Just intonation point - Revision history

FloraC: + "just tuning map" cuz it's linked

2025-12-04T07:39:37Z

+ "just tuning map" cuz it's linked

← Older revision		Revision as of 07:39, 4 December 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP'''), also known as '''just tuning map''', is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

FloraC: Undo revision 192470 by VectorGraphics (talk). Insane linking style. The "guide" is inferior to the Wikipedia article anyway

2025-04-17T11:50:45Z

Undo revision 192470 by VectorGraphics (talk). Insane linking style. The "guide" is inferior to the Wikipedia article anyway

← Older revision		Revision as of 11:50, 17 April 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we ~~[[Mathematical guide/Matrix operations#Dot product\|~~take]] the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

	For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 … }}.		For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 … }}.

VectorGraphics: Add back math guide link

2025-04-16T16:00:04Z

Add back math guide link

← Older revision		Revision as of 16:00, 16 April 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we [[Mathematical guide/Matrix operations#Dot product\|take]] the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

	For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 … }}.		For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 … }}.

FloraC: Style

2025-04-16T13:44:23Z

Style

← Older revision		Revision as of 13:44, 16 April 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic [[interval size unit]]), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.
Line 6:		Line 6:

	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of the ~~JIP—as~~ with a normal (temperament) tuning ~~map—as~~ <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].		It may be helpful to think of the units of each entry of the JIP—as with a normal (temperament) tuning map—as <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

	== Mathematical definition ==		== Mathematical definition ==
	The JIP, commonly denoted J, is a point in ''p''-limit [[~~Vals~~ and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').		The JIP, commonly denoted ''J'', is a point in ''p''-limit [[vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').

	The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then ~~<~~J\|m~~>~~ is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\| e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p'' }}, then J becomes correspondingly the covector {{val\| 1 1 1 … 1 }}.		The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If '''m''' is a monzo, then {{vmprod\|''J''\|'''m'''}} is the untempered JI value of '''m''' measured in octaves. In Tenney-weighted coordinates, where '''m''' = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\| ''e''<sub>2</sub>log<sub>2</sub>2 ''e''<sub>3</sub>log<sub>2</sub>3 ''e''<sub>5</sub>log<sub>2</sub>5 … ''e''<sub>''p''</sub>log<sub>2</sub>''p'' }}, then ''J'' becomes correspondingly the covector {{val\| 1 1 1 … 1 }}.

	As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. ~~<span><math>\frac{~~84}{53} ≈ ~~\frac{~~1.585}{1.000~~}</math></span>~~ and ~~<span><math>\frac{~~123}{53} ≈ ~~\frac{~~2.322}{1.000~~}</math></span>~~.		As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. Equal-temperament maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in ''J'' = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. 84/53 ≈ 1.585/1.000 and 123/53 ≈ 2.322/1.000.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Terms]]		[[Category:Terms]]

FloraC: Link to Wikipedia instead

2025-04-16T13:38:29Z

Link to Wikipedia instead

← Older revision		Revision as of 13:38, 16 April 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the ~~[[Mathematical guide/Matrix operations#Dot product~~\|dot product]] of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the {{w\|dot product}} of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

	For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 ~~...~~}}.		For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 … }}.

	== Units ==		== Units ==
Line 11:		Line 11:
	The JIP, commonly denoted J, is a point in ''p''-limit [[Vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').		The JIP, commonly denoted J, is a point in ''p''-limit [[Vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').

	The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then <J\|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\|e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p''}}, then J becomes correspondingly the covector {{val\| 1 1 1 … 1 }}.		The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then <J\|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\| e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p'' }}, then J becomes correspondingly the covector {{val\| 1 1 1 … 1 }}.

	As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.		As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.

VectorGraphics: we should always say "dot product"

2025-04-16T08:03:21Z

we should always say "dot product"

← Older revision		Revision as of 08:03, 16 April 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the ~~bracket~~ product of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the [[Mathematical guide/Matrix operations#Dot product\|dot product]] of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

	For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 ...}}.		For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 ...}}.
Line 11:		Line 11:
	The JIP, commonly denoted J, is a point in ''p''-limit [[Vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').		The JIP, commonly denoted J, is a point in ''p''-limit [[Vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').

	The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then <J\|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\|e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p''}}, then J becomes correspondingly the ~~bra vector~~ {{val\| 1 1 1 … 1 }}.		The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then <J\|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\|e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p''}}, then J becomes correspondingly the covector {{val\| 1 1 1 … 1 }}.

	As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.		As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.

ArrowHead294: Update links

2024-11-02T18:50:19Z

Update links

← Older revision		Revision as of 18:50, 2 November 2024
Line 6:		Line 6:

	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].		It may be helpful to think of the units of each entry of the JIP—as with a normal (temperament) tuning map—as <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

	== Mathematical definition ==		== Mathematical definition ==

ArrowHead294: Update links

2024-11-02T18:49:42Z

Update links

← Older revision		Revision as of 18:49, 2 November 2024
Line 6:		Line 6:

	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis]].		It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

	== Mathematical definition ==		== Mathematical definition ==

Dave Keenan: Changed {\large\mathsf{¢}} to \mathsf{¢}

2022-12-24T06:00:52Z

Changed {\large\mathsf{¢}} to \mathsf{¢}

← Older revision		Revision as of 06:00, 24 December 2022
Line 6:		Line 6:

	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as <math>~~{\large~~\mathsf{¢}}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis]].		It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as <math>\mathsf{¢}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis]].

	== Mathematical definition ==		== Mathematical definition ==

Cmloegcmluin: /* Units */ update format of units and add link to more information on general topic

2022-12-15T17:46:18Z

Units: update format of units and add link to more information on general topic

← Older revision		Revision as of 17:46, 15 December 2022
Line 6:		Line 6:

	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as c/p (read "cents per prime"), oct/p (read "octaves per prime"), or any other logarithmic pitch unit per prime.		It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as <math>{\large\mathsf{¢}}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis]].

	== Mathematical definition ==		== Mathematical definition ==

← Older revision		Revision as of 08:03, 16 April 2025
Line 1:		Line 1:
	The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).		The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).

	For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the ~~bracket~~ product of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.		For instance, in 5-limit JI, the JIP is {{val \| 1200.000 1901.955 2786.314 }}; if we take the [[Mathematical guide/Matrix operations#Dot product\|dot product]] of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

	For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 ...}}.		For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, {{val \| 1 1 1 ...}}.
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	The JIP, commonly denoted J, is a point in ''p''-limit [[Vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').		The JIP, commonly denoted J, is a point in ''p''-limit [[Vals and tuning space\|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q'').

	The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then <J\|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\|e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p''}}, then J becomes correspondingly the ~~bra vector~~ {{val\| 1 1 1 … 1 }}.		The JIP is the target of optimization in optimized tunings including [[TOP tuning\|TOP]] and [[TE tuning]]. If m is a monzo, then <J\|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo\|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo\|e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p''}}, then J becomes correspondingly the covector {{val\| 1 1 1 … 1 }}.

	As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.		As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as {{val\| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val\| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.