Irvian mode: Difference between revisions
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Even-length Irvian modes with odd number of years per cycle (that is notes) have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the ''K'' is chosen as (''C''-1)/2 instead of ''C''/2, the sequence will be leap, nonleap. Thus it is called ''almost symmetrical'' in the calendar lore. That being said, they fulfill their function just like odd cycles do, and therefore belong in Irvian modes. | Even-length Irvian modes with odd number of years per cycle (that is notes) have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the ''K'' is chosen as (''C''-1)/2 instead of ''C''/2, the sequence will be leap, nonleap. Thus it is called ''almost symmetrical'' in the calendar lore. That being said, they fulfill their function just like odd cycles do, and therefore belong in Irvian modes. | ||
== | ==Example on a standard 12edo piano== | ||
The 12edo piano key layout, which is predominantly use in the world today, is an example of an Irvian mode that is subject to even-length leap rule modification.<blockquote>'''Year is leap if the remainder of (7 x Year + 6) / 12 is less than 7.'''</blockquote>Such a pattern generates keys number '''1-3-5-6-8-10-12-1''' to be the keys on the scale, which is a '''5L 2s''' scale in a pattern of '''LLsLLLs'''. White keys are leap years, and black keys are common years. | The 12edo piano key layout, which is predominantly use in the world today, is an example of an Irvian mode that is subject to even-length leap rule modification.<blockquote>'''Year is leap if the remainder of (7 x Year + 6) / 12 is less than 7.'''</blockquote>Such a pattern generates keys number '''1-3-5-6-8-10-12-1''' to be the keys on the scale, which is a '''5L 2s''' scale in a pattern of '''LLsLLLs'''. White keys are leap years, and black keys are common years. | ||
Years 1,3,6,8,10, that is notes C, D, F, G, A have a long interval - a tone - after them, while E and B, with remainder of 6, have a semitone. When started on C turns out to be plain C major. In this case, the accumulator K is taken to be C/2 instead of (C-1)/2 as with odd cycles, therefore middle of the cycle is nonleap-leap, that is F and F#. Choosing 5 instead of 6 for the K would produce a Lydian scale on C, or a F major scale - patterns of keys are reversed. | Years 1,3,6,8,10, that is notes C, D, F, G, A have a long interval - a tone - after them, while E and B, with remainder of 6, have a semitone. When started on C turns out to be plain C major. In this case, the accumulator K is taken to be C/2 instead of (C-1)/2 as with odd cycles, therefore middle of the cycle is nonleap-leap, that is F and F#. Choosing 5 instead of 6 for the K would produce a Lydian scale on C, or a F major scale - patterns of keys are reversed. | ||
== Other examples == | |||
=== 17edo === | === 17edo === | ||
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L L s L L L s L L s | L L s L L L s L L s | ||
=== 22edo === | === 22edo === | ||
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0-2-4-5-7-9-10-12-14-16-17-19-21-0, proper Irvian mapping as directly taken from the formula. | 0-2-4-5-7-9-10-12-14-16-17-19-21-0, proper Irvian mapping as directly taken from the formula. | ||
{| class="wikitable" | {| class="wikitable" | ||
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Revision as of 18:58, 7 November 2021
Irvian mode is the mode of the maximal evenness scale where the notes are symmetrically arranged.
History
In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's unsmooth distribution of days, weeks, months, and leap years. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.
Such a pattern produces a specific mode of a maximally even scale, which is named an Irvian mode. A stand-alone leap week at the end of year in Sym454 lore is called Irvember, and therefore the constructed name of the mode would be Irvian.
In this paradigm, years correspond to individual steps of the scale, and leap years correspond to steps that are part of the mode. The length of the cycle is the size of an Edo.
The pattern is defined by the following:
Year is leap if the remainder of (L x Y + K)/C is less than L.
L = number of leap years per cycle,
Y = number of the year
C = number of years per cycle
K = (C-1)/2 if C is odd, can choose between (C-1)/2 and C/2 if C is even
The current, "canonical" usage of the cycle is that of 52 leap week years in 293 years - year is leap if the remainder of (52 x Year + 146)/293 is less than 52. Musically, this would correspond to a 33L 19s MOS scale of 293edo. In addition, if the remainder of the leap year is less than the count of long intervals in the MOS, the next year will be in a long interval, otherwise in a short interval. For example here, this means if remainder is less than 33, next leap year (or key) will be 6 years later (6 steps above), otherwise 5 years later.
Even-length Irvian modes with odd number of years per cycle (that is notes) have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the K is chosen as (C-1)/2 instead of C/2, the sequence will be leap, nonleap. Thus it is called almost symmetrical in the calendar lore. That being said, they fulfill their function just like odd cycles do, and therefore belong in Irvian modes.
Example on a standard 12edo piano
The 12edo piano key layout, which is predominantly use in the world today, is an example of an Irvian mode that is subject to even-length leap rule modification.
Year is leap if the remainder of (7 x Year + 6) / 12 is less than 7.
Such a pattern generates keys number 1-3-5-6-8-10-12-1 to be the keys on the scale, which is a 5L 2s scale in a pattern of LLsLLLs. White keys are leap years, and black keys are common years.
Years 1,3,6,8,10, that is notes C, D, F, G, A have a long interval - a tone - after them, while E and B, with remainder of 6, have a semitone. When started on C turns out to be plain C major. In this case, the accumulator K is taken to be C/2 instead of (C-1)/2 as with odd cycles, therefore middle of the cycle is nonleap-leap, that is F and F#. Choosing 5 instead of 6 for the K would produce a Lydian scale on C, or a F major scale - patterns of keys are reversed.
Other examples
17edo
Year is leap if the remainder of (7 x Year + 8) / 17 is less than 7
1-3-6-8-10-13-15
s L s s L s L.
Starting from the other key, it's bayati 3232322. 17edo is the only temperament where bayati is parallel to the Irvian mode.
Year is leap if the remainder of (10 x Year + 8) / 17 is less than 10.
0-2-4-5-7-9-11-12-14-16-17
L L s L L L s L L s
22edo
Year is leap if the remainder of (13 x Year + 11) / 22 is less than 13.
Orwell[13]:
0-2-4-5-7-9-10-12-14-16-17-19-21-0, proper Irvian mapping as directly taken from the formula.
| Name | Formula core |
|---|---|
| Porcupine[15] | (15 x Year + 11) / 22 |
| Superpyth[5] | (5 x Year + 11) / 22 |
| Porcupine[7] | (7 x Year + 11) / 22 |
31edo
31 edo contains the following Irvian modes, derived from ME 31edo MOS scales:
| Name | Formula core | Key layout |
|---|---|---|
| Würschmidt[3] | (3 x Year + 15) / 31 | 6-16-26 |
| Myna[4] | (4 x Year + 15) / 31 | 4-12-20-28 |
| Mothra[5] | (5 x Year + 15) / 31 | 4-10-16-22-28 |
| Hemithirds[6] | (6 x Year + 15) / 31 | 3-8-13-19-24-29 |
| Mohajira[7] | (7 x Year + 15) / 31 | 3-7-12-16-20-25-29 |
| Nusecond[8] | (8 x Year + 15) / 31 | 2-6-10-14-18-22-26-30 |
| Orwell[9] | (9 x Year + 15) / 31 | 2-6-9-13-16-19-23-26-30 |
| Miracle[10] | (10 x Year + 15) / 31 | 2-5-8-11-14-18-21-24-27-30 |
and so on.