stellar

From: Gene Ward Smith (2005-06-03)
Subject: Stellar scale

Here is what I think to be an interesting marvel-tempered scale I'll
call "stellar". It is formed by first taking the stellated hexany, aka
the 2x2x2 chord cube, and projecing it to the 5-limit plane via marvel
tempering. This gives a 14 note scale, which has a convex closure of
16 notes. However, this 16 note scale is not connected in the 5-limit.
We can connect it by adding two more notes, but it seems to me better
to add four more notes symmetrically, given the 3-5 symmetry of
marvel. This gives the scale bilateral symmetry around the secor axis of 
15/14~16/15 steps, and better chords.

The result is a scale with nine major triads, eight of which extend to
otonal tetrads, and nine minor triads, eight of which extend to utonal
tetrads. We also have some 11-limit harmony; in 225/224-tempered marvel,
the chord 5/4-45/32-128/75-15/8-1125/512 tempers to a complete
11-limit otonal chord of good quality, and we likewise get a complete
utonal 11-limit chord. Of course in tunings specifically chosen for
{225/224, 385/384} this is even closer in tune. Naturally we round up
the usual suspects in the form of lots of other kinds of chords.

It is interesting to note that the symmetry of this scale preserves
the symmetry under transformations by a 4 element group which marvel
allows.

Below I give the scale in the 5-limit detempering, which tells exactly
what the notes are, and in 1/4 kleismic tuning, which tells a good way
to tune them.

! stellar5.scl
marvel scale stellar in 5-limit detempering
20
!
16/15
1125/1024
9/8
256/225
75/64
6/5
5/4
675/512
4/3
45/32
64/45
3/2
25/16
8/5
3375/2048
5/3
128/75
225/128
15/8
2

! stellar.scl
stellar scale in 1/4 kleismic marvel tempering
20
!
115.587047
153.211740
200.054240
231.174094
268.798786
315.641287
384.385833
468.853027
499.972880
584.440073
615.559927
700.027120
768.771666
815.614167
853.238860
884.358713
931.201214
968.825906
1084.412953
1200.000000

From: Gene Ward Smith (2005-06-03)
Subject: Stellar scale

Here is what I think to be an interesting marvel-tempered scale I'll
call "stellar". It is formed by first taking the stellated hexany, aka
the 2x2x2 chord cube, and projecing it to the 5-limit plane via marvel
tempering. This gives a 14 note scale, which has a convex closure of
16 notes. However, this 16 note scale is not connected in the 5-limit.
We can connect it by adding two more notes, but it seems to me better
to add four more notes symmetrically, given the 3-5 symmetry of
marvel. This gives the scale bilateral symmetry around the secor axis of 
15/14~16/15 steps, and better chords.

The result is a scale with nine major triads, eight of which extend to
otonal tetrads, and nine minor triads, eight of which extend to utonal
tetrads. We also have some 11-limit harmony; in 225/224-tempered marvel,
the chord 5/4-45/32-128/75-15/8-1125/512 tempers to a complete
11-limit otonal chord of good quality, and we likewise get a complete
utonal 11-limit chord. Of course in tunings specifically chosen for
{225/224, 385/384} this is even closer in tune. Naturally we round up
the usual suspects in the form of lots of other kinds of chords.

It is interesting to note that the symmetry of this scale preserves
the symmetry under transformations by a 4 element group which marvel
allows.

Below I give the scale in the 5-limit detempering, which tells exactly
what the notes are, and in 1/4 kleismic tuning, which tells a good way
to tune them.

! stellar5.scl
marvel scale stellar in 5-limit detempering
20
!
16/15
1125/1024
9/8
256/225
75/64
6/5
5/4
675/512
4/3
45/32
64/45
3/2
25/16
8/5
3375/2048
5/3
128/75
225/128
15/8
2

! stellar.scl
stellar scale in 1/4 kleismic marvel tempering
20
!
115.587047
153.211740
200.054240
231.174094
268.798786
315.641287
384.385833
468.853027
499.972880
584.440073
615.559927
700.027120
768.771666
815.614167
853.238860
884.358713
931.201214
968.825906
1084.412953
1200.000000