The Archipelago
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The archipelago is a rag-tag collection of various regular temperaments of different ranks, inc…
The archipelago is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, 676/675. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The just intonation subgroup generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.
The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer dyad, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an ultramajor triad, with a third sharper even than the 9/7 supermajor third.
kite33
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... 125/63
2/1
Kite notes that in his tuning, the piece pumps the starling comma. To avoid a pi…
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125/63
2/1
Kite notes that in his tuning, the piece pumps the starling comma. To avoid a pitch shift, the tonic drifts steadily upwards 14¢ over the course of the piece. Individual notes have a stable pitch, thus each note has a unique pitch. Thus Kite's tuning of this piece does not use a fixed scale, and this scala file will not tune the Liszt piece properly.
Porcupine Notation
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... BTW with this notation system, one question is - how many lines do you use in a staff? I guess…
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BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you.
Kite Giedraitis's approach
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generator is vM2, and thevM2. The enharmonic is v3A1 (C^3
equals C#).v3A1, thus C^3 equals C#. The alternate
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genchain is
...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - P1 - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...
In C, this would be
Porcupine Notation
edited
... BTW with this notation system, one question is - how many lines do you use in a staff? I guess…
...
BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you.
Kite Giedraitis's approach
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v3A1 (C^3
C#).
equals C#). The alternate
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- E
^^m2.= ^^m2. The genchain
...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - P1 - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...
In C, this would be
Porcupine Notation
edited
... BTW with this notation system, one question is - how many lines do you use in a staff? I guess…
...
BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you.
Kite Giedraitis's approach
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v3A1 (C^3 = C#).
C#). The alternate generator is G - E
^^m2. The genchain
...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - P1 - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...
In C, this would be
...C^ - D - Ev - F^ - G - Av - Bb^ - C - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv...
Unlike the other proposals here, the notes without ups and downs still form the familiar chain of 5ths ...Eb Bb F C G D A E B F# C#..., and interval arithmetic remains unchanged. Staff notation is as usual, with the addition of up and down accidentals before certain notes.
Porcupine[8] in A is A Bv C^ D Ev F^ G Av A. Porcupine[7] would omit Av.
Example comma pump, with brackets indicating an enharmonic equivalence:
C.v -- Av.^m -- Dv.v -- [Bvv=Bb^].^m -- Eb^.v -- Ab^.v -- C.v
Porcupine Notation
edited
... BTW with this notation system, one question is - how many lines do you use in a staff? I guess…
...
BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you.
Kite Giedraitis's approach Ups and downs notation can be used even though we don't know which edo we are in. We know that porcupinePorcupine divides the
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3 equal steps. Also the aug 4thsteps, thus its pergen is always 3 major 2nds by definition. And from P4 to A4(P8, P4/3). Ups and downs notation can be used. The generator is A1. So any edo which divides the aug1 into 3 equal steps will temper out porcupine. These are the edos marked on the scale tree on the linked page as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), A1 = triple-up unison,vM2, and the generatorenharmonic is
generatorv3A1 (C^3 = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^31 / 3 = M2 - ^1 = vM2
For sharp-6 edos (e.g. 30 or 72), the generator is vvM2. Sharp-9 edos are rarely used, but the generator would be v3M2.
For sharp-3 edos, theC#). The genchain is ^1
...^1 - M2
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m7 - v8.v8...
In C,
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would be C^
...C^ - D
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Bb - Cv.
ForCv...
Example comma pump, with brackets indicating an enharmonic equivalence:
C.v -- Av.^m -- Dv.v -- [Bvv=Bb^].^m -- Eb^.v -- Ab^.v -- C.v
See the ups and downs page for an explanation of the chord names.
This is the rank-2 notation, for a generator of indeterminate cents. The edo notation uses ups and downs to represent one EDOstep. If an edo tempers out porcupine, it must be sharp-3, sharp-6, sharp-9, etc. (See the scale tree on the ups and downs page.) The notation is identical for sharp-3 edos (15, 22, 29, etc.). For sharp-6 edos,edos (e.g. 30 or 72), simply double
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ups and downs:downs. The generator is vvM2. The genchain is P1 -
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- vv5... andor C -
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- Gvv...
This assumes a mapping of 5/4 that results in 250/243 mapping to zero edosteps. Since 10/9 should be vM2, 5/4 should be 10/9 + 9/8 = vM2 +M2 = vM3. The obvious mapping often suffices, Sharp-9 edos are rarely used, but sometimes the mapping needs tweaking, as with edos 36c, 43c, 57cc, 58c, 64ccc, 65c and 72cc. If the edo isn't tweaked, the generator won't map to 10/9, and two generators won't map to 6/5. (Arguably, the generated scale is still porcupine-like.)
If we're not in an edo at all, but in a rank-2 tuning with a generator of indeterminate cents, use the sharp-3 notation.would be v3M2.