a way for approximating any m/n-comma meantone in edo:ozan's 112-edo

Hi all

Refering to :

/tuning/topicId_68142.html#68142 </tuning/topicId_68142.html#68142>

using scala , you can approximate any meantone or tuning systems based on fifth-chains in edos :

1- enter fifth of tuning in scala

2- using modify----extend , Change the number of tones to the given size.(i change to 1200)

3- using modify----reduce , Divide all pitches in the current scale by the 2/1 as many

times as needed to make the resulting pitch lower than octave.

Now you see that for any degree of fifth you have an interval size. You must find an interval with small deviation from 0 or 1200 , depending on your accuracy of approximation. The degree related to this interval is cardinality of EDO for approximating.

Examples :

-53 and 665-EDO for Pythagorean scale.

- 1/11 comma meantone -------- fifth= 699.999……for degree of 12 we have an interval with size of 1199.988 , so cardinality of 12 is very good.

- 1/6 comma meantone --------- fifth= 698.3706…..for degree of 55 we have an interval size of 10.383 cent but for 122 we have 1.213 cent , so although 55-edo is historic but 122-EDO is very better.

- 7/29 comma meantone ------- fifth= 696.763…….for degree of 31 we have an interval with size of 1199.678 cent , so cardinality of 31 is very good.

- 2/7 comma meantone --------- fifth= 695.8104 ….. for degree of 50 and 69 is good but 119 is very good ( we have interval size of 1.438)

- 1/3 comma meantone --------- fifth= 694.786 ….. for degree of 19 we have interval size of 0.934. the next cardinalities are 38, 57,76,-EDO ,

- Lucy tuning ------------- fifth= 695.4929… for degree of 88 we have interval size of 3.375 cent , so 88-EDO is very good and the best is 1087- edo ( for interval size of 0.782 cent)

So referring to :

/tuning/topicId_67957.html#68124

ozan's edo choice for a system with meantone fifth around 696.5 is 112-EDO , because we have 112nd degree with an interval size of 8.000 cent which is very better than 224-EDO. Also 31-EDO is good (for 31st degree of chains with interval size of 1191.500 cent.)

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

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

My farsi page in Harmonytalkصفحه 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> <http://www.harmonytalk.com/id/908>

Hi Mohajeri,

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:

> - 1/11 comma meantone -------- fifth= 699.999……for
> degree of 12 we have an interval with size of 1199.988 ,
> so cardinality of 12 is very good.

I have a detailed explanation of 12-edo's extremely close
approximation to 1/11-comma meantone here:

http://www.tonalsoft.com/enc/number/12edo.aspx

-monz
http://tonalsoft.com
Tonescape microtonal music software