Flippery

A fixed-frame flip is a chord progression of the form a:b:c <-> 1/c:1/b:1/a, where the a of the first chord is the same as the 1/c of the second chord.

A fixed-center flip is a chord progression of the form a:b:c <-> 1/c:1/b:1/a, where the b of the first chord is the same as the 1/b of the second chord.

A scale is flippery ^{[idiosyncratic term ]} (term proposed by Praimhin) if every step of the scale is part of either a fixed frame flip or a fixed center flip using only notes in the scale.

Examples

The scale [1/1, 9/8, 6/5, 5/4, 45/32, 3/2, 8/5, 5/3, 15/8, 2/1] is flippery.

• 1/1-9/8: part of the fixed frame flip 1/4:1/3:1/2 -> 2:3:4, where 1/1 serves as the 1/3 of the first chord and 9/8 as the 3 of the second chord.
• 9/8-6/5: part of the fixed center flip 3:4:5 -> 1/5:1/4:1/3, where 9/8 serves as the 3 of the first chord and 6/5 as the 1/5 of the second chord.
• 6/5-5/4: part of the fixed frame flip 1/6:1/5:1/4 -> 4:5:6, where 6/5 serves as the 1/5 of the first chord and 5/4 as the 5 of the second chord.
• 5/4-45/32: part of the fixed frame flip 1/4:1/3:1/2 -> 2:3:4, where 5/4 serves as the 1/3 of the first chord and 45/32 as the 3 of the second chord.
• 45/32-3/2: part of the fixed frame flip 1/5:1/4:1/3 -> 3:4:5, where 45/32 serves as the 1/4 of the first chord and 3/2 as the 4 of the second chord.
• 3/2-8/5: part of the fixed frame flip 1/5:1/4:1/3 -> 3:4:5, where 3/2 serves as the 1/4 of the first chord and 8/5 as the 4 of the second chord.
• 8/5-5/3: part of the fixed center flip 4:5:6 -> 1/6:1/5:1/4, where 8/5 serves as the 1/4 of the first chord and 5/3 as the 1/6 of the second chord.
• 5/3-15/8: part of the fixed frame flip 1/4:1/3:1/2 -> 2:3:4, where 5/3 serves as the 1/3 of the first chord and 15/8 as the 3 of the second chord.
• 15/8-2/1: part of the fixed frame flip 1/5:1/4:1/3 -> 3:4:5, where 15/8 serves as the 1/4 of the first chord and 2/1 as the 4 of the second chord.

In contrast, the scale [1/1, 9/8, 6/5, 5/4, 45/32, 3/2, 8/5, 5/3, 16/9, 2/1] is not flippery; the step 5/3-16/9 is not part of any flip.

• A hypothetical fixed frame flip involving the step 5/3-16/9 would look like 4/3:5/3:20/9 → 4/3:16/9:20/9, but neither the note 4/3 nor 10/9 (octave reduced 20/9) are in the scale.
• A hypothetical fixed center flip involving the step 5/3-16/9 would look either like 1/1:4/3:5/3 → 16/15:4/3:16/9, or 5/3:20/9:50/18 → 16/9:20/9:80/27, which again involve notes outside the scale.