Pinetone: Difference between revisions
→Porcutone Diminished or Porcupine-Diminished: Fixed mistake in diminished chromatic Tags: Mobile edit Mobile web edit |
→Porcutone octatonic scales: added notes names |
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Modifying the right or left-handed Porcutone octatonic by switching the order of adjacent pairs of large and medium steps, i.e., by modifying steps of the scale by the L-M chroma - the difference between the large and medium steps - leads to similar Porcupine[8] detempers. Since L and M temper together under Porcupine tempering, any resulting scale tempers to Porcupine[8] just as before, but the scale it tempers to under Diminished and Father temperaments are modified. | Modifying the right or left-handed Porcutone octatonic by switching the order of adjacent pairs of large and medium steps, i.e., by modifying steps of the scale by the L-M chroma - the difference between the large and medium steps - leads to similar Porcupine[8] detempers. Since L and M temper together under Porcupine tempering, any resulting scale tempers to Porcupine[8] just as before, but the scale it tempers to under Diminished and Father temperaments are modified. | ||
Take the Ultharian dark major mode of the left-handed Porcutone octatonic, for example: LMLsMLML. Raising the sixth and eighth degrees of the scale by the L-M chroma leads to the mode LMLsLMLM. Similarly, taking the Mnarian middle major mode LMLMsLML of the right-handed Porcutone octatonic and | Take the Ultharian dark major mode of the left-handed Porcutone octatonic, for example: LMLsMLML. Raising the sixth and eighth degrees of the scale by the L-M chroma leads to the mode LMLsLMLM. Similarly, taking the Mnarian middle major mode LMLMsLML of the right-handed Porcutone octatonic and lowering the second and the fourth degree by the L-M chroma leads to the mode MLMLsLML. We can see that LMLsLMLM and MLMLsLML are modes of the same scale. We call these modes the dark major diminished and the middle major diminished respectively. | ||
In the Porcutone chromatic with sharps, which contains the left-handed Porcutone octatonic as the naturals plus G♯, the Ultharian dark major mode can be expressed as D E F G G♯ A B C. The Porcutone diminished mode on D, the dark major diminished, is therefore D E F G G♯ A♯ B C♯. In the Porcutone chromatic with flats, which contains the right-handed Porcutone octatonic as the naturals plus A♭, the Mnarian middle major mode can be expressed as D E F G A♭ A B C. The Porcutone diminished mode on D, the middle major diminished, is therefore D E♭ F G♭ A♭ A B C. | |||
We know that this scale tempers to Porcupine[8]; tempering M=s instead leads to LsLsLsLs, i.e., Diminished[8]; and finally tempering L=s leads to LsLLLsLs, a mod of Father[8]. Like the Porcutone chromatic and diatonic scale, this scale is an SN scale, and is therefor achiral. We may name this scale perhaps the Porcupine-Diminished scale, or we may include it in the Porcutone system as the Porcutone diminished scale. It may be more wise to refer to this scale as the Porcupine-Diminished scale to avoid confusion that might result if, while the octatonic scale and the diminished scale are different names for the same scale, the Porcutone octatonic and the Porcutone diminished scale are not. | |||
Every other step of any mode of the Porcutone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Porcutone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Porcutone octatonic. Although the Porcutone Diminished is simpler in this way, the Porcutone octatonic provides more major and minor traids. | Every other step of any mode of the Porcutone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Porcutone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Porcutone octatonic. Although the Porcutone Diminished is simpler in this way, the Porcutone octatonic provides more major and minor traids. | ||
| Line 1,988: | Line 1,992: | ||
|minor 3-step | |minor 3-step | ||
major 3-step | major 3-step | ||
minor 3-step | minor 3-step | ||
| Line 1,993: | Line 1,998: | ||
|minor 3-step | |minor 3-step | ||
minor 3-step | minor 3-step | ||
major 3-step | major 3-step | ||
| Line 1,998: | Line 2,004: | ||
|minor 3-step | |minor 3-step | ||
major-minor 3-step | major-minor 3-step | ||
minor-major 3-step | minor-major 3-step | ||
| Line 2,014: | Line 2,021: | ||
|2 | |2 | ||
1 | 1 | ||
2 | 2 | ||
| Line 2,074: | Line 2,082: | ||
|3 | |3 | ||
2 | 2 | ||
1 | 1 | ||
| Line 2,132: | Line 2,141: | ||
!Mode name | !Mode name | ||
!Step pattern | !Step pattern | ||
!3-step stacked triad on root | |||
!Diminished[8] name | !Diminished[8] name | ||
!Porcupine[8] name | !Porcupine[8] name | ||
| Line 2,139: | Line 2,149: | ||
|Bright quartal diminished | |Bright quartal diminished | ||
|LMLMLMLs | |LMLMLMLs | ||
|G♯-C♯-F, A-D-G♭ | |||
|[8] major | |[8] major | ||
|[8] augmented | |[8] augmented | ||
| Line 2,146: | Line 2,157: | ||
|Bright major diminished | |Bright major diminished | ||
|LMLMLsLM | |LMLMLsLM | ||
|B-E-G♯, C-F-A | |||
|[8] major | |[8] major | ||
|[8] major | |[8] major | ||
| Line 2,153: | Line 2,165: | ||
|Dark quartal diminished | |Dark quartal diminished | ||
|MLMLMLsL | |MLMLMLsL | ||
|A♯-D-G, B-E♭-A♭ | |||
|[8] minor | |[8] minor | ||
|[8] augmented | |[8] augmented | ||
|[8] minor augmented | |[8] minor augmented | ||
|30:39:52 ( | |30:39:52 (16:21:28) | ||
|- | |- | ||
|Dark major diminished | |Dark major diminished | ||
|LMLsLMLM | |LMLsLMLM | ||
|D-G-B, E♭-A♭-C | |||
|[8] major | |[8] major | ||
|[8] major | |[8] major | ||
| Line 2,167: | Line 2,181: | ||
| Middle major diminished | | Middle major diminished | ||
|MLMLsLML | |MLMLsLML | ||
|C♯-F-A♯, D-G♭-B | |||
|[8] minor | |[8] minor | ||
|[8] major | |[8] major | ||
| Line 2,174: | Line 2,189: | ||
| Middle minor diminished | | Middle minor diminished | ||
|LsLMLMLM | |LsLMLMLM | ||
|F-A♯-D, G♭-B-E♭ | |||
|[8] major | |[8] major | ||
|[8] minor | |[8] minor | ||
| Line 2,181: | Line 2,197: | ||
| Bright minor diminished | | Bright minor diminished | ||
|MLsLMLML | |MLsLMLML | ||
|E-G♯-C♯, F-A-D | |||
|[8] minor | |[8] minor | ||
|[8] minor | |[8] minor | ||
| Line 2,188: | Line 2,205: | ||
| Dark minor diminished | | Dark minor diminished | ||
|sLMLMLML | |sLMLMLML | ||
|G-B-E, A♭-C-F | |||
|[8] minor | |[8] minor | ||
|[8] minor | |[8] minor | ||
| Line 2,193: | Line 2,211: | ||
|12:15:20 | |12:15:20 | ||
|} | |} | ||
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Porcutone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. | ||
We can extend the Porcutone Diminished into an alterative chromatic scale: Starting with the bright minor diminished scale, MLsLMLML, we add a small step into the bottom or top of every large step, leading to the scales LsMssMLsMLsM and LMssMsLMsLMs respectively, modes of mirror-inversions of one another. In 5-limit just intonation this pair of scales comprises 3 large steps of 27/25, 4 medium steps of 16/15, and 5 small steps of 25/24, i.e., | We can extend the Porcutone Diminished into an alterative chromatic scale: Starting with the bright minor diminished scale, MLsLMLML, we add a small step into the bottom or top of every large step, leading to the scales LsMssMLsMLsM and LMssMsLMsLMs respectively, modes of mirror-inversions of one another. In 5-limit just intonation this pair of scales comprises 3 large steps of 27/25, 4 medium steps of 16/15, and 5 small steps of 25/24, i.e., 27/25 9/8 6/5 5/4 125/96 25/18 3/2 25/16 5/3 9/5 15/8 2/1 and 27/25 144/125 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1 respectively. | ||
27/25 9/8 6/5 5/4 125/96 25/18 3/2 25/16 5/3 9/5 15/8 2/1 and 27/25 144/ | |||
We could call these scales, which temper to Diminished[12], the left and right-handed Porcutone Diminished chromatic. | We could call these scales, which temper to Diminished[12], the left and right-handed Porcutone Diminished chromatic. | ||