159edo/Interval names and harmonies: Difference between revisions

| As the approximation of both the Pythagorean Minor Second and the Ptolemaic Augmented Prime, this interval is used accordingly; however, it is also worth noting that as a further consequence of the [[schisma]] being tempered out in this system, two of these add up to the approximation of the Ptolemaic Major Second.

| This interval approximates the Ptolemaic Minor Second- that is, the traditional [[5-limit]] leading tone- as well as the Pythagorean Augmented Prime, and thus, is used accordingly; however, this interval is also one of two in this system that are essential in executing the frameshift cadence.

| As one of two Neutral Seconds in this system, this interval is notable for being half of the approximation of the Neo-Gothic Minor Third, though it is also sometimes used in much the same way as 24edo's own Neutral Second.

| In addition to its properties as the interval that most closely resembles the low-complexity JI Submajor Second, this interval serves as both the Ultra-Augmented Prime and as one third of a Perfect Fourth, and is used accordingly.  

| While this interval is large enough to act as a type of whole tone, it is worth noting that two of these add up to the approximation of the low-complexity JI Neutral Third in this system.

| As the approximation of the Ptolemaic Major Second, this interval is used accordingly, though it is worth noting that in this system, two of these add up to the approximation of the thirteenth subharmonic; furthermore, it is also one the intervals in this system that are essential in executing any sort of variation on Jacob Collier's "Four Magical chords" from his rendition of "In the Bleak Midwinter".

| This interval has surprising utility in modulating to keys that are not found on the same circle of fifths owing to both its size and its ease of access through octave-reducing stacks of approximated low-complexity JI intervals.

| This interval is created from stacking two of this system's closet approximation of the 12edo semitone, and thus, it is one of two intervals that come the closest to approximating the 12edo whole tone found in this system.

| This interval is the standard-issue whole tone in this system as it is one of two intervals that come the closest to approximating the 12edo whole tone, and the only one of the two that actually approximates the Pythagorean Major Second; furthermore, it is the whole tone that is used as a reference interval in [[Diatonic, Chromatic, Enharmonic, Subchromatic|diatonic-and-chromatic-style]] interval logic in this system as it pertains to both semitones and quartertones.

| This interval can be interpreted as a type of second on the basis of it approximating the sum of the syntonic comma and the Pythagorean Major Second; it also appears in approximations of 5-limit Neapolitan scales as the interval formed from stacking two Ptolemaic Minor Seconds, making it double as a type of diminished third, and is likely the smallest interval in this system that can be used in chords without causing crowding.

| rm3

| Narrow Minor Third

| F\, Et>↑

| This interval is of particular interest because it is the approximation of the Neo-Gothic Minor Third and is used accordingly; what's more, this interval and the approximation of the Neo-Gothic Major Third add up to make the Perfect Fifth in this system.

| m3

| Pythagorean Minor Third

| F

| This interval approximates the Pythagorean Minor Third, and since this system does not temper out the syntonic comma, this interval- in contrast to the Ptolemaic Minor Third- is very useful as an interpretation of the dissonant Minor Third from [[Wikipedia: Medieval music #Early_polyphony: organum|Medieval music's florid organum]].

| Rm3

| Artomean Minor Third

| F/

| This interval is the closest approximation of the 12edo Minor Third found in this system, and, conveniently enough, it is easily accessed by stacking instances of this system's approximation of the low-complexity JI neutral second.

| rKm3

| Tendomean Minor Third

| F↑\

| The main thing of note concerning this interval is that two of these add up to this system's approximation of the Greater Septimal Tritone.

| Km3

| Ptolemaic Minor Third

| F↑, E#

| As the approximation of the Ptolemaic Minor Third- that is, the traditional 5-limit minor third- this interval is one of four imperfect consonances in this system, and, unsurprisingly, is thus used accordingly; however, one should also note that this interval can be reached by stacking three of this system's approximation of the octave-reduced seventeenth harmonic.

| RKm3, kn3

| Wide Minor Third

| Ft<↓, F↑/

| The main thing of note concerning this interval is that two of these add up to this system's approximation of the Paraminor Fifth.