Full thread (3 messages)
From: Gene Ward Smith (2004-01-07)
Subject: The five or six Schisdia Scales
Five or six, because schisdia6 has already shown up under the name
syndia1, and before that as ramis and tamil_vi. It is the first 5-
limit Fokker block other than Pythagorean to show up under more than
one rubric.
! schisdia1.scl
Schisdia 32805/32768 2048/2025 scale
12
!
16/15
9/8
6/5
81/64
4/3
64/45
3/2
8/5
27/16
3645/2048
256/135
2
! schisdia2.scl
Schisdia 32805/32768 2048/2025 scale
12
!
256/243
10/9
32/27
5/4
4/3
45/32
16384/10935
128/81
5/3
16/9
15/8
2
! schisdia3.scl
Schisdia 32805/32768 2048/2025 scale
12
!
135/128
4096/3645
32/27
5/4
4/3
45/32
3/2
128/81
2048/1215
16/9
15/8
2
! schisdia4.scl
Schisdia 32805/32768 2048/2025 scale
12
!
16/15
9/8
1215/1024
81/64
4/3
64/45
3/2
8/5
27/16
3645/2048
256/135
2
! schisdia5.scl
Schisdia 32805/32768 2048/2025 scale
12
!
135/128
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
27/16
16/9
256/135
2
! schisdia6.scl
Schisdia 32805/32768 2048/2025 scale ~ ramis tamil_vi syndia1
12
!
16/15
9/8
6/5
81/64
4/3
64/45
3/2
8/5
27/16
9/5
256/135
2
From: Andy (2012-04-04)
Subject: 5-limit dodecatonics & Kirnberger's 23-limit comma 25921/25920, was :Re:Schisdia
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@...> wrote:
> ! schisdia2.scl
> Schisdia 32805/32768 2048/2025 scale
> 12
> !
> 256/243
> 10/9
> 32/27
> 5/4
> 4/3
> 45/32
> 16384/10935
> 128/81
> 5/3
> 16/9
> 15/8
> 2
Hi Gene,
how about the following more smooth cirle of partially tempered 4ths?
C: 1/1 = |> unison
F: 4/3 = |2 -1>
Bb 16/9 = |4 -2>
Eb 32/27 = |5 -3>
G# 128/81 = |7 -4>
C# 256/243 = |8 -5>
F# 1,024/729 = |10 -6>
B: 4,096/2,187 = |12 -7>
* 32,805/32,768 = +|-15 8, 1> ! temper up by an 'schisma' ~1.95..cent
E: 5/4 = |-2 0, 1>
* 15,625/15,552 = +|-6 -5, 6> ! temper up by an 'kleisma' ~8.1..cent
A: 72,125/46,646 = |-6 -6, 7>
* 6,115,295,232/6,103,515,625 = +|23 6, -14 > +'vishnu' ~3.3..cent
D: 262,144/234375 = |18 -1, 7>
* 15,625/15,552 = +|-6 -5, 6> ! temper up by an 'kleisma' ~8.1..cent
G: 16,384/10935 = |14 -7, 1>
* 32,805/32,768 = +|-15 8, 1> ! temper up by an 'schisma' ~1.95..cent
C: 1/1 = |> unison
by symmetric subdivision of the PC=|-19 12> into five parts as
PC = (schisma * kleisma)^2 * vishnu
over the five tempered 5ths while keeping the other
remainig seven 5ths in the remote keys just plain 3/2,
when applying the schematic pattern:
C -schisma G -kleisma D -vishnu A -kleisma E -schisma B...
B F# C# G# Eb Bb F C
as inherent staying within 5-limit
when distributing the PC over the cirlce of 5ths,
Especially here attend within the two 5ths C-G & A-E the both
almost ET-like:
http://en.wikipedia.org/wiki/Schisma
"...Kirnberger fifths of 16384:10935 ..." (~700.00128...cents)
indiscernabe from 2^(7/12) at least for most human ears:
! Sp5LimDodek.scl
Sparschuh's 5-limit dodecatonics with two Kirnberger 5ths: C-G & A-E
12
!
256/243 !_______: C# = |8 -5> inital limma
262144/234375 !_: D_ = |18 -1, -7>
32/27 !_________: Eb = |5 -3>
5/4 !___________: E_ = |-2 0, 1>
4/3 !___________: E_ = |2 -1>
1024/729 !______: F# = |10 -6>
16384/10935 !___: G_ = |14 -7, 1>
128/81 !________: G# = |7 -4>
78125/46656 !___: A_ = |-6 -6, 7>
16/9 !__________: Bb = |4 -2>
4096/2187 !_____: B_ = |12 -7> 'leading-tone' concludes by apotome B-C
2/1
!
!
That layout results in an also symmetric progression of the 3rds
relative sharp against proper JI 5/4 = |-2 0,-1> ~386cents
C--E: * 1/1 vs. just 5/4
G--B: * 1/1 vs. just 5/4
D-F#: * 15,552/15,624 !__ = +|-6 -5, 6> kleismatic: ~+ 8.1...cent
A-C#: * 393,625/393,216 ! = +|17 1, -8> Würschmidt: ~+11.4...cent
E-G#: * 2,048/2,025 !____ = +|11 -4,-2> diaschisma: ~+19.5...cent
B-Eb: * 81/80 !_ditone___ = +|-4 4, -1> Synt-Comma: ~+21.5...cent
F#Bb: * 81/80 !_ditone___ = +|-4 4, -1> Synt-Comma: ~+21.5...cent
C#-F: * 81/80 !_ditone___ = +|-4 4, -1> Synt-Comma: ~+21.5...cent
G#-C: * 81/80 !_ditone___ = +|-4 4, -1> Synt-Comma: ~+21.5...cent
Eb-G: * 2,048/2,025 !____ = +|11 -4,-2> diaschisma: ~+19.5...cent
Bb-D: * 393,625/393,216 ! = +|17 1, -8> Würschmidt: ~+11.4...cent
F--A: * 15,552/15,624 !__ = +|-6 -5, 6> kleismatic: ~+ 8.1...cent
C--E: * 1/1 vs. just 5/4
that given devitions imply somehow an intended "key-charateristics"
when modulating 5th by 5th upwards circular through all 24 keys.
http://de.wikipedia.org/wiki/Tonartencharakter
http://eo.wikipedia.org/wiki/Tonalkaraktero
Sorry, that i have to regret,
that there is no wiki-article available about that
funny aspect at the moment:-(
See -due-to-that-deplarable-lack- instead of that pity
for instance the alternative reference:
http://biteyourownelbow.com/keychar.htm
Quote:
"From Christian Schubart's Ideen zu einer Aesthetik der Tonkunst (1806) translated by Rita Steblin in A History of Key Characteristics in the 18th and Early 19th Centuries. UMI Research Press (1983).
C major
Completely pure.
******Comment: !!! ??? hence C-E=5/4 ??? !!!
Its character is: innocence, simplicity, naïvety, children's talk.
C minor
Declaration of love and at the same time the lament of unhappy love. All languishing, longing, sighing of the love-sick soul lies in this key.
Db major
A leering key, degenerating into grief and rapture. It cannot laugh, but it can smile; it cannot howl, but it can at least grimace its crying.--Consequently only unusual characters and feelings can be brought out in this key.
D major
The key of triumph, of Hallejuahs, of war-cries, of victory-rejoicing. Thus, the inviting symphonies, the marches, holiday songs and heaven-rejoicing choruses are set in this key.
D minor
Melancholy womanliness, the spleen and humours brood.
D# minor
Feelings of the anxiety of the soul's deepest distress, of brooding despair, of blackest depresssion, of the most gloomy condition of the soul. Every fear, every hesitation of the shuddering heart, breathes out of horrible
D# minor.
If ghosts could speak, their speech would approximate this key.
Eb major
The key of love, of devotion, of intimate conversation with God.
E major
Noisy shouts of joy, laughing pleasure and not yet complete, full delight lies in E Major.
F major
Complaisance & calm.
F minor
Deep depression, funereal lament, groans of misery and longing for the grave.
F# major
Triumph over difficulty, free sigh of relief utered when hurdles are surmounted; echo of a soul which has fiercely struggled and finally conquered lies in all uses of this key.
F# minor
A gloomy key: it tugs at passion as a dog biting a dress. Resentment and discontent are its language.
G major
Everything rustic, idyllic and lyrical, every calm and satisfied passion, every tender gratitude for true friendship and faithful love,--in a word every gentle and peaceful emotion of the heart is correctly expressed by this key.
G minor
Discontent, uneasiness, worry about a failed scheme; bad-tempered gnashing of teeth; in a word: resentment and dislike.
Ab major
Key of the grave. Death, grave, putrefaction, judgment, eternity lie in its radius.
Ab minor
Grumbler, heart squeezed until it suffocates; wailing lament, difficult struggle; in a word, the color of this key is everything struggling with difficulty.
A major
This key includes declarations of innocent love, satisfaction with one's state of affairs; hope of seeing one's beloved again when parting; youthful cheerfulness and trust in God.
A minor
Pious womanliness and tenderness of character.
Bb major
Cheerful love, clear conscience, hope aspiration for a better world.
Bb minor
A quaint creature, often dressed in the garment of night. It is somewhat surly and very seldom takes on a pleasant countenance. Mocking God and the world; discontented with itself and with everything; preparation for suicide sounds in this key.
B major
Strongly coloured, announcing wild passions, composed from the most glaring coulors. Anger, rage, jealousy, fury, despair and every burden of the heart lies in its sphere.
B minor
This is as it were the key of patience, of calm awaiting ones's fate and of submission to divine dispensation.
Aborginally that ancient remarks about corresponding feelings
were coined then at that time in reference to coeval late Baroque
Kirnberger-2 ratios:
http://en.wikipedia.org/wiki/Kirnberger_temperament
or more precisely en-detail found in the sources:
http://groenewald-berlin.de/text/text_T024.html
http://groenewald-berlin.de/tabellen/TAB-024.html
http://groenewald-berlin.de/graphik-tabelle/GRA-024.html
There given in exact ratios by arithmetic bi-section of the
SC into two consecuting epimoric ratios
81/80 = (162/161)*(161/160)
That got located within the cirle
within the both conscecuting 5ths at positions D~A~E
C-G-D 161/162=~-10.78655...cents A 160/161=--10.71975..cents E B F#
F# 32768/32805=1/schisma C# G# Eb Bb F C
using 23-limit, because 161 when decomposed got:
161=23*7
Now calculate from 161/160 and 162/161
Kirnberger's very tiny 23-limit-comma,
as quotient of that both ratios:
(161/160)/(162/161)=25921/25920 =~0.0668...cents
= 2^-6 * 3^-4 * 5^-1 * 7^2 * 23^2 in prime-factorization
:= |-6 -4, -1 2 0, 0 0 0, 2>
That result amounts barely rougly about 1/15 part of one single cent.
That comma-interval turns out to be even about ~4.35 times larger than his other own even more famous:
"...the tiny interval of 2^161 3^â'84 5^â'12,
the atom of Kirnberger of ~0.01536... cents,..."
:= |161 -84, -12>
Again that other chimaera appears also in the article
http://en.wikipedia.org/wiki/Schisma
as reference too.
so long
bye
Andy
From: genewardsmith (2012-04-04)
Subject: 5-limit dodecatonics & Kirnberger's 23-limit comma 25921/25920, was :Re:Schisdia
--- In tuning-math@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:
> Hi Gene,
> how about the following more smooth cirle of partially tempered 4ths?
Hi, Andy. The two scales are not really comparable, since they are not trying to solve the same problem. Mine is a Fokker block, whereas yours isn't even epimorphic. That's not to take anything away from the scale--it doesn't need to be. It looks like a perfectly viable circulating scale to me.