Spiral tunings
A spiral tuning system[idiosyncratic term] (term proposed by J.B. Cristian) is a layout for string instruments based on any of the diverse configurations of a spiral polygonal chain, known as a spirangle, utilizing the segment's length as source for pitch, either as string length or frequency.
These systems are aperiodic (with exceptions) and possess an infinite range of possibilities. Among these configurations defined by their sides and segments, many prove musically practical, with potential for some to manifest as tangible instruments, such as spiral harps. (An instrument with a single wound string where pitch is linked solely to string length, and tension becomes relative.)
Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.
Theory
Each tuning can be mainly defined by the amount of sides and the margin, and can be named:
S6m1 - "S" for spiral, followed by the number of sides, "m" for margin; if its value is 1, it can be omitted (e.g., S6m1 = S6). [This tuning is of main interest.]
S5.5 - Five-and-a-half-sided spiral with margin 1 (omitted).
S1m1.05946 - One-sided spiral with a margin of the twelfth root of 2.
S7x1.1y1c1 - Seven-sided spiral with a margin of 1 (omitted), starting away from the center at x = 1.1, y = 1, and constant increment c = 1. When omitted, spirals start at x = 0, y = 0, c = 1.
iS6m1 - Inverted six-sided spiral with a margin of 1.
The parameters affecting the resulting relative segment length progression are:
Amount of sides: from 0 to infinity.
Margin: usually 1 (to mimic spider-webs). This property can be (unnecessarily) employed to generate equal-division systems. For example, the angle is calculated with "PI * 2 / spiralSides," so when sides are 1, 1/2, or 1/4, etc., it leaves the margin as the sole control for segment length increase. For instance, a one-sided spiral with a radius of approximately 1.05946 (twelfth root of 2) generates a 12 equal division system. From this perspective, equal-division systems can be seen as a subset of spirals.
Starting coordinates: usually 0, 0. Starting the spiral away from the center opens another dimension of progression; however, it seems to mostly affect the initial segments, and the rest of the spiral converges quickly with its version starting at 0, 0.
Inversion: This parameter doesn't affect the progression but rather how the progression is treated, as string length or as frequency.
Examples
Spiral polygonal chains with different margins.
Spiral polygonal chains with different sides.
Graphs
Unwound spirals next to each other, firsts 10 segments. With margin 1. From 0.5 to 2 sides (150 spirals, in 0.01 step) First segment from each spiral is normalized to the same length. Segments are colored by octave, this means, every red is the same chroma. The 1-sided spiral has all its segments of the same length in this configuration with margin 1.
Unwound spirals next to each other, from 0 sides to 2 sides, in step 0.004, 20 first strings, colored by equave 2:1
Unwound spirals next to each other, from 0 sides to 10 sides, in step 0.02, 20 first strings
Unwound spirals next to each other, from 0 sides to 20 sides, in step 0.01, 20 first strings
Todo: clarify
Add a description of what the three graphs above represent, especially the axes (which is which) and the color coding.
Construction
Starting from the center, and considering the segment's length as string length, the first being the shortest, becomes the highest pitch so the tunings are defined inversely. Since, in most cases, they are aperiodic, the system sizes are infinite, it will depend on how many notes one wants to calculate. Most spiral settings cover the audible range with less than 300 segments.
For instance, a six-sided spiral harp with margin 1, comprised of 120 segments spans approximately five octaves.
The spiral can be of any size, a diameter, or scale property, while changing the length of the segments, won't alter their relative length.(if started at 0,0).
We assign a frequency to the first segment, e.g. 8000hz, and the rest of the notes are calculated from it.
Algorithm for calculating segment length
Given:
- Radius [math]r = (z \times n) \times (m^n)[/math] where 'z' is the constant size increment, 'm' is margin, and 'n' is the point's index, starting at 0.
- Angle [math]a = \frac{2 \pi}{s}[/math] where 's' is the amount of sides of the spiral
The x-coordinate and y-coordinate of a point on the spiral are calculated using:
[math]x = r \times \cos(a)[/math]
[math]y = r \times \sin(a)[/math]
The distance between two consecutive points in Cartesian coordinates (x1,y1) and (x2,y2) is calculated using the Euclidean distance formula:
[math]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/math]
Properties
One significant characteristic that differs from most tunings is that each successive lower octave has more notes. At first sight, the different progressions don't seem to say much. It helps to analyze each tuning by looking at its full interval matrix, revealing that some spirals have many more types of minor thirds, while others have more fifths. Some completely dodge certain harmonics, regardless of how many "strings" you add; some combinations just never happen.
TODO: Add images!
Considerations for spiral tunings and possible harps
For a real spiral, a logical number of sides starts at 3 (greater than 2, avoiding string overlap) and ends at some point depending on the expected range (e.g., 12 sides). However, a real harp beyond this will have too many or too short strings to be practical. Regarding the margin, the value is usually 1; going too far away from this eliminates the possibility of the spiral as an instrument, and so does this tuning inversion, the progression as frequency.