User:Contribution/Chanofreq Project - Revision history

Contribution at 01:09, 21 January 2025

2025-01-21T01:09:37Z

← Older revision		Revision as of 01:09, 21 January 2025
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			== WARNING! ==
			'''WARNING!''' The Chanofreq project has been replaced by the [[User:Contribution/Chanopif Project\|Chanopif project]]!

	== Chanofreq Project ==		== Chanofreq Project ==
	'''Objective:''' The Chanofreq project seeks to create a standardized file format that allows precise tuning of each note on every MIDI channel to a specific frequency. The aim is to achieve this using a minimalistic data structure that prioritizes efficiency and simplicity.		'''Objective:''' The Chanofreq project seeks to create a standardized file format that allows precise tuning of each note on every MIDI channel to a specific frequency. The aim is to achieve this using a minimalistic data structure that prioritizes efficiency and simplicity.

Contribution: /* File Structure */

2024-09-22T00:43:19Z

File Structure

← Older revision		Revision as of 00:43, 22 September 2024
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	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using Mode B or C as a mediator for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places rounding in cents. However, using Mode B or C as a mediator for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''

Contribution: /* File Structure */

2024-09-22T00:37:43Z

File Structure

← Older revision		Revision as of 00:37, 22 September 2024
Line 58:		Line 58:
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using Mode B or C as ~~an intermediary~~ for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using Mode B or C as a mediator for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''

Contribution: /* File Structure */

2024-09-22T00:33:05Z

File Structure

← Older revision		Revision as of 00:33, 22 September 2024
Line 58:		Line 58:
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using Mode B or C as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''

Contribution: /* File Structure */

2024-09-22T00:29:29Z

File Structure

← Older revision		Revision as of 00:29, 22 September 2024
Line 58:		Line 58:
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to ~~a rounding at~~ 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''

Contribution: /* File Structure */

2024-09-22T00:26:28Z

File Structure

← Older revision		Revision as of 00:26, 22 September 2024
Line 58:		Line 58:
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to a rounding at 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''

Contribution: /* File Structure */

2024-09-22T00:22:54Z

File Structure

← Older revision		Revision as of 00:22, 22 September 2024
Line 56:		Line 56:

	'''Answer:'''		'''Answer:'''
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[~~100000edo~~\|~~100~~,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.~~0000069314958282~~) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.5 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[125000edo\|125,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.000005545192819) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.04 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[~~100000edo~~\|~~100~~,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[125000edo\|125,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''

Contribution: /* File Structure */

2024-09-22T00:12:40Z

File Structure

← Older revision		Revision as of 00:12, 22 September 2024
Line 56:		Line 56:

	'''Answer:'''		'''Answer:'''
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[100000edo\|100,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.0000069314958282) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.5 days between two adjacent 64-bit float factor values ~~(1.0000069314958282 and 1.0000069314958284)~~. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[100000edo\|100,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.0000069314958282) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.5 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[100000edo\|100,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[100000edo\|100,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

Contribution: /* File Structure */

2024-09-22T00:11:52Z

File Structure

← Older revision		Revision as of 00:11, 22 September 2024
Line 56:		Line 56:

	'''Answer:'''		'''Answer:'''
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[~~16808edo~~]] is programmed from a base frequency (20 Hz) and a factor (1.~~0000412399734142~~) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 15 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[100000edo\|100,000-EDO]] is programmed from a base frequency (20 Hz) and a factor (1.0000069314958282) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 2.5 days between two adjacent 64-bit float factor values (1.0000069314958282 and 1.0000069314958284). For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[100000edo\|100,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[100000edo\|100,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

Contribution: /* File Structure */

2024-09-22T00:07:21Z

File Structure

← Older revision		Revision as of 00:07, 22 September 2024
Line 58:		Line 58:
	* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[16808edo]] is programmed from a base frequency (20 Hz) and a factor (1.0000412399734142) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 15 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.		* It should be fine to use 64-bit float, as long as no rounding is applied, we stay within the human hearing range, and the number of notes remains reasonable. For example, if [[16808edo]] is programmed from a base frequency (20 Hz) and a factor (1.0000412399734142) with exponents to map the entire multi-channel keyboard, after 10 octaves (20480 Hz), one phase error occurs every 15 days between two adjacent 64-bit float factor values. For the [[402653184edo\|402,653,184-EDO]] tuning scheme within chanofreq using factor and exponents, the phase precision between 64-bit float steps is only 78.9 seconds: 1/(440(2(-69/12))1.0000000017214499*(402653184(32/3)) - 440(2(-69/12))1.0000000017214496*(402653184(32/3))) = 78.9 seconds.
	In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.		In comparison, the phase precision between 32-bit steps in 402,653,184-EDO is 12.86 hours, as calculated here: 1/(4402((127-69)/12)2*(1/402653184)-4402*((127-69)/12))/(6060) = 12.86 hours.
	This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[100000edo\|100,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for 32-bit MIDI tuning should be avoided.		This shows that even with 64-bit floating-point precision, the factor and exponent solution is far less efficient than other tuning methods, but remains practical for moderately sized musical scales up to approximately [[100000edo\|100,000-EDO]], a size that still ensure a precision equivalent to 6 decimal places in cents. However, using it as an intermediary for [[402653184edo\|32-bit MIDI tuning]] should be avoided.

	'''Open Questions:'''		'''Open Questions:'''