Kite's thoughts on pergens: Difference between revisions
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→Addenda (Spring 2026): alternative algorithm for finding EUs -- look for a nearby small scale step and equate that to an up-arrow |
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As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. | As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave. | ||
The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma. | The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma. | ||
| Line 4,800: | Line 4,800: | ||
!cents | !cents | ||
!genspan | !genspan | ||
! | !notes per octave | ||
|- | |- | ||
!1 | !1 | ||
| Line 4,810: | Line 4,808: | ||
|±12G | |±12G | ||
|12 | |12 | ||
|- | |- | ||
!2 | !2 | ||
| Line 4,819: | Line 4,815: | ||
|±6G | |±6G | ||
|12 | |12 | ||
|- | |- | ||
!3 | !3 | ||
| Line 4,828: | Line 4,822: | ||
|5G | |5G | ||
|5 | |5 | ||
|- | |- | ||
!4 | !4 | ||
| Line 4,837: | Line 4,829: | ||
|7G | |7G | ||
|7 | |7 | ||
|- | |- | ||
!5 | !5 | ||
| Line 4,845: | Line 4,835: | ||
|5G | |5G | ||
|10 | |10 | ||
|- | |- | ||
!6 | !6 | ||
| Line 4,854: | Line 4,842: | ||
|±4G | |±4G | ||
|12 | |12 | ||
|- | |- | ||
!7 | !7 | ||
| Line 4,863: | Line 4,849: | ||
| -7G | | -7G | ||
|7 | |7 | ||
|- | |- | ||
!8 | !8 | ||
| Line 4,872: | Line 4,856: | ||
| -5G | | -5G | ||
|5 | |5 | ||
|- | |- | ||
!9 | !9 | ||
| Line 4,881: | Line 4,863: | ||
|2G | |2G | ||
|2 | |2 | ||
|- | |- | ||
!10 | !10 | ||
| Line 4,890: | Line 4,870: | ||
|3G | |3G | ||
|9 | |9 | ||
|- | |- | ||
!11 | !11 | ||
| Line 4,899: | Line 4,877: | ||
|1G | |1G | ||
|3 | |3 | ||
|- | |- | ||
!12 | !12 | ||
| Line 4,907: | Line 4,883: | ||
| -7G | | -7G | ||
|14 | |14 | ||
|- | |- | ||
!13 | !13 | ||
| Line 4,915: | Line 4,889: | ||
| -5G | | -5G | ||
|10 | |10 | ||
|- | |- | ||
!14 | !14 | ||
| Line 4,924: | Line 4,896: | ||
|1G | |1G | ||
|2 | |2 | ||
|- | |- | ||
!15 | !15 | ||
| Line 4,932: | Line 4,902: | ||
| -7G | | -7G | ||
|21 | |21 | ||
|- | |- | ||
!16 | !16 | ||
| Line 4,941: | Line 4,909: | ||
|±3G | |±3G | ||
|12 | |12 | ||
|- | |- | ||
!17 | !17 | ||
| Line 4,949: | Line 4,915: | ||
|10G | |10G | ||
|10 | |10 | ||
|- | |- | ||
!18 | !18 | ||
| Line 4,958: | Line 4,922: | ||
|7G | |7G | ||
|7 | |7 | ||
|- | |- | ||
!19 | !19 | ||
| Line 4,967: | Line 4,929: | ||
| -17G | | -17G | ||
|17 | |17 | ||
|- | |- | ||
!20 | !20 | ||
| Line 4,976: | Line 4,936: | ||
| -5G | | -5G | ||
|5 | |5 | ||
|- | |- | ||
!21 | !21 | ||
!(P8/4, P4/2) | !(P8/4, P4/2) | ||
| | |M2/4 | ||
| | |50¢ + c/2 | ||
| | |G | ||
| | |4 | ||
|- | |- | ||
!22 | !22 | ||
| Line 4,994: | Line 4,950: | ||
|G | |G | ||
|2 | |2 | ||
|- | |- | ||
!23 | !23 | ||
!(P8/2, P4/4) | !(P8/2, P4/4) | ||
| | |m2/4 | ||
| | |25¢ - 1.25c | ||
| | |5G | ||
| | |10 | ||
|- | |- | ||
!24 | !24 | ||
!(P8/2, P5/4) | !(P8/2, P5/4) | ||
| | | colspan="2" |''same as #18 (P8, P5/4)'' | ||
|7G | |||
|14 | |||
| | |||
| | |||
| | |||
|- | |- | ||
!25 | !25 | ||
!(P8/4, P4/3) | !(P8/4, P4/3) | ||
| | |d4/12 | ||
|33.3¢ - 0.67c | |||
|2G | |||
|8 | |||
|- | |||
| | |||
| | |||
|} | |} | ||
The initial comma of (P8, P11/3) is a rather large 67¢, but if there are more than 2 notes per 8ve, the L/s ratio becomes enormous! | The initial comma of (P8, P11/3) is a rather large 67¢, but if there are more than 2 notes per 8ve, the L/s ratio becomes enormous! | ||
Note the unusability of certain pergens such as (P8/2, P11/3). | Note the unusability of certain pergens such as (P8/2, P11/3). | ||
Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma. | |||
This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step. | |||
For example, the pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us whole 4ths, and multiplying P by 4 gives us whole octaves. Thus we must multiply the up-arrow by 12 to get a 3-limit interval. Thus 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v<sup>12</sup>d4, and ^<sup>12</sup>C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^<sup>12</sup>d<sup>9</sup>4. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation. | |||
True doubles require finding two small steps. | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Notation]] | [[Category:Notation]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||