Pinetone: Difference between revisions
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Tables 1. and 2. show the modes of [[Porcupine]][7], and [[Meantone]][7], respectively, in the [[5-limit]]. Given that intervals of tempered scales represent more than a single [[Just intonation|JI]] interval each, modes are described in their ''JI pre-image,'' the simplest [[Just intonation|JI]] ratios each interval above the tonic represents. Along with the step pattern and mode number, the modes' [[UDP]] are shown. A mode's [[Modal UDP notation|UDP]] shows the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Table 3. shows the modes of the [[5-limit]] Pinetone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively. | Tables 1. and 2. show the modes of [[Porcupine]][7], and [[Meantone]][7], respectively, in the [[5-limit]]. Given that intervals of tempered scales represent more than a single [[Just intonation|JI]] interval each, modes are described in their ''JI pre-image,'' the simplest [[Just intonation|JI]] ratios each interval above the tonic represents. Along with the step pattern and mode number, the modes' [[UDP]] are shown. A mode's [[Modal UDP notation|UDP]] shows the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Table 3. shows the modes of the [[5-limit]] Pinetone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively. | ||
In Tables 1.-4., Modes marked with '*' have a consonant triad on their root. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Table 1. Modes of 5-limit Porcupine[7] | |+Table 1. Modes of 5-limit Porcupine[7] | ||
| Line 36: | Line 38: | ||
|Lssssss | |Lssssss | ||
|<nowiki>6|0</nowiki> | |<nowiki>6|0</nowiki> | ||
|Bright major | |Bright major* | ||
|- | |- | ||
|2 | |2 | ||
| Line 42: | Line 44: | ||
|sLsssss | |sLsssss | ||
|<nowiki>5|1</nowiki> | |<nowiki>5|1</nowiki> | ||
|Dark major | |Dark major* | ||
|- | |- | ||
|1 | |1 | ||
| Line 48: | Line 50: | ||
|ssLssss | |ssLssss | ||
|<nowiki>4|2</nowiki> | |<nowiki>4|2</nowiki> | ||
|Bright minor | |Bright minor* | ||
|- | |- | ||
|0 | |0 | ||
| Line 54: | Line 56: | ||
|sssLsss | |sssLsss | ||
|<nowiki>3|3</nowiki> | |<nowiki>3|3</nowiki> | ||
|Symmetric minor | |Symmetric minor* | ||
|- | |- | ||
| -1 | | -1 | ||
| Line 86: | Line 88: | ||
|LLLsLLs | |LLLsLLs | ||
|<nowiki>6|0</nowiki> | |<nowiki>6|0</nowiki> | ||
|Lydian | |Lydian* | ||
|- | |- | ||
|2 | |2 | ||
| Line 92: | Line 94: | ||
|LLsLLLs | |LLsLLLs | ||
|<nowiki>5|1</nowiki> | |<nowiki>5|1</nowiki> | ||
|Ionian | |Ionian* | ||
|- | |- | ||
|1 | |1 | ||
| Line 98: | Line 100: | ||
|LLsLLsL | |LLsLLsL | ||
|<nowiki>4|2</nowiki> | |<nowiki>4|2</nowiki> | ||
|Mixolydian | |Mixolydian* | ||
|- | |- | ||
|0 | |0 | ||
| Line 104: | Line 106: | ||
|LsLLLsL | |LsLLLsL | ||
|<nowiki>3|3</nowiki> | |<nowiki>3|3</nowiki> | ||
|Dorian | |Dorian* | ||
|- | |- | ||
| -1 | | -1 | ||
| Line 110: | Line 112: | ||
|LsLLsLL | |LsLLsLL | ||
|<nowiki>2|4</nowiki> | |<nowiki>2|4</nowiki> | ||
|Aeolian | |Aeolian* | ||
|- | |- | ||
| -2 | | -2 | ||
| Line 116: | Line 118: | ||
|sLLLsLL | |sLLLsLL | ||
|<nowiki>1|5</nowiki> | |<nowiki>1|5</nowiki> | ||
|Phrygian | |Phrygian* | ||
|- | |- | ||
| -3 | | -3 | ||
| Line 142: | Line 144: | ||
|sLsssss | |sLsssss | ||
|Dark major | |Dark major | ||
|[https://xenpaper.com/#%7B1%2F1_10%2F9_5%2F4_25%2F18_3%2F2_5%2F3_50%2F27_2%2F1_20%2F9_5%2F2_25%2F9_3%2F1_10%2F3_100%2F27%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--- Lydian dark major] | |[https://xenpaper.com/#%7B1%2F1_10%2F9_5%2F4_25%2F18_3%2F2_5%2F3_50%2F27_2%2F1_20%2F9_5%2F2_25%2F9_3%2F1_10%2F3_100%2F27%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--- Lydian dark major]* | ||
|- | |- | ||
|2 | |2 | ||
| Line 151: | Line 153: | ||
|Lssssss | |Lssssss | ||
|Bright major | |Bright major | ||
|[https://xenpaper.com/#%7B1%2F1_9%2F8_5%2F4_27%2F20_3%2F2_5%2F3_9%2F5_2%2F1_9%2F4_5%2F2_27%2F10_3%2F1_10%2F3_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--- Mixolydian bright major] | |[https://xenpaper.com/#%7B1%2F1_9%2F8_5%2F4_27%2F20_3%2F2_5%2F3_9%2F5_2%2F1_9%2F4_5%2F2_27%2F10_3%2F1_10%2F3_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--- Mixolydian bright major]* | ||
|- | |- | ||
|1 | |1 | ||
| Line 169: | Line 171: | ||
|sssLsss | |sssLsss | ||
|Symmetric minor | |Symmetric minor | ||
|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_3%2F2_5%2F3_9%2F5_2%2F1_20%2F9_12%2F5_8%2F3_3%2F1_10%2F3_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--- Dorian symmetric minor] | |[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_3%2F2_5%2F3_9%2F5_2%2F1_20%2F9_12%2F5_8%2F3_3%2F1_10%2F3_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--- Dorian symmetric minor]* | ||
|- | |- | ||
| -1 | | -1 | ||
| Line 178: | Line 180: | ||
|ssLssss | |ssLssss | ||
|Bright minor | |Bright minor | ||
|[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_27%2F20_3%2F2_81%2F50_9%2F5_2%2F1_54%2F25_12%2F5_27%2F10_3%2F1_81%2F25_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--- Phrygian bright minor] | |[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_27%2F20_3%2F2_81%2F50_9%2F5_2%2F1_54%2F25_12%2F5_27%2F10_3%2F1_81%2F25_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--- Phrygian bright minor]* | ||
|- | |- | ||
| -2 | | -2 | ||
| Line 206: | Line 208: | ||
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]]. | 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" | {| class="wikitable" | ||
|+Modes of the ptolemismic Pinetone diatonic | |+Table 4. Modes of the ptolemismic Pinetone diatonic | ||
!Mode number | !Mode number | ||
!Pinetone diatonic mode | !Pinetone diatonic mode | ||
| Line 214: | Line 216: | ||
|- | |- | ||
|3 | |3 | ||
|Lydian dark major | |Lydian dark major* | ||
|mLmsmms | |mLmsmms | ||
|~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1 | |~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1 | ||
| Line 220: | Line 222: | ||
|- | |- | ||
|2 | |2 | ||
|Mixolydian bright minor | |Mixolydian bright minor* | ||
|Lmsmmsm | |Lmsmmsm | ||
|~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1 | |~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1 | ||
| Line 232: | Line 234: | ||
|- | |- | ||
|0 | |0 | ||
|Dorian symmetric minor | |Dorian symmetric minor* | ||
|msmLmsm | |msmLmsm | ||
|~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1 | |~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1 | ||
| Line 238: | Line 240: | ||
|- | |- | ||
| -1 | | -1 | ||
|Phrygian bright minor | |Phrygian bright minor* | ||
|smLmsmm | |smLmsmm | ||
|~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1 | |~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1 | ||