malcolmm

From: Gene Ward Smith (2004-07-11)
Subject: Malcolm's monochord revisted

If we Tenney reduce 12 notes of 5-limit 12-et over the range -600 to
600 cents, we get 

1, 16/15, 9/8, 6/5, 5/4, 4/3, 25/18, 3/2, 8/5, 5/3, 16/9, 15/8

I was a little surprised to discover that this scale wasn't already in
my collection of Scala scales. Since 25/18 and 18/25 have the same
Tenney height, the inverted version of this might also be cosidered.

The biggest deviation from 12-equal occurs with 25/18, which is 31.3
cents flat. If we band-limit the scale, so that the biggest deviation
cannot be greater than the 15.6 cents of 6/5, we get something more
familiar:

1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 16/9, 15/8

This is Malcolm's monochord, which has already popped up in several
connections. Since 45/32 and 32/45 have the same Tenney height, we can
put 64/45 in place of 45/32, in which case we get Riley's New Albion
scale. All of these scales are suitable candidates for tempering via
marvel (225/224-planar) where a considerable amount of 7-limit harmony
appears.

If we play the same game in the 7-limit, we get

1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15

The biggest deviation here is now for 8/7; if we band-limit this we
get instead of 8/7 and 7/4 9/8 and 16/9.

Here's Malcolm's monochord in tempered in marvel, where its six major
and six minor triads are joined by two major and two minor tetrads,
three supermajor and two subminor triads, three 1-6/5-7/5 diminished
triads, three 1-5/4-14/9 augmented triads, two 1-9/7-3/2-9/5
supermajor tetrads and two 1-7/6-3/2-5/3 subminor tetrads, all in
marvel tuning which isn't bad. If you'd rather have subminor triads
than supermajor triads inverting it will give marvel-tempered New
Albion instead.

! malcolmm.scl
Malcolm's monochord in 1/4-kleisma marvel
12
!
115.587047
200.054240
315.641288
384.385833
499.972880
584.440075
700.027121
815.614167
884.358713
999.945760
1084.412954
1200.000000

Here's the marvel-tempered verison of the Tenney-reduced scale without 
band limitation, so that we have a 7/5. To five major and five minor
triads (excuding the 12-et like ones) we can add two major and one
minor tetrad, two supermajor and two subminor triads, two 1-6//5-7/5
diminished triads, two 1-7/6-7/5 diminished triads, three 1-5/4-14/9
augmented triads, two 1-9/7-3/2-9/5 supermajor tetrads and two 
1-7/6-3/2-5/3 subminor tetrads. The greater irregularity of this scale
is not netting any benefits; harmonically it is slightly less rich.

! tenred5_12m.scl
Tenney reduced in 1/4-kleisma marvel
12
!
115.587047
200.054240
315.641288
384.385833
499.972880
568.717426
700.027121
815.614167
884.358713
999.945760
1084.412954
1200.000000

From: Gene Ward Smith (2004-07-11)
Subject: Malcolm's monochord revisted

If we Tenney reduce 12 notes of 5-limit 12-et over the range -600 to
600 cents, we get 

1, 16/15, 9/8, 6/5, 5/4, 4/3, 25/18, 3/2, 8/5, 5/3, 16/9, 15/8

I was a little surprised to discover that this scale wasn't already in
my collection of Scala scales. Since 25/18 and 18/25 have the same
Tenney height, the inverted version of this might also be cosidered.

The biggest deviation from 12-equal occurs with 25/18, which is 31.3
cents flat. If we band-limit the scale, so that the biggest deviation
cannot be greater than the 15.6 cents of 6/5, we get something more
familiar:

1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 16/9, 15/8

This is Malcolm's monochord, which has already popped up in several
connections. Since 45/32 and 32/45 have the same Tenney height, we can
put 64/45 in place of 45/32, in which case we get Riley's New Albion
scale. All of these scales are suitable candidates for tempering via
marvel (225/224-planar) where a considerable amount of 7-limit harmony
appears.

If we play the same game in the 7-limit, we get

1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15

The biggest deviation here is now for 8/7; if we band-limit this we
get instead of 8/7 and 7/4 9/8 and 16/9.

Here's Malcolm's monochord in tempered in marvel, where its six major
and six minor triads are joined by two major and two minor tetrads,
three supermajor and two subminor triads, three 1-6/5-7/5 diminished
triads, three 1-5/4-14/9 augmented triads, two 1-9/7-3/2-9/5
supermajor tetrads and two 1-7/6-3/2-5/3 subminor tetrads, all in
marvel tuning which isn't bad. If you'd rather have subminor triads
than supermajor triads inverting it will give marvel-tempered New
Albion instead.

! malcolmm.scl
Malcolm's monochord in 1/4-kleisma marvel
12
!
115.587047
200.054240
315.641288
384.385833
499.972880
584.440075
700.027121
815.614167
884.358713
999.945760
1084.412954
1200.000000

Here's the marvel-tempered verison of the Tenney-reduced scale without 
band limitation, so that we have a 7/5. To five major and five minor
triads (excuding the 12-et like ones) we can add two major and one
minor tetrad, two supermajor and two subminor triads, two 1-6//5-7/5
diminished triads, two 1-7/6-7/5 diminished triads, three 1-5/4-14/9
augmented triads, two 1-9/7-3/2-9/5 supermajor tetrads and two 
1-7/6-3/2-5/3 subminor tetrads. The greater irregularity of this scale
is not netting any benefits; harmonically it is slightly less rich.

! tenred5_12m.scl
Tenney reduced in 1/4-kleisma marvel
12
!
115.587047
200.054240
315.641288
384.385833
499.972880
568.717426
700.027121
815.614167
884.358713
999.945760
1084.412954
1200.000000