blocks ,
> You see a well-temperament with constant maj.thirds of 400 c and
min.sixths of 800 c in interval matrix . 3 Fifth sizes are :
>
> 701.955
> 701.5225
> 695.
>
> And scale is :
>
> 0.
> 98.4775
> 203.4775
> 301.955
> 400.
> 498.4775
> 603.4775
> 701.955
> 800.
> 898.4775
> 1003.4775
> 1101.955
> 1200.
>
> Shaahin Mohajeri

Since your B and C intervals are very, very close, what
happens if we choose an average of them, weighted by
their occurrences? The pattern BABC has twice as many
Bs as Cs, so the required average is
X = (2B + C)/3
= (2*98.4775 + 98.0450)/3
= 295/3
= 98.3333

The intervallic pattern BABC becomes XAXX, or:
0 98.3333 203.3333 301.6667 400
so the scale is now:
0
98.3333
203.3333
301.6667
400
498.3333
603.3333
701.6667
800
898.3333
1003.3333
1101.6667
1200

The fifths have sizes:
3 small: 695 (D to A, F# to C#, Bb to F) and
9 big : 701.6667 (the rest).

Your major thirds are still exactly one-third of
an octave, 400 cents. And your minor sixths and
augmented fifths are still exactly two-thirds of
an octave, 800 cents.

Eight notes of this scale differ from yours by
0.1442 cents and the remaining four notes by
0.2883 cents - quite possibly an inaudible
difference.

E&OE!

Regards,
Yahya

Hi yaha
yes , it is the same as you consider X=(1200-(3*A))/9.
in /tuning/topicId_67904.html#67904 </tuning/topicId_67904.html#67904> from wendy carlos , we see these divisions:

A=16/15
B=135/128
C=256/243

3 divisions are based on 270-EDL and resulted from 270/256(135/128),270/243(10/9),270/240(9/8). scale has an intervallic pattern as : BCABABACBABA

1: 135/128 92.179 major chroma, major limma
2: 256/243 90.225 limma, Pythagorean minor second
3: 16/15 111.731 minor diatonic semitone
4: 135/128 92.179 major chroma, major limma
5: 16/15 111.731 minor diatonic semitone
6: 135/128 92.179 major chroma, major limma
7: 16/15 111.731 minor diatonic semitone
8: 256/243 90.225 limma, Pythagorean minor second
9: 135/128 92.179 major chroma, major limma
10: 16/15 111.731 minor diatonic semitone
11: 135/128 92.179 major chroma, major limma
12: 16/15 111.731 minor diatonic semitone

now by considering (270/243)^(1/2),we have B=91.2018560670301 and
A=(1200-(7*B))/5= 112.317401506158
so our scale will be BBABABABBABA:
0
91.20185607
182.4037121
294.7211136
385.9229697
498.2403712
589.4422273
701.7596288
792.9614849
884.1633409
996.4807424
1087.682598
1200.
with max. and min. errors around 0.97 and -0.78 cent.
now consider 16/15 as B(111.73128 cent) and A=(1200-(5*))/7=91.62051052 cent and the scale with the same pattern is :
0.
91.62051052
183.241021
294.9723063
386.5928168
498.3241021
589.9446126
701.6758979
793.2964084
884.9169189
996.6482042
1088.268715
1200.
with max. and min. errors around 0.558 and -1.11 cent.
i believe this method is an equal temperament of two very near divisions.

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web site?? ???? ????? ?????? <http://240edo.googlepages.com/>

My farsi page in Harmonytalk ???? ??????? ?? ??????? ??? <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia ????? ?????? ??????? ??????? ???? ???? <http://en.wikipedia.org/wiki/Shaahin_mohajeri>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of yahya_melb
Sent: Thursday, December 28, 2006 6:27 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: A well-temperament with constant maj.third 0f 400 and min.sixth of 800

Hi Shahin,

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , "Mohajeri Shahin"wrote:
> With 3 divisions as:
>
> A=105.000
> B=98.477
> C=98.045
>
> And Making an intervallic pattern as : BABC BABC BABC which has 3
repeating blocks ,
> You see a well-temperament with constant maj.thirds of 400 c and
min.sixths of 800 c in interval matrix . 3 Fifth sizes are :
>
> 701.955
> 701.5225
> 695.
>
> And scale is :
>
> 0.
> 98.4775
> 203.4775
> 301.955
> 400.
> 498.4775
> 603.4775
> 701.955
> 800.
> 898.4775
> 1003.4775
> 1101.955
> 1200.
>
> Shaahin Mohajeri

Since your B and C intervals are very, very close, what
happens if we choose an average of them, weighted by
their occurrences? The pattern BABC has twice as many
Bs as Cs, so the required average is
X = (2B + C)/3
= (2*98.4775 + 98.0450)/3
= 295/3
= 98.3333

The intervallic pattern BABC becomes XAXX, or:
0 98.3333 203.3333 301.6667 400
so the scale is now:
0
98.3333
203.3333
301.6667
400
498.3333
603.3333
701.6667
800
898.3333
1003.3333
1101.6667
1200

The fifths have sizes:
3 small: 695 (D to A, F# to C#, Bb to F) and
9 big : 701.6667 (the rest).

Your major thirds are still exactly one-third of
an octave, 400 cents. And your minor sixths and
augmented fifths are still exactly two-thirds of
an octave, 800 cents.

Eight notes of this scale differ from yours by
0.1442 cents and the remaining four notes by
0.2883 cents - quite possibly an inaudible
difference.

E&OE!

Regards,
Yahya