Pinetone: Difference between revisions

Line 1,871:

Let's start with the just left-handed Porcutone octatonic: 27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1 in the Kadathian bright major mode. It contains 4 large steps of 10/9, 3 medium steps of 25/24 and 1 small step of 25/24. Putting a small step into the bottom of each medium and large step results in the 15-note scale

25/24 27/25 9/8 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 125/72 9/5 15/8 2/1, with step pattern msmLmLmsmLmmsmL, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625. This is right-handed Porcutone-15

25/24 27/25 9/8 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 125/72 9/5 15/8 2/1, with step pattern msmLmLmsmLmmsmL, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625. This is right-handed Porcutone-15.

~~Staring~~ instead with the just right-handed Porcutone octatonic: 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 in the Mnarian middle major mode, leads to the scale

Starting instead with the just right-handed Porcutone octatonic: 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 in the Mnarian middle major mode, leads to the scale

25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 25/16 5/3 125/72 9/5 15/8 2/1, with step pattern mLmsmLmsmmLmsmL. We can see that this scale is the inverse of right-handed Porcutone-15; accordingly it is left-handed Porcutone-15.

Line 1,891:

With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. The Porcutone-15 right and left-handed scales comprise

4 large steps of 109.12557c, approximating 16/15,

4 large steps of 109.12557c, approximating 16/15;

3 medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25, and

3 medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25; and

8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26

8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.

In cents, TE 2.3.5.11.13 ptolemismic right and left-handed Porcutone-15 (in the above modes) are

66.766 142.775 209.542 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.992 956.002 1022.768 1089.534 1198.660 as smsLsLsmsLssmsL

66.766 142.775 209.542 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.992 956.002 1022.768 1089.534 1198.660 as smsLsLsmsLssmsL and

66.766 175.892 242.658 318.667 385.443 494.559 561.325 637.334 704.101 770.867 879.992 956.002 1022.768 1089.534 1198.660 as sLsmsLsmssLsmsL respectively

66.766 175.892 242.658 318.667 385.443 494.559 561.325 637.334 704.101 770.867 879.992 956.002 1022.768 1089.534 1198.660 as sLsmsLsmssLsmsL respectively.

~~from~~ smsLsLsmsLssmsL, tempering m = s would lead to sssLsLsssLssssL, which is a MODMOS of Hanson[15]

From smsLsLsmsLssmsL, tempering m = s would lead to sssLsLsssLssssL, which is a MODMOS of Hanson[15];

tempering L = m would lead to sLsLsLsLsLssLsL, which is Porcupine[15]

tempering L = m would lead to sLsLsLsLsLssLsL, which is Porcupine[15];

tempering L = s would lead to LsLLLLLsLLLLsLL, which is a MODMOS of Augmented[15]

tempering L = s would lead to LsLLLLLsLLLLsLL, which is a MODMOS of Augmented[15];

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

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

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

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

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

== Comma pump ==

@@ Line 1,871: / Line 1,871: @@
 Let's start with the just left-handed Porcutone octatonic: 27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1 in the Kadathian bright major mode. It contains 4 large steps of 10/9, 3 medium steps of 25/24 and 1 small step of 25/24. Putting a small step into the bottom of each medium and large step results in the 15-note scale
-/24 27/25 9/8 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 125/72 9/5 15/8 2/1, with step pattern msmLmLmsmLmmsmL, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625. This is right-handed Porcutone-15
+/24 27/25 9/8 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 125/72 9/5 15/8 2/1, with step pattern msmLmLmsmLmmsmL, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625. This is right-handed Porcutone-15.
-Staring instead with the just right-handed Porcutone octatonic: 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 in the Mnarian middle major mode, leads to the scale
+Starting instead with the just right-handed Porcutone octatonic: 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 in the Mnarian middle major mode, leads to the scale
 /24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 25/16 5/3 125/72 9/5 15/8 2/1, with step pattern mLmsmLmsmmLmsmL. We can see that this scale is the inverse of right-handed Porcutone-15; accordingly it is left-handed Porcutone-15.
@@ Line 1,891: / Line 1,891: @@
 With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. The Porcutone-15 right and left-handed scales comprise
-large steps of 109.12557c, approximating 16/15,
+large steps of 109.12557c, approximating 16/15;
-medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25, and
+medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25; and
-small steps of 66.76626c, approximating 25/24, 33/32, and 27/26
+small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.
 In cents, TE 2.3.5.11.13 ptolemismic right and left-handed Porcutone-15 (in the above modes) are
-.766 142.775 209.542 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.992 956.002 1022.768 1089.534 1198.660 as smsLsLsmsLssmsL
+.766 142.775 209.542 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.992 956.002 1022.768 1089.534 1198.660 as smsLsLsmsLssmsL and
-.766 175.892 242.658 318.667 385.443 494.559 561.325 637.334 704.101 770.867 879.992 956.002 1022.768 1089.534 1198.660 as sLsmsLsmssLsmsL respectively
+.766 175.892 242.658 318.667 385.443 494.559 561.325 637.334 704.101 770.867 879.992 956.002 1022.768 1089.534 1198.660 as sLsmsLsmssLsmsL respectively.
-from smsLsLsmsLssmsL, tempering m = s would lead to sssLsLsssLssssL, which is a MODMOS of Hanson[15]
+From smsLsLsmsLssmsL, tempering m = s would lead to sssLsLsssLssssL, which is a MODMOS of Hanson[15];
-tempering L = m would lead to sLsLsLsLsLssLsL, which is Porcupine[15]
+tempering L = m would lead to sLsLsLsLsLssLsL, which is Porcupine[15];
-tempering L = s would lead to LsLLLLLsLLLLsLL, which is a MODMOS of Augmented[15]
+tempering L = s would lead to LsLLLLLsLLLLsLL, which is a MODMOS of Augmented[15];
-tempering out s would lead to sLLsLsL, which is Dicot[7]
+tempering out s would lead to sLLsLsL, which is Dicot[7];
 tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
-Accordingly the Porcutone-15 scales would temper reduce to two steps in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
+Accordingly the Porcutone-15 scales would temper to two steps in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
 == Comma pump ==