Pinetone: Difference between revisions

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|[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_4%2F3_36%2F25_8%2F5_9%2F5_2%2F1_54%2F25_12%2F5_8%2F3_72%2F25_16%2F5_18%2F5%7D0_1_2_3_4_5_6_7_6_5_4_3_2_1_0-_0-_1-_%5B0_2%5D-_%5B1_3%5D-_%5B0_2_4%5D-_%5B1_3_5%5D-_%5B0_2_4_6%5D-_%5B1_3_5_7%5D-_%5B0_2_4_6%5D-_%5B1_3_5%5D-_%5B0_2_4%5D-_%5B1_3%5D-_%5B0_2%5D-_1-_0-.._%5B0_2_4%5D-_%5B1_3_5%5D-_%5B2_4_6%5D-_%5B3_5_7%5D-_%5B4_6_8%5D-_%5B5_7_9%5D-_%5B6_8_10%5D-_%5B7_9_11%5D-_%5B6_8_10%5D-_%5B5_7_9%5D-_%5B4_6_8%5D-_%5B3_5_7%5D-_%5B2_4_6%5D-_%5B1_3_5%5D-_%5B0_2_4%5D-.._%5B0_2_4_6%5D-_%5B1_3_5_7%5D-_%5B2_4_6_8%5D-_%5B3_5_7_9%5D-_%5B4_6_8_10%5D-_%5B5_7_9_11%5D-_%5B6_8_10_12%5D-_%5B7_9_11_13%5D-_%5B6_8_10_12%5D-_%5B5_7_9_11%5D-_%5B4_6_8_10%5D-_%5B3_5_7_9%5D-_%5B2_4_6_8%5D-_%5B1_3_5_7%5D-_%5B0_2_4_6%5D--- Locrian dark diminished]

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Like [[Meantone]][7] and [[Porcupine]][7], and unlike the Ptolemy/Zarlino just major scale, the Pinetone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and the mirror inverse of ~~the~~ Lydian ~~mode is the Locrian mode~~.

Like [[Meantone]][7] and [[Porcupine]][7], and unlike the Ptolemy/[[Zarlino]] just major scale, the Pinetone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and that the Locrian mode is the mirror inverse of Lydian.

Something to note - the [[Meantone]] diatonic scale is ''generated'' by the perfect fifth, [[3/2]], which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. [[Porcupine]][7], on the other hand, is generated by [[10/9]], so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of [[9/8]] then separates G from A. The Zarlino/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of [[3/2]], i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a [[3/2]] perfect fifth, but a wolf fifth of [[40/27]]. The Pinetone diatonic is not a [[generator-offset]] scale. Setting the scale to the naturals, D E F G A B C D, [[3/2]] perfect fifths are available above D, E, F, and C, so there are 1 fewer [[3/2]] perfect fifths in the Pinetone diatonic scale than in the Zarlino/Ptolemy just major scale, and two fewer than in the typical diatonic scale. [[Porcupine]][7] also has [[3/2]] fifths only above D, E, F, and G. It is because [[3/2]] perfect fifths are available above D, E, F, and G in both [[Meantone]][7] and [[Porcupine]][7] that they are available above D, E, F, and G in the Pinetone diatonic.

Something to note - the [[Meantone]] diatonic scale is ''generated'' by the perfect fifth, [[3/2]], which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. [[Porcupine]][7], on the other hand, is generated by [[10/9]], so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of [[9/8]] then separates G from A. The [[Zarlino]]/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of [[3/2]], i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a [[3/2]] perfect fifth, but a wolf fifth of [[40/27]]. The Pinetone diatonic is not a [[generator-offset]] scale. Setting the scale to the naturals, D E F G A B C D, [[3/2]] perfect fifths are available above D, E, F, and C, so there are 1 fewer [[3/2]] perfect fifths in the Pinetone diatonic scale than in the [[Zarlino]]/Ptolemy just major scale, and two fewer than in the typical diatonic scale. [[Porcupine]][7] also has [[3/2]] fifths only above D, E, F, and G. It is because [[3/2]] perfect fifths are available above D, E, F, and G in both [[Meantone]][7] and [[Porcupine]][7] that they are available above D, E, F, and G in the Pinetone diatonic.

The minor tone small step of [[Porcupine]][7] can also represent the neutral seconds [[11/10]] and [[12/11]], since 10/9*11/10*12/11 = 4/3, and [[4/3]] is subtended by 3 small steps of [[Porcupine]][7], tempering out both [[100/99]] and [[121/120]]. [[11/8]] is easily reached in [[Porcupine]][7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of [[Porcupine]][7] represents all of [[10/9]], [[11/10]], [[12/11]] and [[27/25]], in order of largest to smallest. In the Pinetone diatonic, the small step is [[27/25]] and the medium step is [[10/9]]. We can access our 11-limit harmonies in Pinetone by tempering out [[100/99]], which separates [[10/9]] from [[11/10]], as well as [[27/25]] from [[12/11]]. This leads to [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8~[[25/22]], 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic Pinetone diatonic.

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 |[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_4%2F3_36%2F25_8%2F5_9%2F5_2%2F1_54%2F25_12%2F5_8%2F3_72%2F25_16%2F5_18%2F5%7D0_1_2_3_4_5_6_7_6_5_4_3_2_1_0-_0-_1-_%5B0_2%5D-_%5B1_3%5D-_%5B0_2_4%5D-_%5B1_3_5%5D-_%5B0_2_4_6%5D-_%5B1_3_5_7%5D-_%5B0_2_4_6%5D-_%5B1_3_5%5D-_%5B0_2_4%5D-_%5B1_3%5D-_%5B0_2%5D-_1-_0-.._%5B0_2_4%5D-_%5B1_3_5%5D-_%5B2_4_6%5D-_%5B3_5_7%5D-_%5B4_6_8%5D-_%5B5_7_9%5D-_%5B6_8_10%5D-_%5B7_9_11%5D-_%5B6_8_10%5D-_%5B5_7_9%5D-_%5B4_6_8%5D-_%5B3_5_7%5D-_%5B2_4_6%5D-_%5B1_3_5%5D-_%5B0_2_4%5D-.._%5B0_2_4_6%5D-_%5B1_3_5_7%5D-_%5B2_4_6_8%5D-_%5B3_5_7_9%5D-_%5B4_6_8_10%5D-_%5B5_7_9_11%5D-_%5B6_8_10_12%5D-_%5B7_9_11_13%5D-_%5B6_8_10_12%5D-_%5B5_7_9_11%5D-_%5B4_6_8_10%5D-_%5B3_5_7_9%5D-_%5B2_4_6_8%5D-_%5B1_3_5_7%5D-_%5B0_2_4_6%5D--- Locrian dark diminished]
 |}
-Like [[Meantone]][7] and [[Porcupine]][7], and unlike the Ptolemy/Zarlino just major scale, the Pinetone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and the mirror inverse of the Lydian mode is the Locrian mode.
+Like [[Meantone]][7] and [[Porcupine]][7], and unlike the Ptolemy/[[Zarlino]] just major scale, the Pinetone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and that the Locrian mode is the mirror inverse of Lydian.
-Something to note - the [[Meantone]] diatonic scale is ''generated'' by the perfect fifth, [[3/2]], which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. [[Porcupine]][7], on the other hand, is generated by [[10/9]], so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of [[9/8]] then separates G from A. The Zarlino/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of [[3/2]], i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a [[3/2]] perfect fifth, but a wolf fifth of [[40/27]]. The Pinetone diatonic is not a [[generator-offset]] scale. Setting the scale to the naturals, D E F G A B C D, [[3/2]] perfect fifths are available above D, E, F, and C, so there are 1 fewer [[3/2]] perfect fifths in the Pinetone diatonic scale than in the Zarlino/Ptolemy just major scale, and two fewer than in the typical diatonic scale. [[Porcupine]][7] also has [[3/2]] fifths only above D, E, F, and G. It is because [[3/2]] perfect fifths are available above D, E, F, and G in both [[Meantone]][7] and [[Porcupine]][7] that they are available above D, E, F, and G in the Pinetone diatonic.
+Something to note - the [[Meantone]] diatonic scale is ''generated'' by the perfect fifth, [[3/2]], which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. [[Porcupine]][7], on the other hand, is generated by [[10/9]], so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of [[9/8]] then separates G from A. The [[Zarlino]]/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of [[3/2]], i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a [[3/2]] perfect fifth, but a wolf fifth of [[40/27]]. The Pinetone diatonic is not a [[generator-offset]] scale. Setting the scale to the naturals, D E F G A B C D, [[3/2]] perfect fifths are available above D, E, F, and C, so there are 1 fewer [[3/2]] perfect fifths in the Pinetone diatonic scale than in the [[Zarlino]]/Ptolemy just major scale, and two fewer than in the typical diatonic scale. [[Porcupine]][7] also has [[3/2]] fifths only above D, E, F, and G. It is because [[3/2]] perfect fifths are available above D, E, F, and G in both [[Meantone]][7] and [[Porcupine]][7] that they are available above D, E, F, and G in the Pinetone diatonic.
 The minor tone small step of [[Porcupine]][7] can also represent the neutral seconds [[11/10]] and [[12/11]], since 10/9*11/10*12/11 = 4/3, and [[4/3]] is subtended by 3 small steps of [[Porcupine]][7], tempering out both [[100/99]] and [[121/120]]. [[11/8]] is easily reached in [[Porcupine]][7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of [[Porcupine]][7] represents all of [[10/9]], [[11/10]], [[12/11]] and [[27/25]], in order of largest to smallest. In the Pinetone diatonic, the small step is [[27/25]] and the medium step is [[10/9]]. We can access our 11-limit harmonies in Pinetone by tempering out [[100/99]], which separates [[10/9]] from [[11/10]], as well as [[27/25]] from [[12/11]]. This leads to [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8~[[25/22]], 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic Pinetone diatonic.