Pinetone: Difference between revisions

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 Pinetone. 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 [[Pinetone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file).

Additionally available in Pinetone are a two sets of octatonic modes with their own Porcupine-like functional harmony that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. Finally, Pinetone diminished scales combine Porcupine with the familiar diminished scale.  

* [[Pinetone#Pinetone diatonic|Diatonic]]: 4546454, 6567656 (subset of major and minor-harmonic octatonics)

|+Table 2.3. Modes of the just Pinetone diatonic

|+Table 2.4. Modes of the Ptolemismic Pinetone diatonic

 

I find this tuning to be melodically superior, given the small step is 6 cents large, now a sixth tone rather than an eight tone.

 

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 Pinetone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.

 

== Pinetone Aeolian bright and dark minor, Dorian bright minor, and Locrian dark minor scales and their modes ==

Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in Porcupine, so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before), so will call the mode approximating the ratios 10/9 11/9 4/3 3/2 5/3 9/5 2/1 Mixolydian symmetric minor, and use the mode to name the scale as a whole.

To calculate the mode numbers for Tables 4.2-4.5, the mode numbers of their temperings to Porcupine and to Meantone were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties.

|+Table 4.2. Modes of the Ptolemismic Pinetone Aeolian bright minor

!Pinetone diatonic mode

|Ionian bright major*

|G

|~ 9/8 5/4 15/11 3/2 5/3 11/6 2/1

|Lydian bright diminished

|C

|~ 10/9 11/9 110/81 22/15 5/3 11/6 2/1

|Mixolydian dark minor

|Aeolian bright minor*

|Locian dark major

|Phrygian dark diminished

|Dorian magical seventh

|+Table 4.3. Modes of the Ptolemismic Pinetone Aeolian dark minor

!Pinetone diatonic mode

|Ionian dark major*

|Dorian bright major

|Lydian dark diminished

|Mixolydian bright diminished

|Aeolian dark minor*

|Locrian bright minor

|Phrygian magical seventh

Note that the Pinetone melodic major includes a neutral triad 18:22:27 on the root of the Mixolydian symmetric minor, and the Pinetone melodic minor includes it's inverse on the root of the Dorian bright major, a flavour of triad not available in the Pinetone diatonic. Note also though the only one major and one minor triad are available in each of these scales, as opposed to the two each available in the Pinetone diatonic, hence the labelling of these scales as for melodic purposes.  

|+Table 4.4. Modes of the Ptolemismic Pinetone Dorian bright minor

!Pinetone diatonic mode

|Lydian bright major*

|Ionian dark minor

|Dorian bright minor*

|Locrian dark major

|Mixolydian magical seventh

|Phrygian bright diminished

|Aeolian dark diminished

|+Table 4.5. Modes of the Ptolemismic Pinetone Locrian dark minor

!Pinetone diatonic mode

|Mixolydian dark major*

|Lydian bright minor

|~ 10/9 11/9 11/8 3/2 11/8 9/5 2/1

|Aeolian bright major

|Ionian dark diminished

|Dorian bright diminished

|Locrian dark minor*

|Phrygian magical seventh

None of these other diatonic scales are useful as a basis for tonal harmony, since for all of these scales the only two major or minor chords are part of the same minor 7 chord. Instead they may be employed for melodic reasons, or as alternate modes to use with the minor 7 chord.  

Note that the Dorian bright minor and Locrian dark minors are tetrachordal, along with the Dorian dark minor a.k.a. the Pinetone diatonic.  

Starting with the 5-limit diminished tetrad 6/5 36/25 5/3 2/1, putting 10/9 above each note takes us to the Pinetone diminished octatonic 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1. The 5-limit diminished tetrad has 3 large steps of 6/5, and one small step of 125/108. If we only put 10/9 above the large steps, we get the scale 10/9 6/5 4/3 36/25 5/3 50/27 2/1, with step pattern msmsLms, comprising 3 small steps of 27/25, 3 medium steps of 10/9, and one large step of 125/108.   

Tempering the small and medium steps together gives us the scale ssssLss, Porcupine[7]; tempering L=m implies tempering out 25/24, which leads to Dicot[7] as sLsLssL; and tempering s=L gives us sLsLLsL as Sixix[7]. Tempering out 100/99 leads to simplest JI pre-image 10/9 6/5 4/3 16/11 5/3 11/6 2/1, and additionally tempering out 144/143 to 10/9 6/5 4/3 13/9 5/3 11/6 2/1. The Pinetone diminished heptatonic's large step tempers to 55/48 under 2.3.5.11 Ptolemismic, and to 15/13 under 2.3.5.11.13 Ptolemismic. The scale is chiral, with mirror-inverse msmLsms, approximating 12/11 6/5 11/8 3/2 5/3 9/5 2/1, or in other modes, Lmsmsms and smsmsmL, the brightest and darkest modes of the pair of scales, approximating 15/13 33/26 11/8 20/13 5/3 11/6 2/1 and 12/11 6/5 13/10 13/9 39/25 26/15 2/1. The scales, like the Pinetone diatonic, are also trivalent. There's only one major triad and one minor triad available in the scale. In their simplest modes with 3/2 above the root, the pairs of scales in 5-limit JI are 27/25 5/4 25/18 3/2 5/3 9/5 2/1 and 27/25 6/5 25/18 3/2 5/3 2/1, and as Ptolemismic tempered scales their simplest pre-images are 12/11 5/4 11/8 3/2 5/3 9/5 2/1 and 12/11 6/5 11/8 3/2 5/3 9/5 2/1, with step patterns sLmsmsm and smLsmsm respectively.  

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.  

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'').

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].

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 9/5 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.  

2.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).

! Mode (height order)

! Mode (height order)

!Mode

|+Table 9.6. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished

<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. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered to 2.3.5.7 245/243 i.e., Sensamagic temperament.