Pinetone: Difference between revisions
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Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this Porcupine thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing The | Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this Porcupine thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing The Pinetone System. The scales you know and love, with a new-age quirky spin. The perfect mix of consonant and dissonant harmonies, familiar and newfangled. Try it on your keyboard straight away (if you can retune your keyboard using Scala files, grab [[Porcutone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file). | ||
The | The Pinetone system combines [[Porcupine]] – arguably the best way to add the 11th harmonic to major and minor harmonies in a seven-note scale – with [[Meantone]] – the system underpinning most common practice music from the last several hundred years, so all the same scales (diatonic, harmonic minor, pentatonic, chromatic, etc.) are still available, just with a new Porcupine spin, and the 11th harmonic (and the 13th harmonic as well!) | ||
While there aren't as many consonant major and minor triads as we are used to, they are more consonant in | While there aren't as many consonant major and minor triads as we are used to, they are more consonant in Pinetone. | ||
As opposed to in [[12edo]], each key is distinctly different in | As opposed to in [[12edo]], each key is distinctly different in Pinetone, both a blessing and a curse. | ||
Additionally available in | Additionally available in Pinetone are a set of octatonic modes with their own Porcupine functional harmony, that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. | ||
If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the | If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the Pinetone diatonic, a cross between the Meantone diatonic scale and Porcupine[7], and then black keys give the Pinetone pentatonic, which approximates the [[just intonation]] pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G♯/A♭ key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G♯/A♭ key a different colour since that's the one chromatic key used along with the diatonic keys to make the Pinetone octatonic. | ||
If you don't have a [[Lumatone]], no worries, you can set it up just fine on any keyboard! | If you don't have a [[Lumatone]], no worries, you can set it up just fine on any keyboard! | ||
== The | == The Pinetone diatonic == | ||
The diatonic scale has a [[Step pattern|step signature]] of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the [[step pattern]] LsLLLsL (represented in mode 0, Dorian mode). In [[Meantone]][7], the large step represents both [[9/8]] and [[10/9]], the major and minor tones (''tempering out'' the [[81/80]] interval that separates them) hence the name "Meantone". The small step represents [[16/15]] and [[27/25]] (which differ again by [[81/80]]). We write this in the form of a [[Step pattern|step signature]] and ''step mapping'' as [[5L 2s]] = (9/8~10/9, 16/15~27/25). [[Porcupine]][7] instead has a [[Step pattern|step signature]] and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between [[10/9]] and [[27/25]], i.e., [[250/243]], is tempered out. In mode 0 it has [[step pattern]] sssLsss. [[81/80]] is called the [[Meantone comma]], and [[250/243]] is called the [[Porcupine]] comma. | The diatonic scale has a [[Step pattern|step signature]] of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the [[step pattern]] LsLLLsL (represented in mode 0, Dorian mode). In [[Meantone]][7], the large step represents both [[9/8]] and [[10/9]], the major and minor tones (''tempering out'' the [[81/80]] interval that separates them) hence the name "Meantone". The small step represents [[16/15]] and [[27/25]] (which differ again by [[81/80]]). We write this in the form of a [[Step pattern|step signature]] and ''step mapping'' as [[5L 2s]] = (9/8~10/9, 16/15~27/25). [[Porcupine]][7] instead has a [[Step pattern|step signature]] and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between [[10/9]] and [[27/25]], i.e., [[250/243]], is tempered out. In mode 0 it has [[step pattern]] sssLsss. [[81/80]] is called the [[Meantone comma]], and [[250/243]] is called the [[Porcupine]] comma. | ||
We are familiar with the Zarlino/Ptolemy [[just]] major scale: 9/8 5/4 4/3 3/2 5/3 15/8 2/1. This scale has 3 large steps of [[9/8]], 2 medium steps of [[10/9]], and 2 small steps of [[16/15]], with [[step pattern]] LMsLMLs. If we temper out the difference between L and M, we get LLsLLLs, which, as mode 2 of [[Meantone]][7] is the familiar Ionian/major mode. | We are familiar with the Zarlino/Ptolemy [[just]] major scale: 9/8 5/4 4/3 3/2 5/3 15/8 2/1. This scale has 3 large steps of [[9/8]], 2 medium steps of [[10/9]], and 2 small steps of [[16/15]], with [[step pattern]] LMsLMLs. If we temper out the difference between L and M, we get LLsLLLs, which, as mode 2 of [[Meantone]][7] is the familiar Ionian/major mode. | ||
Consider instead the [[just]] scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of [[9/8]], 4 medium steps of [[10/9]], and 2 small steps of [[27/25]], with step pattern MsMLMsM (mode 0). It can be represented with [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our [[just]] | Consider instead the [[just]] scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of [[9/8]], 4 medium steps of [[10/9]], and 2 small steps of [[27/25]], with step pattern MsMLMsM (mode 0). It can be represented with [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our [[just]] Pinetone diatonic. If we temper out the difference between L and M, we get LsLLLsL, [[Meantone]][7] mode 0: Dorian; if we temper out instead the difference between [[10/9]] and [[27/25]], we get sssLsss, [[Porcupine]][7] mode 0, which is referred to as ''symmetric minor''. In this way, the [[just]] Pinetone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Pinetone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode names introduced in Table 1., with the [[Meantone]] diatonic mode names referenced in Table 2., so mode 0 of the Pinetone diatonic is called ''Dorian symmetric minor''. We continue this naming process with the other 6 modes to arrive at the modes shown in Table 3. | ||
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]] | 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. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Table 1. Modes of 5-limit Porcupine[7] | |+Table 1. Modes of 5-limit Porcupine[7] | ||
| Line 122: | Line 122: | ||
|} | |} | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Table 3. Modes of the just | |+Table 3. Modes of the just Pinetone diatonic | ||
!Mode number | !Mode number | ||
!Mode in JI | !Mode in JI | ||
| Line 130: | Line 130: | ||
!Porcupine[7] | !Porcupine[7] | ||
!Porcupine[7] mode | !Porcupine[7] mode | ||
! | !Pinetone diatonic mode | ||
|- | |- | ||
|3 | |3 | ||
| Line 195: | Line 195: | ||
|Locrian dark diminished | |Locrian dark diminished | ||
|} | |} | ||
Like [[Meantone]][7] and [[Porcupine]][7], and unlike the Ptolemy/Zarlino just major scale, the | 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. | ||
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 | 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 | 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. | ||
The modes of the ptolemismic | 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 | |+Modes of the ptolemismic Pinetone diatonic | ||
!Mode number | !Mode number | ||
! | !Pinetone diatonic mode | ||
!Step pattern | !Step pattern | ||
!Mode as simplest JI pre-image | !Mode as simplest JI pre-image | ||
| Line 254: | Line 254: | ||
=== Tuning options === | === Tuning options === | ||
We see 11/8 as the 4th in Lydian dark major. In Meantone[7] this is an augmented fourth. The Meantone extension representing 11/8 with an augmented fourth is called [[Meanenneadecal]], referencing the fact that it is most at home in [[19edo]]. Tuning the scale to 19edo (or 12edo) will collapse it into a Meanenneadecal[7] diatonic scale. Similarly, tuning the scale to 15edo, 22edo, or 29edo will collapse it to Porcupine[7] scale. 27edo, 34edo, and 41edo are good tunings for the | We see 11/8 as the 4th in Lydian dark major. In Meantone[7] this is an augmented fourth. The Meantone extension representing 11/8 with an augmented fourth is called [[Meanenneadecal]], referencing the fact that it is most at home in [[19edo]]. Tuning the scale to 19edo (or 12edo) will collapse it into a Meanenneadecal[7] diatonic scale. Similarly, tuning the scale to 15edo, 22edo, or 29edo will collapse it to Porcupine[7] scale. 27edo, 34edo, and 41edo are good tunings for the Pinetone diatonic if tuning to an edo is desired. | ||
27edo: 1L 4m 2s = (5, 4, 3) = (222.2222c, 177.7778c, 133.3333c) | 27edo: 1L 4m 2s = (5, 4, 3) = (222.2222c, 177.7778c, 133.3333c) | ||
| Line 279: | Line 279: | ||
=== Intervals and chords === | === Intervals and chords === | ||
The table below show the sizes, interval names, ratios approximated, tuning, and occurrence of all intervals of the ptolemismic | The table below show the sizes, interval names, ratios approximated, tuning, and occurrence of all intervals of the ptolemismic Pinetone diatonic scale within an octave, tuned to TE tuning. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Intervals of the | |+Intervals of the Pinetone diatonic | ||
!Interval class | !Interval class | ||
!size | !size | ||
!Meantone[7] name | !Meantone[7] name | ||
!Porcupine[7] name | !Porcupine[7] name | ||
! | !Pinetone diatonic names | ||
!JI ratios approximated | !JI ratios approximated | ||
!size in cents (TE) | !size in cents (TE) | ||
| Line 473: | Line 473: | ||
|} | |} | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Tertian triads of the | |+Tertian triads of the Pinetone diatonic on D | ||
!Root note | !Root note | ||
!Triad notes | !Triad notes | ||
!Meantone triad | !Meantone triad | ||
!Porcupine triad | !Porcupine triad | ||
! | !Pinetone triad | ||
!JI chord approximated by triad | !JI chord approximated by triad | ||
|- | |- | ||
| Line 533: | Line 533: | ||
Porcupine tetrads in the table below are named after the third above the tonic and the third above the fifth, apart from tetrads with a diminished fifth. | Porcupine tetrads in the table below are named after the third above the tonic and the third above the fifth, apart from tetrads with a diminished fifth. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Tertian tetrads of the | |+Tertian tetrads of the Pinetone diatonic on D | ||
!Root note | !Root note | ||
!Triad notes | !Triad notes | ||
!Meantone tetrad | !Meantone tetrad | ||
!Porcupine tetrad | !Porcupine tetrad | ||
! | !Pinetone tetrad | ||
!JI chord approximated by tetrad | !JI chord approximated by tetrad | ||
|- | |- | ||
| Line 593: | Line 593: | ||
==== Quartal Triads ==== | ==== Quartal Triads ==== | ||
Also of interest are the quartal triads of the | Also of interest are the quartal triads of the Pinetone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, with major and minor designating the large and small 3-step intervals (fourths) respectively. This may seem an odd way to describe quartal chords, but it is consistent with the naming scheme I introduce for the Porcupine[7] 3-step (quartal) triads along side the quartal triads of the Pinetone diatonic. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+3-step stacked triads of the | |+3-step stacked triads of the Pinetone diatonic on D | ||
!Root note | !Root note | ||
!Triad notes | !Triad notes | ||
!Meantone triad | !Meantone triad | ||
!Porcupine[7] triad | !Porcupine[7] triad | ||
! | !Pinetone triad | ||
!JI chord approximated by triad | !JI chord approximated by triad | ||
|- | |- | ||
| Line 653: | Line 653: | ||
|} | |} | ||
== The | == The Pinetone pentatonic and the Pinetone chromatic == | ||
We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the | We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the Pinetone pentatonic is a subset of the Pinetone diatonic. We also know that adding a (meantone) pentatonic to a (meantone) diatonic leads to a (meantone) chromatic, i.e., diatonic on white keys + pentatonic on black keys. We can do this with Pinetone. | ||
=== | === Pinetone pentatonic === | ||
Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the | Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the Pinetone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the Pinetone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the Pinetone harmonic minor available, and all flats makes the Pinetone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E. | ||
We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The | We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The Pinetone pentatonic is also a subset of the Pinetone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The Pinetone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the Pinetone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of Pinetone pentatonic scales, the right handed Pinetone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed Pinetone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed Pinetone pentatonic, while G-A-B-D-E, F♯-G♯-A♯-C♯-D♯, and G♭-A♭-B♭-D♭-E♭ are right handed Pinetone pentatonics. | ||
=== | === Pinetone chromatic === | ||
Adding the right handed | Adding the right handed Pinetone pentatonic (on F♯/G♭) to the just Pinetone diatonic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just Pinetone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL. | ||
The now familiar Meantone comma of 81/80 separates the medium step (25/24) from the small step (250/243), so our | The now familiar Meantone comma of 81/80 separates the medium step (25/24) from the small step (250/243), so our Pinetone chromatic is a ''detempering'' of Meantone[12], the meantone chromatic scale, just like how the Pinetone diatonic is a detempering of Meantone[7], the meantone diatonic scale. | ||
The ptolemismic | The ptolemismic Pinetone chromatic has a step signature, mapping, and TE tuning of 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.6352c, 63.1434c, 27.4197c). | ||
Mode -3 approximates the JI ratios: 55/54 10/9 6/5 11/9 4/3 11/8 3/2 55/36 5/3 9/5 11/6 2/1. | Mode -3 approximates the JI ratios: 55/54 10/9 6/5 11/9 4/3 11/8 3/2 55/36 5/3 9/5 11/6 2/1. | ||
| Line 676: | Line 676: | ||
The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269 | The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269 | ||
Note the more complex intervals: 55/54, 55/36, 72/55, and 108/55. If we temper out an additional comma, we can equate these with simpler intervals, adding prime 13: Tempering out 144/143, these four interval approximate 40/39, 20/13, 13/10, and 39/20 respectively. Tempering out 144/143 also means that the small step of the | Note the more complex intervals: 55/54, 55/36, 72/55, and 108/55. If we temper out an additional comma, we can equate these with simpler intervals, adding prime 13: Tempering out 144/143, these four interval approximate 40/39, 20/13, 13/10, and 39/20 respectively. Tempering out 144/143 also means that the small step of the Pinetone diatonic, equivalently the large step of the Pinetone chromatic approximates 13/12, which, when all three are justly tuned, lies between the other intervals approximated by the step - 27/25, and 12/11. | ||
This leads to a step signature, mapping, and TE tuning of 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c). | This leads to a step signature, mapping, and TE tuning of 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c). | ||
| Line 692: | Line 692: | ||
If a full 13-limit tuning is desired, there are two options. The interval approximating 13/10 may either be tempered to approximate 21/16, leading to Supermagic, or 7/6, leading to Thrasher. The Supermagic tuning decreases the size of the small step, and the Starling tuning increases it. The Supermagic tuning reduces to Flattone (where 7/4 is found at a diminished 7th) and Porcupine (where 7/4 is found at a minor seventh), and the Starling tuning reduces to Meanenneadecal and Opossum (both where 7/4 is found at an augmented 6th). If we temper to 13/10 to equate to both 9/7 and 21/16, we get Keema, an extension of Hanson temperament. Keema[7] comprises 4 large steps of 247.695c, and 3 small steps of 69.682c. | If a full 13-limit tuning is desired, there are two options. The interval approximating 13/10 may either be tempered to approximate 21/16, leading to Supermagic, or 7/6, leading to Thrasher. The Supermagic tuning decreases the size of the small step, and the Starling tuning increases it. The Supermagic tuning reduces to Flattone (where 7/4 is found at a diminished 7th) and Porcupine (where 7/4 is found at a minor seventh), and the Starling tuning reduces to Meanenneadecal and Opossum (both where 7/4 is found at an augmented 6th). If we temper to 13/10 to equate to both 9/7 and 21/16, we get Keema, an extension of Hanson temperament. Keema[7] comprises 4 large steps of 247.695c, and 3 small steps of 69.682c. | ||
The ptolemismic | The ptolemismic Pinetone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale. | ||
=== Intervals and triads === | === Intervals and triads === | ||
| Line 703: | Line 703: | ||
Mode 3 has 4:5:6 major triads available above E♭, E, F, G♭, and G. | Mode 3 has 4:5:6 major triads available above E♭, E, F, G♭, and G. | ||
The following tables show the (3, 4) and (4, 3) triads available of mode 3 and mode -3 of the | The following tables show the (3, 4) and (4, 3) triads available of mode 3 and mode -3 of the Pinetone chromatic scale: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+(3, 4) and (4, 3) triads of the | |+(3, 4) and (4, 3) triads of the Pinetone chromatic mode -3 | ||
!Note | !Note | ||
!Triad class | !Triad class | ||
!Triad in meantone | !Triad in meantone | ||
!Triad in porcupine | !Triad in porcupine | ||
! | !Pinetone triad name | ||
!JI triads approximated | !JI triads approximated | ||
!Triads in cents | !Triads in cents | ||
| Line 905: | Line 905: | ||
6L 1s = (10/9~11/10~28/25, 27/25~15/14~12/11~13/12) = (176.8600, 136.3262) as Wollemia. | 6L 1s = (10/9~11/10~28/25, 27/25~15/14~12/11~13/12) = (176.8600, 136.3262) as Wollemia. | ||
We can see that the large step of Tetracot[7] is the medium step of the | We can see that the large step of Tetracot[7] is the medium step of the Pinetone diatonic, and the small step of Tetracot[7] is the small step of the Pinetone diatonic. The large step of the Pinetone diatonic is the augmented second of tetracot[7]. | ||
=== Tuning options === | === Tuning options === | ||
As with the | As with the Pinetone diatonic, tuning the Pinetone chromatic to 19edo collapses it to the Meantone[12] (Flattone[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step signatures, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows: | ||
27edo: 7L 1m 4s = (3, 2, 1) = (133.3333c, 88.8889c, 44.4444c) (dim min 4 is 9/7 - Starling) | 27edo: 7L 1m 4s = (3, 2, 1) = (133.3333c, 88.8889c, 44.4444c) (dim min 4 is 9/7 - Starling) | ||
| Line 916: | Line 916: | ||
41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) (dim min 4 is 21/16 - Supermagic) | 41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) (dim min 4 is 21/16 - Supermagic) | ||
All three of these edos also temper out 243/242, so the major minor and minor major thirds collapse to a single interval - the neutral third, and the | All three of these edos also temper out 243/242, so the major minor and minor major thirds collapse to a single interval - the neutral third, and the Pinetone diatonic can be considered a MODMOS of Tetracot[7] i.e. Pinetone msmLmsm = Tetracot LsLALsL. | ||
And allowing octave stretch, the tuning may be optimized via TE tuning to: | And allowing octave stretch, the tuning may be optimized via TE tuning to: | ||
| Line 930: | Line 930: | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264f&limit=2.3.5.11.13 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c)], | [http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264f&limit=2.3.5.11.13 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c)], | ||
the TE step signature, mapping, and sizes for the 13-limit Supermagic | the TE step signature, mapping, and sizes for the 13-limit Supermagic Pinetone chromatic is | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=7p%261cde%264f&limit=13 7L 1m 4s = (27/25~12/11~13/12~35/32, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39~64/63) = (145.47082c, 58.39270c, 30.85183c)], | [http://x31eq.com/cgi-bin/rt.cgi?ets=7p%261cde%264f&limit=13 7L 1m 4s = (27/25~12/11~13/12~35/32, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39~64/63) = (145.47082c, 58.39270c, 30.85183c)], | ||
and the TE step signature, mapping, and sizes for the 13-limit Thrasher | and the TE step signature, mapping, and sizes for the 13-limit Thrasher Pinetone chromatic is | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=7d%261cdde%264f&limit=13 7L 1m 4s = (27/25~15/14~12/11~13/12, 25/24~21/20~33/32~27/26, 250/243~28/27~55/54~121/120~40/39) = (136.27690c, 81.02531c, 40.63434c)], | [http://x31eq.com/cgi-bin/rt.cgi?ets=7d%261cdde%264f&limit=13 7L 1m 4s = (27/25~15/14~12/11~13/12, 25/24~21/20~33/32~27/26, 250/243~28/27~55/54~121/120~40/39) = (136.27690c, 81.02531c, 40.63434c)], | ||
and if optimization just to the 2.3.5.11 subgroup is desired,TE step signature, mapping, and sizes for the (2.3.5.11) ptolemismic | and if optimization just to the 2.3.5.11 subgroup is desired,TE step signature, mapping, and sizes for the (2.3.5.11) ptolemismic Pinetone chromatic is | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264p&limit=2.3.5.11 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c)]. | [http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264p&limit=2.3.5.11 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c)]. | ||
| Line 946: | Line 946: | ||
7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 ptolemismic. | 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 ptolemismic. | ||
== | == Pinetone octatonic scales == | ||
The porcupine comma is the small step of the scale, so tempering the | The porcupine comma is the small step of the scale, so tempering the Pinetone chromatic scale to porcupine leads from 7L 1m 4s = (27/25, 25/24, 250/243) to 7L 1s = (10/9~27/25, 25/24~81/80), which is Porcupine[8]! The Porcupine[7] scale has its large step between G and A, so the eighth note of Porcupine[8] is either G♯ or A♭, adding another small step of Porcupine[7] below A (for G♯) or above G (A♭). Mode -3 or mode 3 of the Pinetone chromatic scale, respectively, are set to D so that this is preserved in The Pinetone System. This leads to the Pinetone octatonic scales: D E F G G♯/A♭ A B C. In Just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the Pinetone diatonic (the Zarlino/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The mirror inverse of any mode of the Pinetone octatonic with G♯ is a mode of the Pinetone octatonic with A♭. The Pinetone octatonic with G♯ is called the left handed porcupine octatonic, and the Pinetone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]). | ||
On a keyboard with standard (Bosanquet or 12edo) mapping, the | On a keyboard with standard (Bosanquet or 12edo) mapping, the Pinetone octatonic is the C Major bebop scale! On my [[Lumatone]] I chose to colour the G♯/A♭ pink, and the rest of the chromatic notes blue, so the Pinetone octatonic is on the white and pink keys, while there's a Pinetone diatonic on the white keys and a Pinetone pentatonic on the blue and pink keys. | ||
If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' [[UDP]] are show in the table. A mode's 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). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our | If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' [[UDP]] are show in the table. A mode's 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). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our Pinetone chromatic and octatonic scales. | ||
The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80) | The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80) | ||
| Line 1,108: | Line 1,108: | ||
|200:243:324 | |200:243:324 | ||
|} | |} | ||
For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the [[oneirotonic]] mode names associated with Father[8]. Like in the tables of modes of the | For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the [[oneirotonic]] mode names associated with Father[8]. Like in the tables of modes of the Pinetone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Modes of the left handed just | |+Modes of the left handed just Pinetone octatonic | ||
!Mode in JI | !Mode in JI | ||
!Step pattern | !Step pattern | ||
| Line 1,121: | Line 1,121: | ||
!Oneirotonic | !Oneirotonic | ||
mode | mode | ||
! | !Pinetone octatonic | ||
mode | mode | ||
|- | |- | ||
| Line 1,189: | Line 1,189: | ||
|} | |} | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Modes of the right handed just | |+Modes of the right handed just Pinetone octatonic | ||
!Mode in JI | !Mode in JI | ||
!Step pattern | !Step pattern | ||
| Line 1,200: | Line 1,200: | ||
!Oneirotonic | !Oneirotonic | ||
mode | mode | ||
! | !Pinetone octatonic | ||
mode | mode | ||
|- | |- | ||
| Line 1,269: | Line 1,269: | ||
Note that the darkest mode of the LH octatonic is the brightest mode of the RH octatonic, etc. | Note that the darkest mode of the LH octatonic is the brightest mode of the RH octatonic, etc. | ||
Tempering out 100/99, the large step (174.05488c) represents 10/9~11/10, the medium step (146.63528c) represents 27/25~12/11, and the small step (63.14327c) represents 25/24~33/32. The following tables display the JI intervals approximated by the modes of the ptolemismic | Tempering out 100/99, the large step (174.05488c) represents 10/9~11/10, the medium step (146.63528c) represents 27/25~12/11, and the small step (63.14327c) represents 25/24~33/32. The following tables display the JI intervals approximated by the modes of the ptolemismic Pinetone octatonic scales, along with the scale steps in cents. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4p%263p%261ce&limit=2.3.5.11 TE tuning]. | ||
Tempering out 144/143 as well, the large step is tuned to 175.89183c TE, medium step (142.77537c TE) also represents 13/12, and the small step (66.76626c TE) also represents 27/26. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning]. | Tempering out 144/143 as well, the large step is tuned to 175.89183c TE, medium step (142.77537c TE) also represents 13/12, and the small step (66.76626c TE) also represents 27/26. See [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning]. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Modes of the left handed ptolemismic | |+Modes of the left handed ptolemismic Pinetone octatonic | ||
! | !Pinetone ocatonic mode | ||
!Step pattern | !Step pattern | ||
!Mode as simplest JI pre-image | !Mode as simplest JI pre-image | ||
| Line 1,320: | Line 1,320: | ||
|} | |} | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Modes of the right handed ptolemismic | |+Modes of the right handed ptolemismic Pinetone octatonic | ||
! | !Pinetone ocatonic mode | ||
!Step pattern | !Step pattern | ||
!Mode as simplest JI pre-image | !Mode as simplest JI pre-image | ||
| Line 1,368: | Line 1,368: | ||
=== Intervals and chords === | === Intervals and chords === | ||
The following table gives all intervals of the | The following table gives all intervals of the Pinetone octatonic. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Intervals of the | |+Intervals of the Pinetone octatonic | ||
!Interval class | !Interval class | ||
!sizes | !sizes | ||
!Oneirotonic name | !Oneirotonic name | ||
!Porcupine[8] name | !Porcupine[8] name | ||
! | !Pinetone octatonic name | ||
!JI ratios approximated* | !JI ratios approximated* | ||
!size in cents (TE) | !size in cents (TE) | ||
| Line 1,639: | Line 1,639: | ||
1 | 1 | ||
|} | |} | ||
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic). | ||
The following two tables detail the 3-step stacked triads of the left and right handed | The following two tables detail the 3-step stacked triads of the left and right handed Pinetone octatonics: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+3-step stacked triads of the left handed | |+3-step stacked triads of the left handed Pinetone octatonic (G♯-G gamut) | ||
!Mode name | !Mode name | ||
!Step pattern | !Step pattern | ||
| Line 1,649: | Line 1,649: | ||
!Oneirotonic name | !Oneirotonic name | ||
!Porcupine[8] name | !Porcupine[8] name | ||
! | !Pinetone octatonic name | ||
!JI triad approximated* | !JI triad approximated* | ||
|- | |- | ||
| Line 1,716: | Line 1,716: | ||
|12:15:20 | |12:15:20 | ||
|} | |} | ||
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic). | ||
{| class="wikitable" | {| class="wikitable" | ||
|+3-step stacked triads of the right handed | |+3-step stacked triads of the right handed Pinetone octatonic (G-A♭ gamut) | ||
!Mode name | !Mode name | ||
!Step pattern | !Step pattern | ||
| Line 1,724: | Line 1,724: | ||
!Oneirotonic name | !Oneirotonic name | ||
!Porcupine[8] name | !Porcupine[8] name | ||
! | !Pinetone octatonic name | ||
!JI triad approximated* | !JI triad approximated* | ||
|- | |- | ||
| Line 1,792: | Line 1,792: | ||
|} | |} | ||
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. | ||
We could alternatively treat the | We could alternatively treat the Pinetone octatonic as a bebop scale, using 2-step stacked tetrads. Since the scale has 8 notes, there are only 2 different 2-step stacked tetrads. In 12edo these are the major add 6 and the fully diminished tetrads. The meantone C major add 6 tunes to 45:55:66:75 in Pinetone. Using the G♯, as in the left-handed Pinetone octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the right-handed Pinetone octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note). | ||
Unlike the | Unlike the Pinetone diatonic, and chromatic scales, the Pinetone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the Pinetone diatonic. The Pinetone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the Pinetone octatonic is more complex than the Pinetone pentatonic as well. It is left as an exercise for the reader to determine the complexity of the Pinetone chromatic, and compare that to the Pinetone octatonic. | ||
=== | === Pinetone Diminished or Porcupine-Diminished === | ||
Modifying the right or left-handed | Modifying the right or left-handed Pinetone 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 | Take the Ultharian dark major mode of the left-handed Pinetone 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 Pinetone 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 | In the Pinetone chromatic with sharps, which contains the left-handed Pinetone octatonic as the naturals plus G♯, the Ultharian dark major mode can be expressed as D E F G G♯ A B C. The Pinetone diminished mode on D, the dark major diminished, is therefore D E F G G♯ A♯ B C♯. In the Pinetone chromatic with flats, which contains the right-handed Pinetone octatonic as the naturals plus A♭, the Mnarian middle major mode can be expressed as D E F G A♭ A B C. The Pinetone 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 | 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 Pinetone 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 Pinetone system as the Pinetone 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 Pinetone octatonic and the Pinetone diminished scale are not. | ||
Every other step of any mode of the | Every other step of any mode of the Pinetone Diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Pinetone Diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Pinetone octatonic. Although the Pinetone Diminished is simpler in this way, the Pinetone octatonic provides more major and minor triads. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Modes of the just | |+Modes of the just Pinetone Diminished scale (Porcupine-Diminished) | ||
!Mode in JI | !Mode in JI | ||
!Step pattern | !Step pattern | ||
| Line 1,819: | Line 1,819: | ||
!Diminished[8] | !Diminished[8] | ||
step pattern and UDP | step pattern and UDP | ||
! | !Pinetone diminished | ||
mode | mode | ||
|- | |- | ||
| Line 1,879: | Line 1,879: | ||
|} | |} | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Modes of the Ptolemismic | |+Modes of the Ptolemismic Pinetone Diminished scale (Porcupine-Diminished) | ||
!Mode name | !Mode name | ||
!Step pattern | !Step pattern | ||
| Line 1,926: | Line 1,926: | ||
|} | |} | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Intervals of the | |+Intervals of the Pinetone diminished (Porcupine-Diminished) | ||
!Interval class | !Interval class | ||
!sizes | !sizes | ||
!Diminished[8] name | !Diminished[8] name | ||
!Porcupine[8] name | !Porcupine[8] name | ||
! | !Pinetone octatonic name | ||
!JI ratios approximated* | !JI ratios approximated* | ||
!size in cents (TE) | !size in cents (TE) | ||
| Line 2,137: | Line 2,137: | ||
1 | 1 | ||
|} | |} | ||
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic). | ||
{| class="wikitable" | {| class="wikitable" | ||
|+3-step stacked triads of the | |+3-step stacked triads of the Pinetone diminished (Porcupine-Diminished) | ||
!Mode name | !Mode name | ||
!Step pattern | !Step pattern | ||
| Line 2,146: | Line 2,146: | ||
!Diminished[8] name | !Diminished[8] name | ||
!Porcupine[8] name | !Porcupine[8] name | ||
! | !Pinetone octatonic name | ||
!JI triad approximated* | !JI triad approximated* | ||
|- | |- | ||
| Line 2,213: | Line 2,213: | ||
|12:15:20 | |12:15:20 | ||
|} | |} | ||
<nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic | <nowiki>*</nowiki> Non-bracketed JI ratios are those approximated in 2.3.5.11.13 ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. | ||
The following 13 notes are used in total for these scales: E♭, G♭, A♭ D, E, F, G, A, B, C, G♯, A♯, C♯ | The following 13 notes are used in total for these scales: E♭, G♭, A♭ D, E, F, G, A, B, C, G♯, A♯, C♯ | ||
==== | ==== Pinetone Diminished chromatic ==== | ||
We can extend the | We can extend the Pinetone 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. In other modes, they can be expressed as 25/24 10/9 6/5 5/4 4/3 36/25 3/2 8/5 5/3 125/72 50/27 2/1, and 25/24 10/9 125/108 5/4 4/3 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1, i.e., sMLsMLsMssML and sMsLMsLMsLMs respectively. | ||
We could call these scales, which temper to Diminished[12], the left and right-handed | We could call these scales, which temper to Diminished[12], the left and right-handed Pinetone Diminished chromatic. | ||
Tempering M=L alternatively leads to sLLsLLsLssLL and sLsLLsLLsLLs, which are MODMOS of Meantone[12], i.e., C C♯ D E♭ E F G♭ G A♭ A A♯ B and G G♯ A A♯ B C C♯ D E♭ E F G♭. | Tempering M=L alternatively leads to sLLsLLsLssLL and sLsLLsLLsLLs, which are MODMOS of Meantone[12], i.e., C C♯ D E♭ E F G♭ G A♭ A A♯ B and G G♯ A A♯ B C C♯ D E♭ E F G♭. | ||
| Line 2,230: | Line 2,230: | ||
mLm(sm)Lm(sm)Lm(sm)Lm | mLm(sm)Lm(sm)Lm(sm)Lm | ||
mLmsmLmsmLmsmLm, which we later introduce as | mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15. | ||
== Summary for xen-math nerds == | == Summary for xen-math nerds == | ||
The | The Pinetone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note SNS 6/5 3/2 9/5 2/1. | ||
The | The Pinetone chromatic is a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys. | ||
For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue). | For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue). | ||
The | The Pinetone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) comprising 1 large step of 9/8 (''L'' x ''L''), 3 medium steps of 10/9 (''L'' x ''s''), and 3 small steps of 27/25 (''s'' x ''s''). | ||
The | The Pinetone octatonic is an 8-note rank-3 [[Porcupine]][8] x [[Father]][8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 16/15, and 648/625; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''L''). | ||
<nowiki>:</nowiki>The | <nowiki>:</nowiki>The Pinetone diminished scale is a [[step-nested scale]] and a [[Porcupine]][8] x Diminished[8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 648/625, and 16/15; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''s''). | ||
* | * | ||
== | == Pinetone harmonic minor and harmonic major == | ||
Additionally, we have another set of [[Porcupine]][7] modes contained in the | Additionally, we have another set of [[Porcupine]][7] modes contained in the Pinetone octatonic: Replacing the G with the G♯ changes the mode of the Porcupine[7] scale represented, and replaces diatonic with harmonic minor modes for the [[Meantone]][7] scale represented, now a MODMOS. | ||
We note that there are fewer consonant triads available in these scales than in the | We note that there are fewer consonant triads available in these scales than in the Pinetone diatonic and octatonic scales, so they may be useful for melody only. | ||
On D we get the scale: | On D we get the scale: | ||
| Line 2,256: | Line 2,256: | ||
174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 on the notes D E F G♯ A B C D | 174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 on the notes D E F G♯ A B C D | ||
We get the following 7 modes of | We get the following 7 modes of Pinetone harmonic minor scale: | ||
* Lsmsmms Lydian ♯2 bright major | * Lsmsmms Lydian ♯2 bright major | ||
| Line 2,270: | Line 2,270: | ||
174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 | 174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 | ||
Which has | Which has Pinetone harmonic major modes: | ||
* Lsmmsms Lydian Augmented ♯2 bright major | * Lsmmsms Lydian Augmented ♯2 bright major | ||
| Line 2,280: | Line 2,280: | ||
* smmsmsL Locrian magical ♭♭7 | * smmsmsL Locrian magical ♭♭7 | ||
== | == Pinetone hyperchromatic scales == | ||
Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos? | Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos? | ||
Indeed we can have both! | Indeed we can have both! | ||
From the | From the Pinetone chromatic with sharps (mode -3), we add another Pinetone diatonic scale, mode 0 starting on D♭, leading to the left-handed Pinetone hyperchromatic scale, with step pattern, sLsLssLsmLssLsLssLs. | ||
Or, from the | Or, from the Pinetone chromatic with flats (mode 3), we add another Pinetone diatonic scale, mode 0 starting on D♯, leading to the right-handed Pinetone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs. | ||
If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the | If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39. | ||
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c). | We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c). | ||
In 2.3.5.11.13 Tetracot, the left handed | In 2.3.5.11.13 Tetracot, the left handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1. | ||
Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left handed | Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left handed Pinetone hyperchromatic in cents is | ||
33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917, | 33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917, | ||
and the right handed | and the right handed Pinetone hyperchromatic in cents is | ||
33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917. | 33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917. | ||
The | The Pinetone hyperchromatic scales may alternatively be tuned to 27edo, 34edo, or 41edo: | ||
27edo: 7L 1m 11s = (2, 2, 1) = (88.8889c, 88.8889c, 44.4444c) | 27edo: 7L 1m 11s = (2, 2, 1) = (88.8889c, 88.8889c, 44.4444c) | ||
| Line 2,311: | Line 2,311: | ||
41edo: 7L 1m 11s = (4, 2, 1) = (117.0732c, 58.5366c, 29.2683c). | 41edo: 7L 1m 11s = (4, 2, 1) = (117.0732c, 58.5366c, 29.2683c). | ||
== | == Pinetone-15 == | ||
Alternatively, a 15-note scale can be built from the | Alternatively, a 15-note scale can be built from the Pinetone diminished. The resulting scale tempers to Porcupine[15], as well as to Hanson[11]. | ||
Starting instead with the | Starting instead with the Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished. Putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e., | ||
25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 48/25 2/1. | 25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 48/25 2/1. | ||
| Line 2,330: | Line 2,330: | ||
Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 48/25 2/1. | Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 48/25 2/1. | ||
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. | 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. Pinetone-15 comprises | ||
4 large steps of 109.12557c, approximating 16/15; | 4 large steps of 109.12557c, approximating 16/15; | ||
| Line 2,338: | Line 2,338: | ||
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 | In cents, TE 2.3.5.11.13 ptolemismic Pinetone-15, in the symmetric mode, is | ||
66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs. | 66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs. | ||
Accordingly | Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively. | ||
Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 ptolemismic tempering by the inclusion of 18/13 rather than 11/8. | Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 ptolemismic tempering by the inclusion of 18/13 rather than 11/8. | ||