Spiral tunings: Difference between revisions

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[[File:Spiral Harp -01 concept art.png|thumb|Six-sided spiral harp concept art]]

[[File:6-sided spiral tuning melody.wav|thumb|Example melody tuned to a six-sided spiral]]

[[~~File~~:~~Heptagon spiral harp~~.~~png|thumb|An eight-sided spiral polygonal chain~~.]]

A '''spiral tuning system'''{{idiosyncratic}} (term proposed by [[User:Jbcristian|J.B. Cristian]]) is a [[tuning system]] 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.

~~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, either as string length or frequency.

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.

~~For instance, a six-sided spiral harp comprised of 120 segments spans approximately five octaves~~.

Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.

Spiral tuning systems are aperiodic 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. For instance, a six-sided spiral harp comprised of 120 segments spans approximately five octaves. Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.

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.

== Theory ==

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.

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.

Additionally, concerning the margin/radius property, the same algorithm used for calculating the spirals can be (unnecessarily) employed to generate [[Equal-step tuning|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. For instance, a one-sided spiral with a radius of approximately 1.05946 (twelfth root of 2) generates a 12 equal division system.

~~For instance, a one-sided~~ spiral with ~~a radius of approximately 1.05946 (twelfth root of 2) generates a 12 equal division system~~.

=== Examples ===

Spirangles with different margin..png|Spiral polygonal chains with different margins.

Virtual spiral harps configurations.png|Spiral polygonal chains with different sides.

</gallery>

~~An open-source~~, ~~virtual playable spiral harp is accessible~~ in ~~https://kepleriandreams~~.~~github~~.io.

=== Graphs ===

~~[[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.]]

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

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

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>

==~~====Construction:======~~

{{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.}}

~~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~~.

~~'''Algorithm for Calculating Segment Length~~:'''

== Construction ==

[[File:Heptagon spiral harp.png|thumb|An eight-sided spiral polygonal chain.]]

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 aperiodic, 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.

=== 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.

- 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

- 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:

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[~~[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]]~~

== External links ==

[[~~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~~ ]]

* [https://kepleriandreams.github.io An open-source, virtual playable spiral harp] by [[User:Jbcristian|J.B. Cristian]]

[[~~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~~ ]]

[[Category:Tuning]]

@@ Line 1: / Line 1: @@
 [[File:Spiral Harp -01 concept art.png|thumb|Six-sided spiral harp concept art]]
 [[File:6-sided spiral tuning melody.wav|thumb|Example melody tuned to a six-sided spiral]]
-[[File:Heptagon spiral harp.png|thumb|An eight-sided spiral polygonal chain.]]
+A '''spiral tuning system'''{{idiosyncratic}} (term proposed by [[User:Jbcristian|J.B. Cristian]]) is a [[tuning system]] 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.
-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, either as string length or frequency.
-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.
-For instance, a six-sided spiral harp comprised of 120 segments spans approximately five octaves.
-Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.
+Spiral tuning systems are aperiodic 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. For instance, a six-sided spiral harp comprised of 120 segments spans approximately five octaves. Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.
 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.
+== Theory ==
 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.
-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.
+Additionally, concerning the margin/radius property, the same algorithm used for calculating the spirals can be (unnecessarily) employed to generate [[Equal-step tuning|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. For instance, a one-sided spiral with a radius of approximately 1.05946 (twelfth root of 2) generates a 12 equal division system.
-For instance, a one-sided spiral with a radius of approximately 1.05946 (twelfth root of 2) generates a 12 equal division system.
+=== 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>
-An open-source, virtual playable spiral harp is accessible in https://kepleriandreams.github.io.
+=== Graphs ===
-[[File:Spirangles with different margin..png|thumb|Spiral polygonal chains with different margins.]]
+<gallery widths=300 heights=200>
-[[File:Virtual spiral harps configurations.png|thumb|Spiral polygonal chains with different sides.]]
+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
+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
+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>
-======Construction:======
+{{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.}}
-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.
-'''Algorithm for Calculating Segment Length:'''
+== Construction ==
+[[File:Heptagon spiral harp.png|thumb|An eight-sided spiral polygonal chain.]]
+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 aperiodic, 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.
+=== 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.
-- 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
-- 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:
@@ Line 43: / Line 46: @@
 <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]]
+== External links ==
-[[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 ]]
+* [https://kepleriandreams.github.io An open-source, virtual playable spiral harp] by [[User:Jbcristian|J.B. Cristian]]
-[[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 ]]
+[[Category:Tuning]]