Pinetone: Difference between revisions

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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].

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|~ 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 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