User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

The index values correspond most directly to degrees of 31-tET, whose interval names by this method are given in the following table:

{| class="wikitable"

!31-tET degree

!Ratios

=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===

One system which in it's naming of meantone and non-meantone edos is able to conserve interval arithmetic, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of [[64/63]], which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]] respectively. Similarly, 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11.

Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined.

== Premise: ==