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Spiral tunings: Difference between revisions

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[[File:Heptagon spiral harp.png|thumb|An eight-sided spiral polygonal chain.]]
[[File:Spiral Harp -01 concept art.png|thumb|Six-sided spiral harp concept art]]
Spiral tuning systems encompass the diverse configurations of a spiral polygonal chain, known as a "spirangle," utilizing the segment's length as source for pitch, whether as string length or frequency.
[[File:6-sided spiral tuning melody.wav|thumb|Example melody tuned to a six-sided spiral]]
A '''spiral tuning system'''{{idiosyncratic}} (term proposed by [[User:Jbcristian|J.B. Cristian]]) is a layout for string instruments based on any of the diverse configurations of a spiral polygonal chain, known as a {{w|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.
 
'''''S7r2c1''''' - Seven-sided spiral with a margin of 1 (omitted), with an initial radius of 2, and constant increment c = 1. When omitted, spirals initial radius is 0, c = 1.
 
'''''iS6m1''''' - Inverted six-sided spiral with a margin of 1.


These systems are non-periodic 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.
The parameters affecting the resulting relative segment length progression are:


For instance, a six-sided spiral harp comprised of 120 segments spans approximately five octaves.
''Amount of sides:'' from 0 to infinity.


Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.
''Margin:'' usually 1 (to mimic spider-webs). This property can be (unnecessarily) employed to generate [[Equal divisions of the octave|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.


Furthermore, the inversions of these tunings hold musical merit. Instead of the segment's length dictating string length, it determines frequency. However, this approach forfeits the utilization of the spiral as an instrument.
Initial radius: usually 0 Using a different initial radius 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 with radius 0.


Another aspect influencing pitch is the spiral margin. This alteration also sacrifices the characteristic spider-web appearance and eliminates the possibility of a spiral harp.
''Inversion'': This parameter doesn't affect the progression but rather how the progression is treated, as string length or as frequency.


Additionally, concerning the margin/radius property, the same algorithm used for calculating the spirals can be (unnecessarily) employed to generate equal-division systems. For example, when sides are 1, 1/2, or 1/4, etc., the angle is calculated with "PI*2/spiralSides," leaving the margin as the sole control for segment length increase, with the rest of the calculation following Pythagoras' theorem.
=== Examples ===
<gallery widths="300" heights="200">
Spirangles with different margin..png|Spiral polygonal chains with different margins.
Virtual spiral harps configurations.png|Spiral polygonal chains with different sides.
</gallery>


For instance, a one-sided spiral with a radius of approximately 1.05946 (twelfth root of 2) generates a 12 equal division system.
=== Graphs ===
<gallery widths="300" heights="200">
File:Unwound spirals next to each other.png|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.
File:Unwinded spirals next to each other, from 0 sides to 2 sides, in step 0.004, 20 first strings .png|Unwound spirals next to each other, from 0 sides to 2 sides, in step 0.004, 20 first strings, colored by equave 2:1
File:Unwinded spirals next to each other, from 0 sides to 10 sides, in step 0.02, 20 first strings .png|Unwound spirals next to each other, from 0 sides to 10 sides, in step 0.02, 20 first strings
File:Unwound spirals next to each other, from 0 sides to 20 sides, in step 0.01, 20 first strings .png|Unwound spirals next to each other, from 0 sides to 20 sides, in step 0.01, 20 first strings
</gallery>


An open-source, virtual playable spiral harp is accessible in https://kepleriandreams.github.io.
{{todo|clarify|inline=1|comment=Add a description of what the three graphs above represent, especially the axes (which is which) and the color coding.}}
[[File:Spirangles with different margin..png|thumb|Spiral polygonal chains with different margins.]]
[[File:Virtual spiral harps configurations.png|thumb|Spiral polygonal chains with different sides.]]


======Construction:======
== Construction ==
Each tuning can be defined by the amount of sides and the margin. Starting from the center, and considering the segment's length as string length, the first being the shortest becomes the highest pitch, so in this case the tunings are defined inversely. Since they are non-periodic, the system sizes are infinite; it will depend on how many notes one wants to calculate. Most harp settings cover the audible range with fewer than 300 strings. The spiral can be of any size, a diameter, or scale property, while changing the length of the segments won't alter their relative lengths. We assign a frequency to the first segment, e.g., 8000 Hz, and the rest of the notes are calculated from it.
[[File:Heptagon spiral harp.png|thumb|An eight-sided spiral polygonal chain.]]
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.


'''Algorithm for Calculating Segment Length:'''
For instance, a six-sided spiral harp with margin 1, comprised of 120 segments spans approximately five octaves.


Given:
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).


- 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.
We assign a frequency to the first segment, e.g. 8000hz, and the rest of the notes are calculated from it.


- Angle <math>a = \frac{2 \pi}{s}</math> where 's' is the amount of sides of the spiral
=== 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:
The x-coordinate and y-coordinate of a point on the spiral are calculated using:
Line 41: Line 72:
<math>d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}</math>
<math>d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}</math>


[[File:Unwinded spirals next to each other, from 0 sides to 2 sides, in step 0.004, 20 first strings .png|thumb|Unwound spirals next to each other, from 0 sides to 2 sides, in step 0.004, 20 first strings, colored by equave 2:1]]
== Properties ==
[[File:Unwinded spirals next to each other, from 0 sides to 10 sides, in step 0.02, 20 first strings .png|thumb|Unwound spirals next to each other, from 0 sides to 10 sides, in step 0.02, 20 first strings ]]
 
[[File:Unwound spirals next to each other, from 0 sides to 20 sides, in step 0.01, 20 first strings .png|thumb|Unwound spirals next to each other, from 0 sides to 20 sides, in step 0.01, 20 first strings ]]
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 strings 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.
 
The most important part is the number of sides, which will expose different chords at each row, some configurations make progressions more intuitive. The most altered part of the progression are the initial segments, since its the result of truncating the spiral, they are the most affected. from these different truncation different patterns still emerge. The rest of the progression are mostly equal, relatively, in most configurations. From a string length perspective,(already away from the center, with less error in the truncation) the progression is almost arithmetic, seems to increase at constant but there is an increasing unnoticeable ratio.
 
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.
 
== External links ==
* [https://kepleriandreams.github.io An open-source, virtual playable spiral harp] by [[User:Jbcristian|J.B. Cristian]]
 
[[Category:Instruments]]
[[Category:Tuning]]
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