Proposal of an Improved Degree Expression

Note: There are some new definitions (written in italics) in this proposal. They only apply to this proposal and should not be referred to as terms in Shasavic music theory.

For a CJK-style notation, please refer to the Chinese version of this proposal.

Rules

P1

The interval of 0 difference is notated as P1, in which the number “1” is called a degree number, and the letter “P” is called a dimension-polar name.

0 = P1

2D Accumulation

When accumulating an interval of the frequency ratio 3/2, move the degree number left (accumulating downward) or right (accumulating upward) by one item in the following cycling sequence.

…, 3, 7, 4, 1, 5, 2, 6, 3, 7, 4, 1, …

Let the cycle index of P1 be 0. For each time accumulating the aforementioned 2D interval, increment the cycle index by 1 if the degree number moves up from 7 to 4, and decrement the cycle index by 1 if the degree number moves down from 4 to 7. The cycle index stays unchanged otherwise.

When the cycle index is not 0, prepend an accidental prefix to the degree number. A positive cycle index n is notated as a prefix “♯ⁿ”; a negative cycle index n is notated as a prefix “♭ⁿ”.

“♯¹” can be simplified to “♯”, and “♯²” to “𝄪”. “♭¹” can be simplified to “♭”, and “♭²” to “𝄫”。

The rules above make the following sequence.

…, 𝄫7, ♭4, …, ♭7, 4, 1, 5, 2, 6, 3, 7, ♯4, …, ♯7, 𝄪4, …

Every item in the sequence is a degree, consisting a degree number and an optional accidental prefix. This sequence is referred to as the degree sequence.

• (3/2)^-23 = P♭⁴7
• (3/2)^-16 = P♭³7
• (3/2)^-9 = P𝄫7
• (3/2)^-8 = P♭4
• 2ddd = P♭3
• 2dd = P♭7
• 2d = P4
• 0 = P1
• 2u = P5
• 2uu = P2
• 2uuu = P6
• 2uuuu = P3
• 2uuuuu = P7
• 2uuuuuu = P♯4
• 2uuuuuuu = P♯1
• (3/2)¹² = P♯7
• (3/2)¹³ = P𝄪4
• (3/2)²⁰ = P♯³4
• (3/2)²⁷ = P♯⁴4

Single N-D Accumulation (N≥3)

When accumulating intervals of 3D or higher dimension, change the dimension-polar name based on the dimension and the accumulating direction.

When accumulating one such interval, change the degree based on its logarithmic value according to the following table.

The logarithmic value $l$ of an interval of the frequency ratio $r$ is defined as $l=12\cdot \left({\log }_{2}rmod1\right)$.

Here $amodd$ is defined as $a-d\cdot \mathrm{floor}\left(\frac{a}{d}\right)$.

• 3u = F3
• 3d = T♭6
• 4u = D♭7
• 4d = S2
• 5u = C♯4
• 5d = V♭5
• 6u = ?
• 6d = ?
• 3d4u = ?
• 3u4d = ?

Multiple N-D Accumulation (N≥3)

The degree expression in this proposal, as the stratal expression, only supports intervals with multiple accumulations of a same N-D interval, but not those of two different N-D intervals.

When accumulating the same N-D intervals again, move the degree by the same distance between 1 and the degree of the single N-D interval in the degree sequence.

Let the guest multiplier of P1 be 0. For each time accumulating an N-D interval, increment the guest multiplier by 1 when accumulating upwards, and decrement the guest multiplier by 1 when accumulating downwards.

When the absolute value of the guest multiplier is greater than 1, append a multipler suffix to the dimension-polar name. An absolute value n is notated as a suffix “ⁿ”。

It's also doable to double the dimension-polar name for the value 2 instead of writing the suffix.

• 3uu = FF♯5
• 3dd = TT♭4
• 3uuu = F³♯7
• 3ddd = T³𝄫2
• 4uu = DD♭6
• 4dd = SS3
• 5uu = CC♯7
• 5dd = VV𝄫2
• 6uu = ?
• 6dd = ?
• 3dd4uu = ?
• 3uu4dd = ?

Reference Table