Topic: Rank 3 to rank 2

1 scales

Thread (3 messages)

From: genewardsmith (2010-08-03)
Subject: Rank 3 to rank 2

Here are some examples of the phenomenon I mentioned, where a rank three temperament isn't much harmed by adding another comma and using a rank two tuning for it. Below I give a 7-limit comma defining a 7-limit rank three temperament, an associated rank two temmperament, and an equal temperament which serves to tune both.

126/125: valentine, 185et
1728/1715: semisept, 142et
1029/1024: unidec, 190et
225/224: catakleismic, 197et
3136/3125: parakleismic, 415et

From: genewardsmith (2010-08-04)
Subject: Re: Rank 3 to rank 2

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Here are some examples of the phenomenon I mentioned, where a rank three temperament isn't much harmed by adding another comma and using a rank two tuning for it. 

Here are more rank two temperaments which can serve as surrogates for rank three temperaments, this time 11-limit:

{225/224, 441/440} prodigy: miracle 72 (with 540/539)
{126/125, 176/175} thrush: myna 89 (with 540/539)
{121/120, 176/175} zeus: hitchcock/amity 99 (with 2200/2187)
{225/224, 385/384} marvel: wizard 166 (with 4375/4374)
{385/384, 441/440} portent: unidec 190 (with 4375/4374)
{243/242, 441/440} jove: harry 346 (with 4000/3993)

From: Andy (2010-08-06)
Subject: Double epimoric 11-limit dodecatonics, was: Re: Rank 3 to rank 2

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>....11-limit:
> 
> {225/224, 441/440} prodigy: miracle 72 (with 540/539)
> {126/125, 176/175} thrush: myna 89 (with 540/539)
> {121/120, 176/175} zeus: hitchcock/amity 99 (with 2200/2187)
> {225/224, 385/384} marvel: wizard 166 (with 4375/4374)
> {385/384, 441/440} portent: unidec 190 (with 4375/4374)
> {243/242, 441/440} jove: harry 346 (with 4000/3993)
>
Hi Gene & all others,
just some of that epimoric 11-limit ratios can be used  
in order to construct an funny dodecatonics:

Chain of 5ths:

Eb 384/385 Bb-F-C 539/540 G 384/385 D 440/441 A-E 539/540 B 384/385 F#
F#-C#-G# 4375/4374 Eb

so choosen, that even all the biasses of the 3rds 
also turn out to be epimoric too:

Eb 126/125 G 176/175 B  100/99 Eb
Bb 126/125 D 176/175 F# 100/99 Bb   
F  176/175 A 126/125 C# 100/99 F
C  176/175 E 126/125 G# 100/99 C

Each row represents an superparticular tri-section of the diesis

128/125 = (176/175)(126/125)(100/99) = |7,0,-3>

Hence there exist only three seizes of sharpnessess for the 3rds

176/175 = |4,0,-2,-1,1>  ~+9.86..Cents for F-A, C-E, G-B and D-F#
126/125 = |1,2,-3,1>    ~+13.8...Cents for Eb-G, Bb-D A-C# and E-G#  
100/99  = |2,-2,2,0,-1> ~+17.4...Cents for B-Eb, F#-Bb, C#-F and G#-C

That ratios yield as 'Scala' file-format:

! SpDoubEpi11lim.scl
Sparschuh's [2010] double (5ths & 3rds) epimoric 11-lim. dodecatonics
12
132/125 ! C# |2,1,-3,0,1>
28/25   ! D  |2,0,-2,1>
385/324 ! Eb |-2,-4,1,1,1> or (D# = 297/250)*(4375/4374 Ragisma)
44/35   ! E  |-2,0,-1,-1,1>
4/3     ! F  |-2,1>
176/125 ! F# |4,0,-3,0,1>
539/360 ! G  |-3,-2,-1,1,1> 
198/125 ! G# |1,2,-3,0,1> or (Ab = 385/243)*(4374/4375 Ragisma)
176/105 ! A  |4,-1,-1,-1,1>
16/9    ! Bb |4,-2>
847/450 ! B  |-1,-2,-2,1,2>
2/1     
!
![eof]

Attend the Ragismatic enharmonics!
That results in absolute pitch as frequencies in Hertzians or [cps]:

c' 262.5 middle_C4
#' 277.2
d' 294
#' 311+199/216
e' 330
f' 350
#' 369.6
g' 393+1/16
#' 415.8
a' 440 Hz
#' 466+2/3
b' 494+1/12
c" 525 tenor_C5

Have a lot of fun when playing that!
Andy