Tuning-Math Digests messages 11427 - 11427

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Message: 11427

Date: Sat, 17 Jul 2004 07:13:20

Subject: Re: names and definitions: meantone

From: monz

thanks, Gene ... this is great!

it looks to me like this tells the whole story
... but others out there will know better than i.

can you make some like this for all of the other
named temperaments?  i'm in no hurry, but folks
who are confused about the names might appreciate it.


-monz


--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 
> > family name: meantone
> > period: 2:1 ratio
> > generator: a flat fifth or sharp fourth
> > wedge product: eh?
> 
> 5-limit: 81/80
> mapping: [<1 2 4|, <0 -1 -4|]
> poptimal generator: 34/81
> MOS: 5, 7, 12, 19, 31, 50
> 
> 7-limit:
> 
> name: meantone, standard septimal meantone, 12&19
> wedgie: <1 4 10 4 13 12| 
> mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor)
> 7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone)
> 7 limit poptimal: 86/205
> 9 limit poptimal: 47/112
> TM basis: {81/80, 126/125}
> MOS: 5, 7, 12, 19, 31, 50
> 
> name: dominant, dominant sevenths, 5&12
> wedgie: <1 4 -2 4 -6 -16|
> mapping: [<1 2 4 2|, <0 -1 -4 2|]
> 7&9 limit copoptimal generator: 29/70
> TM basis: {36/35, 64/63}
> MOS: 5, 7, 12, 17, 29
> 
> name: flattone, 19&26
> wedgie: <1 4 -9 4 -17 -32|
> mapping: [<1 2 4 -1|, <0 -1 -4 9|]
> 7 limit poptimal: 46/109
> 9 limit poptimal: 27/64
> TM basis: {81/80, 525/512}
> MOS: 5, 7, 12, 19, 26, 45
> 
> 11 limit
> 
> name: meantone, meanpop, 31&19
> wedgie: <1 4 10 -13 4 13 -24 12 -44 -71|
> mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|]
> poptimal generator: 47/112
> TM basis: {81/80, 126/125, 385/384}
> MOS: 5, 7, 12, 19, 31, 50, 81
> 
> name: huygens, fokker, 31&12
> wedgie: <1 4 10 18 4 13 25 12 28 16|
> mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|]
> poptimal generator: 13/31
> TM basis: {81/80, 126/125, 99/98}
> MOS: 5, 7, 12, 19, 31, 43
> 
> name: meanertone
> wedgie: <1 4 3 -1 4 2 -5 -4 -16 -13|
> mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|]
> poptimal generator: 13/31
> TM basis: {27/25, 21/20, 33/32}
> MOS: 5, 7, 12, 19, 31, 43


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