Pinetone: Difference between revisions

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{| class="wikitable"

|+Table 2.3. Modes of the just Pinetone diatonic

|+Table 2.3. Modes of the [[SNS (2/1, 3/2, 6/5)-7|just Pinetone diatonic]]

!Mode number

!Mode in JI

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The modes of the Ptolemismic Pinetone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called [[TE tuning]].

{| class="wikitable"

|+Table 2.4. Modes of the Ptolemismic Pinetone diatonic

|+Table 2.4. Modes of the [[SNS (2/1, 3/2, 6/5: 100/99)-7|Ptolemismic Pinetone diatonic]]

!Mode number

!Pinetone diatonic mode

Line 1,063:

The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269

{| class="wikitable"

|+(2.3.5.11) Ptolemismic Pinetone Chromatic

|+(2.3.5.11) [[Ptolemismic Pinetone chromatic|Ptolemismic Pinetone Chromatic]]

!Mode number

!Mode as simplest JI pre-image

Line 1,511:

{| class="wikitable"

|+Table 4.1.

! colspan="8" |Scale (mode -3 subset)

! colspan="8" |Scale (mode 3 subset)

~~! colspan="8" |Scale (mode -3 subset)~~

!JI ratios approximated

!Step pattern

Line 1,639:

The scale D E F G A B C has step pattern MsMLMsM, which tempers to sssLsss under Porcupine (M=s) and LsLLLsL under Meantone (M=L).

Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before)~~, so will call the mode approximating the ratios 10/9 11/9~~ 4/3 3/~~2 5/3 9/5 2/1 Mixolydian symmetric minor~~, ~~and use~~ the ~~mode~~ to ~~name the scale as~~ a ~~whole~~.

Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before). The modes of this scale are detailed in Table 4.3. Similarly, lowering B to B♭ lowers by ~55/54, leading to the Duradian ♯5 shown in Table 4.4, which I consider to be a really beautiful minor mode.

To calculate the mode numbers for Tables 4.3-4.5, the mode numbers of their temperings to Porcupine and to [[Meantone]] were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties. Modes are named via the [[Tetracot]][7] [[MODMOS]] they temper to when [[243/242]] is tempered out (i.e., the difference between [[11/9]] and [[27/22]]), as in [[27edo]], [[34edo]], and [[41edo]]. Table 4.2 introduces the modes of Tetracot[7]. The Pinetone diatonic, therefore, is also a detempering of a Tetracot MODMOS, with generator chain equivalent to that of the double harmonic major scale (a MODMOS of the Meantone diatonic scale). Tetracot[7] mode names used the Archeotonic mode names [[6L 1s#Proposed names|here]] as a basis, with the substitution of Azurian, Duradian, and Phyradian as Tetracot specific mode names from [https://www.youtube.com/watch?v=xYZwye9PWSo here].

Line 1,742:

|~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1

|}

~~Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.~~

{| class="wikitable"

|+Table 4.4. Modes of the Ptolemismic Pinetone Duradian dark minor

Line 3,469:

tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].

tempering out s would lead to ~~sLLsLsL~~, which is ~~Dicot~~[7];

tempering out s would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].

tempering out m would lead to ~~ssLsLssLsssL~~, which is ~~a MODMOS of Diminished~~[12].

tempering out m would lead to sLLsLsL, which is Dicot[7];

Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1.

Line 3,489:

Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.

Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8.

2.3.5.13 325/324 may be ~~better~~ tuned to 46edo, with (L, m, s) = (4, 2, 1).

In this tuning the medium and small steps are within 1c of the same size. Tempering them together results in Cata temperament, an extension of Hanson. 2.3.5.13 325/324 may alternatively be tuned to 46edo, with (L, m, s) = (4, 2, 3), or as Cata[15] in 53edo, 72edo, or 87edo as (5, 3, 3), (7, 4, 4), or (8, 5, 5).

Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name.

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{| class="wikitable"

|+Table 9.1. Modes of the just Pinetone harmonic diminished

! Mode ~~(height order)~~

!Mode number

! Mode name

!Step pattern

!Oneirotonic step pattern

Line 3,514:

Line 3,515:

! Comments

|-

|4

|Hlanithian diminished

|AsLMLMLs

Line 3,521:

Line 3,523:

|

|-

|3

|Mnarian diminished*

|LMLMLsAs

Line 3,528:

Line 3,531:

|

|-

|2

|Celephaïsian diminished*

|LMLsAsLM

Line 3,535:

Line 3,539:

|

|-

|1

|Sarnathian diminished†

|MLMLsAsL

Line 3,542:

Line 3,547:

|10:12:15 on the root

|-

| -1

|Dylathian diminished*

|LsAsLMLM

Line 3,549:

Line 3,555:

|

|-

| -2

|Kadathian diminished*††

|MLsAsLML

Line 3,556:

Line 3,563:

|root 4:5:6,10:12:15

|-

| -3

|Ultharian diminished*††

|sAsLMLML

Line 3,563:

Line 3,571:

|root 4:5:6,10:12:15

|-

| -4

|Illarnekian diminished*†

|sLMLMLsA

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{| class="wikitable"

|+Table 9.2. Modes of the Ptolemismic Pinetone harmonic diminished

! Mode ~~(height order)~~

!Mode number

! Mode name

!Step pattern

!Mode as simplest JI pre-image 5-limit JI

Line 3,579:

Line 3,589:

!Comments

|-

|4

|Hlanithian diminished

|AsLMLMLs

Line 3,585:

Line 3,596:

|

|-

|3

|Mnarian diminished*

|LMLMLsAs

Line 3,591:

Line 3,603:

|

|-

|2

|Celephaïsian diminished*

|LMLsAsLM

Line 3,597:

Line 3,610:

|

|-

|1

|Sarnathian diminished†

|MLMLsAsL

Line 3,603:

Line 3,617:

|10:12:15 on the root

|-

| -1

|Dylathian diminished*

|LsAsLMLM

Line 3,609:

Line 3,624:

|

|-

| -2

|Kadathian diminished*††

|MLsAsLML

Line 3,615:

Line 3,631:

|root 4:5:6,10:12:15

|-

| -3

|Ultharian diminished*††

|sAsLMLML

Line 3,621:

Line 3,638:

|root 4:5:6,10:12:15

|-

| -4

|Illarnekian diminished*†

|sLMLMLsA

Line 3,810:

Line 3,828:

|+

Table 9.4. Intervals of modes of the Pinetone harmonic diminished

!Mode

!Mode (height order)

!step

!2-step

Line 3,936:

Line 3,954:

|12:15:20

|}

<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 ~~or 245/243~~ as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered ~~to 2.3.5.~~7 ~~245/243 i.e., Sensamagic temperament~~.

<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered 7-limit rather than 13-limit Supermagic.

==Comma pump==

@@ Line 150: / Line 150: @@
 |}
 {| class="wikitable"
-|+Table 2.3. Modes of the just Pinetone diatonic
+|+Table 2.3. Modes of the [[SNS (2/1, 3/2, 6/5)-7|just Pinetone diatonic]]
 !Mode number
 !Mode in JI
@@ Line 239: / Line 239: @@
 The modes of the Ptolemismic Pinetone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called [[TE tuning]].
 {| class="wikitable"
-|+Table 2.4. Modes of the Ptolemismic Pinetone diatonic
+|+Table 2.4. Modes of the [[SNS (2/1, 3/2, 6/5: 100/99)-7|Ptolemismic Pinetone diatonic]]
 !Mode number
 !Pinetone diatonic mode
@@ Line 1,063: / Line 1,063: @@
 The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269
 {| class="wikitable"
-|+(2.3.5.11) Ptolemismic Pinetone Chromatic
+|+(2.3.5.11) [[Ptolemismic Pinetone chromatic|Ptolemismic Pinetone Chromatic]]
 !Mode number
 !Mode as simplest JI pre-image
@@ Line 1,511: / Line 1,511: @@
 {| class="wikitable"
 |+Table 4.1.
+! colspan="8" |Scale (mode -3 subset)
 ! colspan="8" |Scale (mode 3 subset)
-! colspan="8" |Scale (mode -3 subset)
 !JI ratios approximated
 !Step pattern
@@ Line 1,639: / Line 1,639: @@
 The scale D E F G A B C has step pattern MsMLMsM, which tempers to sssLsss under Porcupine (M=s) and LsLLLsL under Meantone (M=L).
-Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before), so will call the mode approximating the ratios 10/9 11/9 4/3 3/2 5/3 9/5 2/1 Mixolydian symmetric minor, and use the mode to name the scale as a whole.
+Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before). The modes of this scale are detailed in Table 4.3. Similarly, lowering B to B♭ lowers by ~55/54, leading to the Duradian ♯5 shown in Table 4.4, which I consider to be a really beautiful minor mode.
 To calculate the mode numbers for Tables 4.3-4.5, the mode numbers of their temperings to Porcupine and to [[Meantone]] were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties. Modes are named via the [[Tetracot]][7] [[MODMOS]] they temper to when [[243/242]] is tempered out (i.e., the difference between [[11/9]] and [[27/22]]), as in [[27edo]], [[34edo]], and [[41edo]]. Table 4.2 introduces the modes of Tetracot[7]. The Pinetone diatonic, therefore, is also a detempering of a Tetracot MODMOS, with generator chain equivalent to that of the double harmonic major scale (a MODMOS of the Meantone diatonic scale). Tetracot[7] mode names used the Archeotonic mode names [[6L 1s#Proposed names|here]] as a basis, with the substitution of Azurian, Duradian, and Phyradian as Tetracot specific mode names from [https://www.youtube.com/watch?v=xYZwye9PWSo here].
@@ Line 1,742: / Line 1,742: @@
 |~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1
 |}
-Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.
 {| class="wikitable"
 |+Table 4.4. Modes of the Ptolemismic Pinetone Duradian dark minor
@@ Line 3,469: / Line 3,469: @@
 tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].
-tempering out s would lead to sLLsLsL, which is Dicot[7];
+tempering out s would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
-tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
+tempering out m would lead to sLLsLsL, which is Dicot[7];
 Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1.
@@ Line 3,489: / Line 3,489: @@
 Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
 Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8.
-.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).
+In this tuning the medium and small steps are within 1c of the same size. Tempering them together results in Cata temperament, an extension of Hanson. 2.3.5.13 325/324 may alternatively be tuned to 46edo, with (L, m, s) = (4, 2, 3), or as Cata[15] in 53edo, 72edo, or 87edo as (5, 3, 3), (7, 4, 4), or (8, 5, 5).
 Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name.
@@ Line 3,507: / Line 3,507: @@
 {| class="wikitable"
 |+Table 9.1. Modes of the just Pinetone harmonic diminished
-! Mode (height order)
+!Mode number
+! Mode name
 !Step pattern
 !Oneirotonic step pattern
@@ Line 3,514: / Line 3,515: @@
 ! Comments
 |-
+|4
 |Hlanithian diminished
 |AsLMLMLs
@@ Line 3,521: / Line 3,523: @@
 |
 |-
+|3
 |Mnarian diminished*
 |LMLMLsAs
@@ Line 3,528: / Line 3,531: @@
 |
 |-
+|2
 |Celephaïsian diminished*
 |LMLsAsLM
@@ Line 3,535: / Line 3,539: @@
 |
 |-
+|1
 |Sarnathian diminished<sup>†</sup>
 |MLMLsAsL
@@ Line 3,542: / Line 3,547: @@
 |10:12:15 on the root
 |-
+| -1
 |Dylathian diminished*
 |LsAsLMLM
@@ Line 3,549: / Line 3,555: @@
 |
 |-
+| -2
 |Kadathian diminished*<sup>††</sup>
 |MLsAsLML
@@ Line 3,556: / Line 3,563: @@
 |root 4:5:6,10:12:15
 |-
+| -3
 |Ultharian diminished*<sup>††</sup>
 |sAsLMLML
@@ Line 3,563: / Line 3,571: @@
 |root 4:5:6,10:12:15
 |-
+| -4
 |Illarnekian diminished*<sup>†</sup>
 |sLMLMLsA
@@ Line 3,573: / Line 3,582: @@
 {| class="wikitable"
 |+Table 9.2. Modes of the Ptolemismic Pinetone harmonic diminished
-! Mode (height order)
+!Mode number
+! Mode name
 !Step pattern
 !Mode as simplest JI pre-image 5-limit JI
@@ Line 3,579: / Line 3,589: @@
 !Comments
 |-
+|4
 |Hlanithian diminished
 |AsLMLMLs
@@ Line 3,585: / Line 3,596: @@
 |
 |-
+|3
 |Mnarian diminished*
 |LMLMLsAs
@@ Line 3,591: / Line 3,603: @@
 |
 |-
+|2
 |Celephaïsian diminished*
 |LMLsAsLM
@@ Line 3,597: / Line 3,610: @@
 |
 |-
+|1
 |Sarnathian diminished<sup>†</sup>
 |MLMLsAsL
@@ Line 3,603: / Line 3,617: @@
 |10:12:15 on the root
 |-
+| -1
 |Dylathian diminished*
 |LsAsLMLM
@@ Line 3,609: / Line 3,624: @@
 |
 |-
+| -2
 |Kadathian diminished*<sup>††</sup>
 |MLsAsLML
@@ Line 3,615: / Line 3,631: @@
 |root 4:5:6,10:12:15
 |-
+| -3
 |Ultharian diminished*<sup>††</sup>
 |sAsLMLML
@@ Line 3,621: / Line 3,638: @@
 |root 4:5:6,10:12:15
 |-
+| -4
 |Illarnekian diminished*<sup>†</sup>
 |sLMLMLsA
@@ Line 3,810: / Line 3,828: @@
 |+
 Table 9.4. Intervals of modes of the Pinetone harmonic diminished
-!Mode
+!Mode (height order)
 !step
 !2-step
@@ Line 3,936: / Line 3,954: @@
 |12:15:20
 |}
-<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered to 2.3.5.7 245/243 i.e., Sensamagic temperament.
+<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered 7-limit rather than 13-limit Supermagic.
 ==Comma pump==