A Cyclic Subtraction Model for Diatonic Interval Arithmetic

Abstract

This paper develops an algebraic model of diatonic interval numbers using the cyclic group of order seven. We begin with the traditional ascending and descending interval operators, denoted $\uparrow$ and $\downarrow$, and show how they can be unified into a single abstract operator $\ominus$. This operator computes interval numbers by performing cyclic subtraction on the 7‑cycle of diatonic letter‑names, with inclusive counting incorporated via a +1 shift. We construct the Cayley table for $\ominus$, show how $\uparrow$ and $\downarrow$ arise as special cases, and derive the inverse operation corresponding to cyclic addition of interval numbers.

1. The Diatonic 7‑Cycle

We encode the seven diatonic letter‑names as integers modulo 7:

$C=0,$ $D=1,$ $E=2,$ $F=3,$ $G=4,$ $A=5,$ $B=6.$

All arithmetic is performed modulo 7, written as:

\[x \bmod 7.\]

This encoding preserves the cyclic order of the diatonic scale.

2. Ascending and Descending Interval Operators

Traditional music theory distinguishes between:

Ascending intervals: $X \uparrow Y$
Descending intervals: $X \downarrow Y$

We define these operators algebraically as:

\[X \uparrow Y = \bigl( (Y - X) \bmod 7 \bigr) + 1,\] \[X \downarrow Y = \bigl( (X - Y) \bmod 7 \bigr) + 1.\]

Thus:

$X \uparrow Y$ measures the inclusive distance from $X$ to $Y$ going forward around the 7‑cycle.
$X \downarrow Y$ measures the inclusive distance from $X$ to $Y$ going backward around the 7‑cycle.

Example:

\[B \uparrow C = 2,\qquad B \downarrow C = 7.\]

These two operators are direction‑specific but structurally identical except for argument order.

3. Unifying Both Directions: The Circled‑Minus Operator

To eliminate the need for two operators, we define a single binary operator:

\[X \ominus Y = \bigl( (Y - X) \bmod 7 \bigr) + 1.\]

This operator returns the interval number from $X$ to $Y$ when moving forward around the cycle.

The key observation:

\[X \uparrow Y = X \ominus Y,\] \[X \downarrow Y = Y \ominus X.\]

Thus:

Ascending interval = row $X$, column $Y$
Descending interval = row $Y$, column $X$

The operator $\ominus$ is therefore a direction‑agnostic interval operator whose direction is encoded purely by argument order.

4. Cayley Table for the Operator $\ominus$

Using the encoding $C=0,\dots,B=6$, the Cayley table for $\ominus$ is:

$X \ominus Y$	C	D	E	F	G	A	B
C	1	2	3	4	5	6	7
D	7	1	2	3	4	5	6
E	6	7	1	2	3	4	5
F	5	6	7	1	2	3	4
G	4	5	6	7	1	2	3
A	3	4	5	6	7	1	2
B	2	3	4	5	6	7	1

Interpretation:

Entry at row $X$, column $Y$ = ascending interval $X \uparrow Y$.
Entry at row $Y$, column $X$ = descending interval $X \downarrow Y$.

Thus the table simultaneously encodes both directions.

5. $\ominus$ as Cyclic Subtraction

Define the cyclic difference:

\[\Delta(X,Y) = (Y - X) \bmod 7.\]

Then:

\[X \ominus Y = \Delta(X,Y) + 1.\]

This shows:

$\ominus$ is subtraction on the 7‑cycle,
followed by inclusive counting.

This is exactly what interval numbers are.

6. The Inverse Operation: Adding an Interval to a Letter

Since $\ominus$ computes the interval between two letters, its inverse should compute the resulting letter when an interval is added to a starting letter.

Define:

$X \oplus n =$ the letter obtained by ascending an $n\text{th}$ from $X.$

Algebraically:

\[X \oplus n = (X + n - 1) \bmod 7.\]

This satisfies:

If $Y = X \oplus n$, then $X \ominus Y = n$.
If $X \ominus Y = n$, then $Y = X \oplus n$.

Thus:

$\ominus$ = cyclic subtraction + 1
$\oplus$ = cyclic addition − 1

They are perfect inverses.

7. Conclusion

We have shown that:

Diatonic interval numbers arise naturally from cyclic subtraction on the 7‑letter cycle.
The traditional operators $\uparrow$ and $\downarrow$ are special cases of a single operator $\ominus$.
The Cayley table of $\ominus$ encodes all ascending and descending intervals.
The inverse operation $\oplus$ corresponds to adding an interval to a letter.

This framework provides a clean, unified, and mathematically precise model of diatonic interval arithmetic.