Yahoo Tuning Groups Ultimate Backup tuning A useful tool to recognize consonance.

Dear friends,:

Until now the 624 elements of the Progression of cells were used to solve 2 equations with four variables and get three Piagui scales with tone frequencies near to 12edo and four ones with about 4 cents far from this scale. I don´t know if some member of the list applied the progression for other purposes. It has interesting properties like the coincidence between any cell´s fourth/fifth and a higher cell. The Progression was deposited by Steve Martin in his folder.

Due to its roots, all the progression elements are consonant figures, so if some body of the list work with consonances, he can copy the whole progression. Also, he can prepare a table to include both, the cells and their corresponding consonant ratio.
I checked the group of consonances given below and all of them were confirmed; the contrary case, when we want to know if any common fraction is a consonant ratio, the progression data gives the information.

THE EXTENDED SCALE OF PYTHAGORAS
NOTATION RELATIVE FREQUENCY

C 1 = 1

B# (81/80) (32805 / 32768) = 1.01364326477

Db (256 / 243) = 1.05349794239

C# (2187 / 2048) = 1.06787109375

D (9/8) = 1.125

Eb (32/27) = 1.185185185...

D# (19683 / 16384) = 1.20135498047

Fb (8192 / 6561) = 1.24859015394

E (81/64) = 1.265625

F (4/3) = 1.333333...

E# (177147 / 131072) = 1.351524353

Gb (1024 / 729) = 1.40466392318

F# (729 / 512) = 1.423828125

G (3/2) = 1.5

Ab (128 / 81) = 1.58024691358

G# (6561 / 4096) = 1.60180664063

A (27/16) = 1.6875

Bb (16/9) = 1.777777...

A# (59049 / 32768) = 1.8020324707

2Cb (4096 / 2187) = 1.8728852309

B (243 / 128) = 1.8984375

2C 2 = 2

THE EXTENDED SCALE OF ARISTOXENUS-ZARLINO

NOTATION RELATIVE FREQUENCY

C 1 = 1

C# (25/24) = 1.0416666...

Db (16/15) = 1.0666666...

D (9/8) = 1.125

D# (75/64) = 1.171875

Eb (6/5) = 1.2

E (5/4) = 1.25

Fb (32/25) = 1.28

E# (125 / 96) = 1.302083333...

F (4/3) = 1.333333...

F# (45/32) = 1.40625

Gb (36/25) = 1.44

G (3/2) = 1.5

G# (25/16) = 1.5625

Ab (8/5) = 1.6

A (5/3) = 1.666666...

A# (225 / 128) = 1.7578125

Bb (9/5) = 1.8

B (15/8) = 1.875

2Cb (48/25) = 1.92

B# (125 / 64) = 1.953125

2C 2 = 2

I hope this information is useful to you.

Mario

January, 5.

🔗Mike Battaglia <battaglia01@...>

Are 7/4 or 11/8 consonant in your scheme? I tend to enjoy those
interval, especially in the context of 4:5:6:7:9:11 chords.

-Mike

On Thu, Jan 5, 2012 at 9:47 PM, Mario Pizarro <piagui@ec-red.com> wrote:
> Dear friends,:
>
> Until now the 624 elements of the Progression of cells were used to solve 2
> equations with four variables and get three Piagui scales with tone
> frequencies near to 12edo and four ones with about 4 cents far from this
> scale. I don´t know if some member of the list applied the progression for
> other purposes. It has interesting properties like the coincidence
> between any cell´s fourth/fifth and a higher cell. The Progression was
> deposited by Steve Martin in his folder.
>
> Due to its roots, all the progression elements are consonant figures, so if
> some body of the list work with consonances, he can copy the whole
> progression. Also, he can prepare a table to include both, the cells and
> their corresponding consonant ratio.
> I checked the group of consonances given below and all of them were
> confirmed; the contrary case, when we want to know if any common fraction is
> a consonant ratio, the progression data gives the information.
>
> THE EXTENDED SCALE OF PYTHAGORAS
>
> NOTATION                          RELATIVE FREQUENCY
>
>        C                        1                                    = 1
>
>        B#                       (81/80) (32805 / 32768)     = 1.01364326477
>
>        Db                       (256 / 243)                       =
> 1.05349794239
>
>        C#                       (2187 / 2048)                   =
> 1.06787109375
>
>        D                        (9/8)                                = 1.125
>
>        Eb                       (32/27)                            =
> 1.185185185...
>
>        D#                       (19683 / 16384)                =
> 1.20135498047
>
>        Fb                       (8192 / 6561)                   =
> 1.24859015394
>
>        E                        (81/64)                            =
> 1.265625
>
>        F                        (4/3)                                =
> 1.333333...
>
>        E#                       (177147 / 131072)            = 1.351524353
>
>        Gb                      (1024 / 729)                     =
> 1.40466392318
>
>        F#                       (729 / 512)                       =
> 1.423828125
>
>        G                        (3/2)                                = 1.5
>
>        Ab                       (128 / 81)                        =
> 1.58024691358
>
>        G#                      (6561 / 4096)                   =
> 1.60180664063
>
>        A                        (27/16)                            = 1.6875
>
>        Bb                       (16/9)                              =
> 1.777777...
>
>        A#                       (59049 / 32768)                =
> 1.8020324707
>
>        2Cb                     (4096 / 2187)                   =
> 1.8728852309
>
>        B                        (243 / 128)                       =
> 1.8984375
>
>        2C                      2                                    = 2
>
>
>
> THE EXTENDED SCALE OF ARISTOXENUS–ZARLINO
>
> NOTATION                          RELATIVE FREQUENCY
>
>        C                        1                                    =
>         1
>
>        C#                       (25/24)                            =
> 1.0416666...
>
>        Db                       (16/15)                            =
> 1.0666666...
>
>        D                        (9/8)                                =
>         1.125
>
>        D#                       (75/64)                            =
> 1.171875
>
>        Eb                       (6/5)                                =
>         1.2
>
>        E                        (5/4)                                =
>         1.25
>
>        Fb                       (32/25)                            =
> 1.28
>
>        E#                       (125 / 96)                        =
> 1.302083333...
>
>        F                        (4/3)                                =
>         1.333333...
>
>        F#                       (45/32)                            =
> 1.40625
>
>        Gb                      (36/25)                            =
> 1.44
>
>        G                        (3/2)                                =
>         1.5
>
>        G#                      (25/16)                            =
> 1.5625
>
>        Ab                       (8/5)                                =
>         1.6
>
>        A                        (5/3)                                =
>         1.666666...
>
>        A#                       (225 / 128)                       =
> 1.7578125
>
>        Bb                       (9/5)                                =
>         1.8
>
>        B                        (15/8)                              =
>         1.875
>
>        2Cb                     (48/25)                            =
> 1.92
>
>        B#                       (125 / 64)                        =
> 1.953125
>
>        2C                      2                                    =
>         2
>
>
>
>    I hope this information is useful to you.
>
> Mario
>
> January, 5.

🔗Mario Pizarro <piagui@...>

To the tuning list:
Ratios 7/4 (= 1.75) and 11/8 (=1.175) are used by Mike. According to the progression of cells, both ratios are not consonant. Nearest consonances found in the progression are:

J 142 1.17452815104
M 143 1.17585436996
M 144 1.17718208638

M 493 1.74791410141
M 494 1.74988775930
J 495 1.75186753174

The numbers at the central column are the cells positions in the first set of 624 elements.

Mario

January, 6

<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
----- Original Message ----- From: "Mike Battaglia" <battaglia01@...>
To: <tuning@yahoogroups.com>
Sent: Thursday, January 05, 2012 11:08 PM
Subject: [tuning] Re: A useful tool to recognize consonance.

> Are 7/4 or 11/8 consonant in your scheme? I tend to enjoy those
> interval, especially in the context of 4:5:6:7:9:11 chords.
>
> -Mike
>
>
>
> On Thu, Jan 5, 2012 at 9:47 PM, Mario Pizarro <piagui@...> wrote:
>> Dear friends,:
>>
>> Until now the 624 elements of the Progression of cells were used to solve >> 2
>> equations with four variables and get three Piagui scales with tone
>> frequencies near to 12edo and four ones with about 4 cents far from this
>> scale. I donï¿½t know if some member of the list applied the progression >> for
>> other purposes. It has interesting properties like the coincidence
>> between any cellï¿½s fourth/fifth and a higher cell. The Progression was
>> deposited by Steve Martin in his folder.
>>
>> Due to its roots, all the progression elements are consonant figures, so >> if
>> some body of the list work with consonances, he can copy the whole
>> progression. Also, he can prepare a table to include both, the cells and
>> their corresponding consonant ratio.
>> I checked the group of consonances given below and all of them were
>> confirmed; the contrary case, when we want to know if any common fraction >> is
>> a consonant ratio, the progression data gives the information.
>>
>> THE EXTENDED SCALE OF PYTHAGORAS
>>
>> NOTATION RELATIVE FREQUENCY
>>
>> C 1 = 1
>>
>> B# (81/80) (32805 / 32768) = 1.01364326477
>>
>> Db (256 / 243) =
>> 1.05349794239
>>
>> C# (2187 / 2048) =
>> 1.06787109375
>>
>> D (9/8) = 1.125
>>
>> Eb (32/27) =
>> 1.185185185...
>>
>> D# (19683 / 16384) =
>> 1.20135498047
>>
>> Fb (8192 / 6561) =
>> 1.24859015394
>>
>> E (81/64) =
>> 1.265625
>>
>> F (4/3) =
>> 1.333333...
>>
>> E# (177147 / 131072) = 1.351524353
>>
>> Gb (1024 / 729) =
>> 1.40466392318
>>
>> F# (729 / 512) =
>> 1.423828125
>>
>> G (3/2) = 1.5
>>
>> Ab (128 / 81) =
>> 1.58024691358
>>
>> G# (6561 / 4096) =
>> 1.60180664063
>>
>> A (27/16) = 1.6875
>>
>> Bb (16/9) =
>> 1.777777...
>>
>> A# (59049 / 32768) =
>> 1.8020324707
>>
>> 2Cb (4096 / 2187) =
>> 1.8728852309
>>
>> B (243 / 128) =
>> 1.8984375
>>
>> 2C 2 = 2
>>
>>
>>
>> THE EXTENDED SCALE OF ARISTOXENUSï¿½ZARLINO
>>
>> NOTATION RELATIVE FREQUENCY
>>
>> C 1 =
>> 1
>>
>> C# (25/24) =
>> 1.0416666...
>>
>> Db (16/15) =
>> 1.0666666...
>>
>> D (9/8) =
>> 1.125
>>
>> D# (75/64) =
>> 1.171875
>>
>> Eb (6/5) =
>> 1.2
>>
>> E (5/4) =
>> 1.25
>>
>> Fb (32/25) =
>> 1.28
>>
>> E# (125 / 96) =
>> 1.302083333...
>>
>> F (4/3) =
>> 1.333333...
>>
>> F# (45/32) =
>> 1.40625
>>
>> Gb (36/25) =
>> 1.44
>>
>> G (3/2) =
>> 1.5
>>
>> G# (25/16) =
>> 1.5625
>>
>> Ab (8/5) =
>> 1.6
>>
>> A (5/3) =
>> 1.666666...
>>
>> A# (225 / 128) =
>> 1.7578125
>>
>> Bb (9/5) =
>> 1.8
>>
>> B (15/8) =
>> 1.875
>>
>> 2Cb (48/25) =
>> 1.92
>>
>> B# (125 / 64) =
>> 1.953125
>>
>> 2C 2 =
>> 2
>>
>>
>>
>> I hope this information is useful to you.
>>
>> Mario
>>
>> January, 5.
>
>
> ------------------------------------
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🔗Mario Pizarro <piagui@...>

To the members of the tuning list.

ï¿½ï¿½.....CELL #

J ï¿½ 142 - 1.17452815104 = 36529/31101

M ï¿½ 143 - 1.17585436996 = 6847/5823

M ï¿½ 144 - 1.17718208638 = 3601/3059

M ï¿½ 493 - 1.74791410141 = 12779/7311

M ï¿½ 494 - 1.74988775930 = 42874/24501

J ï¿½ 495 - 1.75186753174 = 14071/8032

1.75 and 1.175 are not cells so they are not consonant.

Mario

January, 6

<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
----- Original Message ----- From: "Mike Battaglia" <battaglia01@...>
To: <tuning@yahoogroups.com>
Sent: Thursday, January 05, 2012 11:08 PM
Subject: [tuning] Re: A useful tool to recognize consonance.

🔗Keenan Pepper <keenanpepper@...>

🔗Mike Battaglia <battaglia01@...>

🔗Mario Pizarro <piagui@...>

Dear members,
Mike,

By error I didnï¿½t send you the consonance information of (11/8) = 1.375. Instead, I sent it of 1.175. Below you can see that 1.375 is not consonant since it is not a cell; the close to 1.375 is numbered 281.

M 279 1.37151045250
M 280 1.37305909407
J 281 1.37461253344
J 282 1.37616773034
U 283 1.37783798031
U 284 1.37951025746

-------------------------------------------------------------------
J ï¿½ 142 - 1.17452815104 = 36529/31101
M ï¿½ 143 - 1.17585436996 = 6847/5823
M ï¿½ 144 - 1.17718208638 = 3601/3059 ............ Therefore 1.175 is not consonant.

M ï¿½ 493 - 1.74791410141 = 12779/7311
M ï¿½ 494 - 1.74988775930 = 42874/24501
J ï¿½ 495 - 1.75186753174 = 14071/8032

1.75 and 1.175 are not cells so they are not consonant.

I am completing some definitions regarding a group of non consonant ratios like
1.175, 1.75, 9/7, 14/9. These ratios and many other ones seem to work in a parallel set of elements.

Mario

January, 7

<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
----- Original Message ----- From: "Mike Battaglia" <battaglia01@...>
To: <tuning@yahoogroups.com>
Sent: Thursday, January 05, 2012 11:08 PM
Subject: [tuning] Re: A useful tool to recognize consonance.