Pentatonic Functional Just System - Revision history

Overthink: /* Higher limits */ slight corrections

2026-02-07T04:43:31Z

Higher limits: slight corrections

← Older revision		Revision as of 04:43, 7 February 2026
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	\| [[8/7]] \|\| 231.2 \|\| <sub>5</sub>m2<sub>7</sub>		\| [[8/7]] \|\| 231.2 \|\| <sub>5</sub>m2<sub>7</sub>
	\|-		\|-
	\| [[15/13]] \|\| 247.8 \|\| <sub>5</sub>m2<sup>5</sup><sub>13</sub>		\| [[15/13]] \|\| 247.7 \|\| <sub>5</sub>m2<sup>5</sup><sub>13</sub>
	\|-		\|-
	\| [[7/6]] \|\| 266.9 \|\| <sub>5</sub>M2<sup>7</sup>		\| [[7/6]] \|\| 266.9 \|\| <sub>5</sub>M2<sup>7</sup>
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	\| [[11/9]] \|\| 347.4 \|\| <sub>5</sub>s3<sup>11</sup>		\| [[11/9]] \|\| 347.4 \|\| <sub>5</sub>s3<sup>11</sup>
	\|-		\|-
	\| [[16/13]] \|\| 359.3 \|\| <sub>5</sub>M2<sub>13</sub>		\| [[16/13]] \|\| 359.5 \|\| <sub>5</sub>M2<sub>13</sub>
	\|-		\|-
	\| [[5/4]] \|\| 386.3 \|\| <sub>5</sub>s3<sup>5</sup>		\| [[5/4]] \|\| 386.3 \|\| <sub>5</sub>s3<sup>5</sup>

Overthink: more correct

2026-02-06T05:42:24Z

more correct

← Older revision		Revision as of 05:42, 6 February 2026
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	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page. All of these terms were coined by [[User:Overthink]].}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page. All of these terms were coined by [[User:Overthink]].}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 ~~notes~~ per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 interval classes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.

Xenllium at 10:57, 5 February 2026

2026-02-05T10:57:15Z

← Older revision		Revision as of 10:57, 5 February 2026
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	\| [[27/14]] \|\| 1137.0 \|\| <sub>5</sub>s6<sub>7</sub>		\| [[27/14]] \|\| 1137.0 \|\| <sub>5</sub>s6<sub>7</sub>
	\|-		\|-
	\| [[63/32]] \|\| ~~1200~~.0 \|\| <sub>5</sub>P6<sup>7</sup>		\| [[63/32]] \|\| 1172.7 \|\| <sub>5</sub>P6<sup>7</sup>
	\|}		\|}
	</div><div style="display: inline-grid; margin-right: 25px;">		</div><div style="display: inline-grid; margin-right: 25px;">

Overthink: /* Higher limits */ grammar

2026-01-27T00:26:28Z

Higher limits: grammar

← Older revision		Revision as of 00:26, 27 January 2026
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	== Higher limits ==		== Higher limits ==
	We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use different formal commas ~~than~~ in the FJS: The formal comma for 11 is [[729/704\|704/729]] (11/8 is <sub>5</sub>s4<sup>11</sup>), and the formal comma for 13 is [[27/26\|26/27]] (13/8 is <sub>5</sub>m5<sup>13</sup>).		We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use different formal commas from in the FJS: The formal comma for 11 is [[729/704\|704/729]] (11/8 is <sub>5</sub>s4<sup>11</sup>), and the formal comma for 13 is [[27/26\|26/27]] (13/8 is <sub>5</sub>m5<sup>13</sup>).

	<div><div style="display: inline-grid; margin-right: 25px;">		<div><div style="display: inline-grid; margin-right: 25px;">

Overthink: give credit for terms

2026-01-15T04:10:06Z

give credit for terms

← Older revision		Revision as of 04:10, 15 January 2026
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	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page. All of these terms were coined by [[User:Overthink]].}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.

Overthink: /* Ratios of 5 */ a few more rewordings

2026-01-11T01:32:43Z

Ratios of 5: a few more rewordings

← Older revision		Revision as of 01:32, 11 January 2026
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	One can see that the ratios of 5 are further from 5edo intervals than ratios of 7. Thus, the 5-limit intervals can now be considered "subminor" and "supermajor", compared to the intervals of 7 in diatonic. Using the <sub>5</sub>fifth construction, we get the [[3:5:9]] subminor and [[5:9:15\|1/(9:5:3) = 5:9:15]] supermajor chords, the compact voicings of which are [[9:10:12]] and [[15:18:20]] respectively.		One can see that the ratios of 5 are further from 5edo intervals than ratios of 7. Thus, the 5-limit intervals can now be considered "subminor" and "supermajor", compared to the intervals of 7 in diatonic. Using the <sub>5</sub>fifth construction, we get the [[3:5:9]] subminor and [[5:9:15\|1/(9:5:3) = 5:9:15]] supermajor chords, the compact voicings of which are [[9:10:12]] and [[15:18:20]] respectively.

	If we try to construct 5-limit triads the normal way, the [[4:5:6]] major triad becomes <sub>5</sub>P1–<sub>5</sub>s3<sup>5</sup>–<sub>5</sub>P4, and the [[10:12:15]] minor triad becomes <sub>5</sub>P1–<sub>5</sub>M2<sub>5</sub>–<sub>5</sub>P4. ~~Now we~~ see why it was a good idea refer to augmented and diminished as "super" and "sub"; these intervals occur so much more often. However, ~~now~~ the [[4:5:6]] and [[10:12:15]] triads ~~aren't~~ classified by the same interval categories, while they are in diatonic.		If we try to construct 5-limit triads the normal way, the [[4:5:6]] major triad becomes <sub>5</sub>P1–<sub>5</sub>s3<sup>5</sup>–<sub>5</sub>P4, and the [[10:12:15]] minor triad becomes <sub>5</sub>P1–<sub>5</sub>M2<sub>5</sub>–<sub>5</sub>P4. We now see why it was a good idea refer to augmented and diminished as "super" and "sub"; these intervals occur much more often in a pentic system. However, the [[4:5:6]] and [[10:12:15]] triads are no longer classified by the same interval categories, while they are in diatonic.

	The [[7/5]] and [[10/7]] intervals are not included in the above tables due to containing factors of both 5 and 7; 7/5 is written as <sub>5</sub>S3<sup>7</sup><sub>5</sub>, while 10/7 is written as <sub>5</sub>s4<sup>5</sup><sub>7</sub>. An advantage of pentic notation is that these intervals are in the right order in terms of interval categories, unlike in traditional diatonic-based FJS, where 7/5 is d5<sup>7</sup><sub>5</sub> and 10/7 is A4<sup>5</sup><sub>7</sub>.		The [[7/5]] and [[10/7]] intervals are not included in the above tables due to containing factors of both 5 and 7; 7/5 is written as <sub>5</sub>S3<sup>7</sup><sub>5</sub>, while 10/7 is written as <sub>5</sub>s4<sup>5</sup><sub>7</sub>. An advantage of pentic notation is that these intervals are in the right order in terms of interval categories, unlike in traditional diatonic-based FJS, where 7/5 is d5<sup>7</sup><sub>5</sub> and 10/7 is A4<sup>5</sup><sub>7</sub>.

Overthink: more rewordings

2026-01-11T01:28:45Z

more rewordings

← Older revision		Revision as of 01:28, 11 January 2026
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	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. ~~In this~~ page~~, we will develop~~ a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we ~~will~~ omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.

Overthink: reword statement

2026-01-11T01:26:31Z

reword statement

← Older revision		Revision as of 01:26, 11 January 2026
Line 2:		Line 2:
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates ~~in 2026~~.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.

	== The 3-limit ==		== The 3-limit ==

Overthink: note buzzard

2026-01-10T21:05:16Z

note buzzard

← Older revision		Revision as of 21:05, 10 January 2026
Line 1:		Line 1:
	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates in 2026.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates in 2026.

Overthink: + note names

2026-01-10T21:04:03Z

+ note names

← Older revision		Revision as of 21:04, 10 January 2026
Line 1:		Line 1:
	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates in 2026.

	== The 3-limit ==		== The 3-limit ==

← Older revision		Revision as of 05:42, 6 February 2026
Line 1:		Line 1:
	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page. All of these terms were coined by [[User:Overthink]].}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page. All of these terms were coined by [[User:Overthink]].}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 ~~notes~~ per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 interval classes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.

← Older revision		Revision as of 01:28, 11 January 2026
Line 1:		Line 1:
	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. ~~In this~~ page~~, we will develop~~ a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we ~~will~~ omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates made later.

← Older revision		Revision as of 21:05, 10 January 2026
Line 1:		Line 1:
	{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}		{{Idiosyncratic terms\|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}
	Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.		Traditionally, we use a [[5L 2s\|diatonic]] system of interval classification. This works well in the [[5-limit]], and in [[meantone]] systems. However, in other systems like [[superpyth]] or [[buzzard]], a pentatonic system of classification based on the [[2L 3s\|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup\|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 notes per octave rather than the traditional 7, we will omit the notes F and B, and only use C, D, E, G, and A.

	The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates in 2026.		The PFJS was devised by [[User:Overthink\|Overthink]] in 2025, with updates in 2026.