The twelve schismas scale

To the tuning members,

A new scale proposal.

The twelve frequency tones of the scale were determined

by using only two values: 9/8 and the square of the schisma:

[(32805/32768)^2] = M^2

Then, the twelve schismas scale is formed by:

C = 1

C# = C [(9/8)^(1/2)] / M^2]=1.05826893295

D = C#[(9/8)^(1/2)] = 1.12246370821.

Eb = D [(9/8)^(1/2)] / M^2] = 1.18786320090.

E = Eb [(9/8)^(1/2)] = 1.25991918672

F = E [(9/8)^(1/2)] / M^2] = (4/3)

F# = F [(9/8)^(1/2)] = (2)^(1/2)

G = F# [(9/8)^(1/2)] = (3/2)

Ab = G [(9/8)^(1/2)] / M^2] = 1.58740339942

A = Ab [(9/8)^(1/2)] = 1.68369556231

Bb = A [(9/8)^(1/2)] / M^2] = 1.78179480135

B = Bb [(9/8)^(1/2)] = 1.88987878008

2C = B [(9/8)^(1/2)] / M^2] = 2

Total number of schismas : 12

More information is given as follows:

C = 1........... = 0 Cents

C# = 1.05826893294 .. = 98.04756. = About -2 cents from ETT.

D = 1.12246370821.. = 200.00256. = About 0 cents from ETT.

Eb = 1.18786320091.. = 298.04243. = About -2 cents from ETT.

E = 1.25991918672 .. = 399.997439 = About 0 cents from ETT.

F = 1.33333333333.. = 498.045.. = About -2 cents from ETT.

F# = 1.41421356237.. = 600..... = 0 cents from ETT.

G = 1.5 ......... = 701.955.. = About +2 cents from ETT.

Ab = 1.58740339942.. = 800.00256 = About 0 cents from ETT.

A = 1.68369556231.. = 901.95756. = About +2 cents from ETT.

Bb = 1.78179480135... = 999.99743 = About 0 cents from ETT.

B = 1.88987878008.. = 1101.95244 = About +2 cents from ETT.

2C = 2........... = 1200.... = 0 cents from ETT.

Note that the twelve tone relative frequencies comply with the following ratios:

(D/C#) = (E/Eb) = (F#/F) = (G/F#) = (A/Ab) = (B/Bb) = (9/8)^(1/2)

= 1.06066017178

C# = (Eb/D) = (F/E) = (Ab/G) = (Bb/A) = (2C/B) =[(9/8)^(1/2)]/[(Schisma)^2]

= 1.05826893295

All twelve tone scales operate with narrow fifths and the Pythagorean

comma which are given by either common numbers or cents.

The twelve schismas scale presents interesting features which might be signs

of the searched scale. The scale works with six narrow fifths; each one is

complemented by a factor to give 1.5. One of its interesting features concerns

to the six complementary factors which are not common numbers; rather,

the square of M and the square of J factors produce the Pythagorean comma:

(M^2)(M^2)(M^2)(M^2)(J^2)(J^2) = (M^8)(J^4) = Pythagorean comma =

1.013643264.....

M and J are the corresponding Schisma and comma that work in the six

narrow fifths detailed below and have the following values:

M = (32805/32768) = 1.00112915039 = 1.95372 cents = Schisma.

J = [(33554432 x 2^1/4)/39858075] = 1.0011313711 = 1.957561 cents.

The first six lines of the following tabulation correspond to the six narrow fifths.

(2D/G) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2E/A) = 1.496611638 and (1.5/1.496611638)=1,002264022= J^2.

(2F#/B) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2C#/F#) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2Eb/Ab) = 1.496611638 and (1.5/1.496611638)=1,002264022= J^2.

(2F/Bb) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(G/C) = 1.5 Pure fifth

(A/D) = 1.5 Pure fifth

(B/E) = 1.5 Pure fifth

(Ab/C#) = 1.5 Pure fifth

(Bb/Eb) = 1.5 Pure fifth

(2C/F) = 1.5 Pure fifth

More information regarding the six narrow fifths:

1,00225957576 = M^2 = 3,90744157266 Cents

1,00226402222 = J^2 = 3,91512206388 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

1.00226402222 = J^2 = 3,91512206388 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

Total = (M^8) (J^4) = Pythagorean comma = 23,4600104 Cents

The proximity of this scale to equal tempered scale is understandable since the ETS

imperfection is not much appreciable. It is expected that the tone relations of the twelve schismas scale define properly the intervals and chords.

Thanks

Mario Pizarro

piagui@...

Lima, February 04, 2009

Dear Mario,

this tuning alternates - with one exception - pure fifths with fifths that
are, to all practical purposes, 1/6 Pythagorean-comma fifths. It is thus
virtually identical to the "Fifths Circle No 3" published by Neidhardt in
his well known work on temperaments of 1732. (see the Unequal Temperaments
book, http://temper.braybaroque.ie/, p.391).

Kind regards

Claudio Di Veroli

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Mario Pizarro
Sent: 04 February 2009 23:16
To: tuning yahoogroups
Subject: [tuning] The twelve schismas scale

To the tuning members,

A new scale proposal.

The twelve frequency tones of the scale were determined

by using only two values: 9/8 and the square of the schisma:

[(32805/32768)^2] = M^2

Then, the twelve schismas scale is formed by:

C = 1

C# = C [(9/8)^(1/2)] / M^2]=1.05826893295

D = C#[(9/8)^(1/2)] = 1.12246370821.

Eb = D [(9/8)^(1/2)] / M^2] = 1.18786320090.

E = Eb [(9/8)^(1/2)] = 1.25991918672

F = E [(9/8)^(1/2)] / M^2] = (4/3)

F# = F [(9/8)^(1/2)] = (2)^(1/2)

G = F# [(9/8)^(1/2)] = (3/2)

Ab = G [(9/8)^(1/2)] / M^2] = 1.58740339942

A = Ab [(9/8)^(1/2)] = 1.68369556231

Bb = A [(9/8)^(1/2)] / M^2] = 1.78179480135

B = Bb [(9/8)^(1/2)] = 1.88987878008

2C = B [(9/8)^(1/2)] / M^2] = 2

Total number of schismas : 12

More information is given as follows:

C = 1........... = 0 Cents

C# = 1.05826893294 .. = 98.04756. = About -2 cents from ETT.

D = 1.12246370821.. = 200.00256. = About 0 cents from ETT.

Eb = 1.18786320091.. = 298.04243. = About -2 cents from ETT.

E = 1.25991918672 .. = 399.997439 = About 0 cents from ETT.

F = 1.33333333333.. = 498.045.. = About -2 cents from ETT.

F# = 1.41421356237.. = 600..... = 0 cents from ETT.

G = 1.5 ......... = 701.955.. = About +2 cents from ETT.

Ab = 1.58740339942.. = 800.00256 = About 0 cents from ETT.

A = 1.68369556231.. = 901.95756. = About +2 cents from ETT.

Bb = 1.78179480135... = 999.99743 = About 0 cents from ETT.

B = 1.88987878008.. = 1101.95244 = About +2 cents from ETT.

2C = 2........... = 1200.... = 0 cents from ETT.

Note that the twelve tone relative frequencies comply with the following
ratios:

(D/C#) = (E/Eb) = (F#/F) = (G/F#) = (A/Ab) = (B/Bb) = (9/8)^(1/2)

= 1.06066017178

C# = (Eb/D) = (F/E) = (Ab/G) = (Bb/A) = (2C/B) =[(9/8)^(1/2)]/[(Schisma)^2]

= 1.05826893295

All twelve tone scales operate with narrow fifths and the Pythagorean

comma which are given by either common numbers or cents.

The twelve schismas scale presents interesting features which might be signs

of the searched scale. The scale works with six narrow fifths; each one is

complemented by a factor to give 1.5. One of its interesting features
concerns

to the six complementary factors which are not common numbers; rather,

the square of M and the square of J factors produce the Pythagorean comma:

(M^2)(M^2)(M^2)(M^2)(J^2)(J^2) = (M^8)(J^4) = Pythagorean comma =

1.013643264.....

M and J are the corresponding Schisma and comma that work in the six

narrow fifths detailed below and have the following values:

M = (32805/32768) = 1.00112915039 = 1.95372 cents = Schisma.

J = [(33554432 x 2^1/4)/39858075] = 1.0011313711 = 1.957561 cents.

The first six lines of the following tabulation correspond to the six narrow
fifths.

(2D/G) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2E/A) = 1.496611638 and (1.5/1.496611638)=1,002264022= J^2.

(2F#/B) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2C#/F#) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2Eb/Ab) = 1.496611638 and (1.5/1.496611638)=1,002264022= J^2.

(2F/Bb) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(G/C) = 1.5 Pure fifth

(A/D) = 1.5 Pure fifth

(B/E) = 1.5 Pure fifth

(Ab/C#) = 1.5 Pure fifth

(Bb/Eb) = 1.5 Pure fifth

(2C/F) = 1.5 Pure fifth

More information regarding the six narrow fifths:

1,00225957576 = M^2 = 3,90744157266 Cents

1,00226402222 = J^2 = 3,91512206388 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

1.00226402222 = J^2 = 3,91512206388 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

Total = (M^8) (J^4) = Pythagorean comma = 23,4600104 Cents

The proximity of this scale to equal tempered scale is understandable since
the ETS

imperfection is not much appreciable. It is expected that the tone relations
of the twelve schismas scale define properly the intervals and chords.

Thanks

Mario Pizarro

piagui@ec-red. <mailto:piagui@...> com

Lima, February 04, 2009