User:Eliora/Standard deviation - Revision history

Eliora at 22:19, 8 April 2023

2023-04-08T22:19:57Z

← Older revision		Revision as of 22:19, 8 April 2023
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	An interval which consists of edosteps is an irrational number, which occasionally comes close rational numbers. Null hypothesis is therefore that a typical interval is dissonant.		An interval which consists of edosteps is an irrational number, which occasionally comes close rational numbers. Null hypothesis is therefore that a typical interval is dissonant.

	== Center for measurement ==		=== Center for measurement ===

	Therefore, counting the null hypothesis above, we assume 50% of error on edostep to be the center of our measuring, and an exact landing on the edostep by JI interval (which never happens) as infinitely away from the center of the error curve. Since it never happens, this means every interval approximation is finite number of standard deviations away.		Therefore, counting the null hypothesis above, we assume 50% of error on edostep to be the center of our measuring, and an exact landing on the edostep by JI interval (which never happens) as infinitely away from the center of the error curve. Since it never happens, this means every interval approximation is finite number of standard deviations away.

	This means that an edo can be said to approximate interval within one standard deviation, if the relative error is smaller than the right-tailed p-value.		This means that an edo can be said to approximate interval within one standard deviation, if the relative error is smaller than the right-tailed p-value.
			{\| class="wikitable"
			\|+
			!SD
			!Relative cents
			\|-
			\|1
			\|15.87
			\|-
			\|2
			\|2.275
			\|-
			\|3
			\|0.135
			\|}

			== Examples ==

	=== 12edo ===		=== 12edo ===

	Assume we want to measure the strength of 3/2 in 12edo this way. The value is 701.955c just. First, take the center - 750 cents. Then take the two semitone borders - 700 and 800 cents.		Assume we want to measure the strength of 3/2 in 12edo this way. The value is 701.955c just. First, take the center - 750 cents. Then take the two semitone borders - 700 and 800 cents. Since the difference between just and 12edo fifth is 1.955 cent, we take two-tailed z-score of 1.955 / 50, or alternately single-tailed z-score for 1.955 / 100.

	~~Since~~ the ~~difference between just and 12edo fifth is~~ 1.955 ~~cent, we take two-tailed z-score~~ of ~~1.955 / 50, or alternately single-tailed z-score for 1.955 / 100.~~		Intuitively speaking, what is the standard deviation of something which has a 1.955% chance of happening?

	https://www.wolframalpha.com/input?i=z-score+0.0195500086538741774&assumption=%7B%22MC%22%2C+%22%22%7D+-%3E+%7B%22Formula%22%7D&assumption=%7B%22FSVar%22%2C+%22Q%22%2C+%7B%229%22%2C+%2229%22%7D%2C+%220.0195500086538741774%22%7D+-%3E+%7B%22NormalProbabilities%22%2C+%22pr%22%7D&assumption=%22FSelect%22+-%3E+%7B%7B%22NormalProbabilities%22%7D%7D		https://www.wolframalpha.com/input?i=z-score+0.0195500086538741774&assumption=%7B%22MC%22%2C+%22%22%7D+-%3E+%7B%22Formula%22%7D&assumption=%7B%22FSVar%22%2C+%22Q%22%2C+%7B%229%22%2C+%2229%22%7D%2C+%220.0195500086538741774%22%7D+-%3E+%7B%22NormalProbabilities%22%2C+%22pr%22%7D&assumption=%22FSelect%22+-%3E+%7B%7B%22NormalProbabilities%22%7D%7D

	The value is '''2.063''', which means that 12edo represents the expected consonance of 3/2 with about two standard deviations away from the average dissonant irrational sound a random edostep interval will make. Two standard deviations is a point of significance in many places of research, therefore [[12edo]] can be said to provide great approximation to 3/2.		The value is '''2.063''', which means that 12edo represents the expected consonance of 3/2 with about two standard deviations away from the average dissonant irrational sound a random edostep interval will make. Two standard deviations is a point of significance in many places of research, therefore [[12edo]] can be said to provide great approximation to 3/2.

			=== 665edo ===
			665edo is known for its extremely precise 3/2.

			It differs from the just by 6.298 x 10^-5 edosteps. The standard deviation of such number is '''3.834''', which is almost four. This makes 665edo's approximation of 3/2 exceptionally good, as this kind of confidence interval is rarely used even in science.

Eliora at 22:13, 8 April 2023

2023-04-08T22:13:33Z

← Older revision		Revision as of 22:13, 8 April 2023
Line 8:		Line 8:

	Therefore, counting the null hypothesis above, we assume 50% of error on edostep to be the center of our measuring, and an exact landing on the edostep by JI interval (which never happens) as infinitely away from the center of the error curve. Since it never happens, this means every interval approximation is finite number of standard deviations away.		Therefore, counting the null hypothesis above, we assume 50% of error on edostep to be the center of our measuring, and an exact landing on the edostep by JI interval (which never happens) as infinitely away from the center of the error curve. Since it never happens, this means every interval approximation is finite number of standard deviations away.

			This means that an edo can be said to approximate interval within one standard deviation, if the relative error is smaller than the right-tailed p-value.

	=== 12edo ===		=== 12edo ===
Line 17:		Line 19:
	https://www.wolframalpha.com/input?i=z-score+0.0195500086538741774&assumption=%7B%22MC%22%2C+%22%22%7D+-%3E+%7B%22Formula%22%7D&assumption=%7B%22FSVar%22%2C+%22Q%22%2C+%7B%229%22%2C+%2229%22%7D%2C+%220.0195500086538741774%22%7D+-%3E+%7B%22NormalProbabilities%22%2C+%22pr%22%7D&assumption=%22FSelect%22+-%3E+%7B%7B%22NormalProbabilities%22%7D%7D		https://www.wolframalpha.com/input?i=z-score+0.0195500086538741774&assumption=%7B%22MC%22%2C+%22%22%7D+-%3E+%7B%22Formula%22%7D&assumption=%7B%22FSVar%22%2C+%22Q%22%2C+%7B%229%22%2C+%2229%22%7D%2C+%220.0195500086538741774%22%7D+-%3E+%7B%22NormalProbabilities%22%2C+%22pr%22%7D&assumption=%22FSelect%22+-%3E+%7B%7B%22NormalProbabilities%22%7D%7D

	The value is '''2.063''', which means that 12edo represents the expected consonance of 3/2 with about two standard deviations away from the average dissonant irrational sound a random edostep interval will make.		The value is '''2.063''', which means that 12edo represents the expected consonance of 3/2 with about two standard deviations away from the average dissonant irrational sound a random edostep interval will make. Two standard deviations is a point of significance in many places of research, therefore [[12edo]] can be said to provide great approximation to 3/2.

Eliora: Created page with "Standard deviation is a way of assessing harmonic strength. == Null hypothesis == An interval which consists of edosteps is an irrational number, which occasionally comes cl..."

2023-04-08T22:11:26Z

Created page with "Standard deviation is a way of assessing harmonic strength. == Null hypothesis == An interval which consists of edosteps is an irrational number, which occasionally comes cl..."

New page

Standard deviation is a way of assessing harmonic strength.

== Null hypothesis ==

An interval which consists of edosteps is an irrational number, which occasionally comes close rational numbers. Null hypothesis is therefore that a typical interval is dissonant.

== Center for measurement ==

Therefore, counting the null hypothesis above, we assume 50% of error on edostep to be the center of our measuring, and an exact landing on the edostep by JI interval (which never happens) as infinitely away from the center of the error curve. Since it never happens, this means every interval approximation is finite number of standard deviations away.

=== 12edo ===

Assume we want to measure the strength of 3/2 in 12edo this way. The value is 701.955c just. First, take the center - 750 cents. Then take the two semitone borders - 700 and 800 cents.

Since the difference between just and 12edo fifth is 1.955 cent, we take two-tailed z-score of 1.955 / 50, or alternately single-tailed z-score for 1.955 / 100.

https://www.wolframalpha.com/input?i=z-score+0.0195500086538741774&assumption=%7B%22MC%22%2C+%22%22%7D+-%3E+%7B%22Formula%22%7D&assumption=%7B%22FSVar%22%2C+%22Q%22%2C+%7B%229%22%2C+%2229%22%7D%2C+%220.0195500086538741774%22%7D+-%3E+%7B%22NormalProbabilities%22%2C+%22pr%22%7D&assumption=%22FSelect%22+-%3E+%7B%7B%22NormalProbabilities%22%7D%7D

The value is '''2.063''', which means that 12edo represents the expected consonance of 3/2 with about two standard deviations away from the average dissonant irrational sound a random edostep interval will make.