159edo/Interval names and harmonies: Difference between revisions

[[159edo]] contains all the intervals of [[53edo]], however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed.  It should be noted that since 159edo does a better job of representing the 2.3.11 subgroup than [[24edo]], some of the chords listed on the page for [[24edo interval names and harmonies]] carry over to this page, even though the exact sets of enharmonics differ between the two systems.

|+ style=white-space:nowrap | Table of 159edo intervals

! Step

! Cents

! 5 limit

! 7 limit

! 13 limit

! colspan="3"| Interval Names

! Notes

| The root of any chord, as well as the base representation of the Tonic.

| As the approximation of both the [[rastma]] and the [[marvel comma]], this interval is useful for slight dissonances that convey something less than satisfactory, however it can also add to the bandwidth of a sound.

| As the approximation of the [[ptolemisma]] and the [[biyatisma]], this interval is useful for slight dissonances that create more noticeable tension.

| As the approximation of the [[syntonic comma]] and the [[Pythagorean comma]], this interval is useful for appoggiaturas, acciaccaturas, and quick passing tones.

| As the approximation of [[septimal comma]] and the [[telepathma]], this interval is useful for various types of subchromatic gestures, as well as for appoggiaturas, acciaccaturas, and quick passing tones.

| By default, this interval is a type of paradiatonic quartertone, and indeed, the [[11-limit]] ratio this interval approximates is the namesake of [[24edo]]'s own Inframinor Second; in addition, it is also the closest approximation of the [[31edo]] Superprime found in this system.

| Wide Inframinor Second, Narrow Ultraprime, Semilimma

| This interval is particularly likely to be used as a cross between an Ultraprime and an Inframinor Second; furthermore, as the name "Semilimma" suggests, this interval is one half of a Pythagorean Minor Second.

| By default, this interval is a type of parachromatic quartertone- specifically, the representation of the Al-Farabi Quartertone- and is thus used in much the same way as 24edo's own Ultraprime; what might be surprising is that five of these add up to this system's approximation of the Septimal Subminor Third.

| Although this interval can act as a leading tone, it can also act as a trienstone- that is, a third of a tone- since it's one third of the Ptolemaic Major Second.

| Eb↓, Dt<↑\, D#↓↓

| Although this interval frequently acts as the Classic Chroma due to consistently approximating it, it can also act as a trienstone- that is, a third of a tone- since it's one third of the Pythagorean Major Second.

| This interval acts as a type of semitone, however, whether it's a diatonic or chromatic semitone depends on the situation; in addition, it is also the closest approximation of the 31edo Subminor Second found in this system.

| Not only does this interval serve as a type of leading tone due to it being the approximation of the Septimal Minor Semitone, but it should be noted that six of these add up to a Perfect Fourth.

| 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 is one of two in this system that are essential in executing the [[Frameshift comma #Frameshift cedence|frameshift cadence]]; it is also the closest approximation of the [[12edo]] Minor Second found in this system.

| As the approximation of both the [[octave-reduced]] seventeenth harmonic and the interval formed from stacking two Ultraprimes, this interval is used accordingly.

| 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 useful for evoking the feel of 31edo due to approximating that system's Minor Second, and is also one of two in this system that are essential in executing the frameshift cadence.

| In addition to being the approximation of the Septimal Major Semitone, this interval is also one third of a Lesser Submajor Third in this system, and is thus used accordingly.

| In addition to its properties as a type of supraminor second, this interval is also one third of a Ptolemaic Major Third in this system and is thus used accordingly.

| Ed<\, Eb↑↑, D#↑  

| This interval is not only both two thirds of Pythagorean Major Second and the approximation of the Large Limma or Diptolemaic Limma in this system, but also a type of supraminor second, and is thus used accordingly.

| n2, SA1, kUA1

| 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.

| As one of two neutral seconds in this system, this interval is the one that most closely resembles the [[low-complexity JI]] neutral second, and thus, it is frequently used in much the same way as 24edo's own Neutral Second.

| Lesser Submajor Second, Diretroptolemaic Augmented Prime

| In addition to being a type of Submajor Second and the closest approximation of the 31edo Middle Second found in this system, two of these add up to the approximation of the Ptolemaic Minor Third.

| In addition to its properties as the interval that most closely resembles the Undecimal 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.

| In addition to being the closest approximation of the 31edo Major Second found in this system, it is one of two intervals that come the closest to approximating the 12edo Major Second 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 Major Second, 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 is interesting on the basis that it is formed by stacking two instances of the octave-reduced approximation of the seventeenth harmonic.

| This interval is interesting not only because it is utilized in approximations of the [[17-odd-limit]], but also because it is the whole tone found in this system's Superpyth scale, and is of such quality that two of these add up to this system's approximation of the Septimal Supermajor Third.

| 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.

| As the approximation of the octave-reduced seventh subharmonic- that is, the Septimal Supermajor Second- this interval is used accordingly; in fact, since three of these add up to a Perfect Fifth in this system, there are multiple ways this interval can be used in chords to great effect.

| The 11-limit ratio this interval approximates is the namesake of 24edo's own Inframinor Third; however, in a higher-fidelity system such as this, one will notice that this is a syntactic third that sounds more like a second.

| This interval is particularly likely to be used as a cross between an Ultramajor Second and an Inframinor Third; furthermore, as the name "Semifourth" suggests, this interval is one half of a Perfect Fourth, and used in exactly the same way as 24edo's own Semifourth, right down to the [[Low-complexity JI|low-complexity]] [[13-limit]] interpretation.

| The 11-limit ratio this interval approximates is the namesake of 24edo's own Ultramajor Second; however, in a higher-fidelity system such as this, one will notice that this is a syntactic second that sounds more like a third.

| As the approximation of the Septimal Subminor Third, those who are not already familiar with septimal harmony will find this interval useful in forming not only strident-sounding triads framed by the Perfect Fifth, but also other, ambisonant triads framed by the Perfect Fourth; in addition, three of these add up to the Pythagorean Minor Sixth.

| This interval is useful for evoking the feel of 31edo due to approximating that system's Subminor Third, and even approximates the result of subtracting a syntonic comma from a Pythagorean Minor Third; however, it most frequently appears in approximations of [[5-limit]] Harmonic scales as the interval between the Ptolemaic Minor Sixth and the Ptolemaic Major Seventh, making it double as a type of augmented second.

| This interval is utilized in approximations of the [[17-odd-limit]], courtesy of acting as the [[fourth complement]] to the Narrow Supermajor Second; it is also good for evoking the feel of [[17edo]] due to approximating that system's Minor Third.

| 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.

| 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]], and can thus be used in creating a subtle instability in certain Diatonic harmonies.

|-  

| 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.

| In addition to being the closest approximation of the 31edo Minor Third found in this system, this interval is also half of this system's approximation of the Greater Septimal Tritone and is thus used accordingly as part of a triad.

| 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.

| The main thing of note concerning this interval is that two of these add up to this system's approximation of the Paraminor Fifth, thus facilitating the formation of strange-sounding triads

| This interval is mainly of interest due to the fact that it's exactly twice the size of it's fourth complement- the approximation of the Undecimal Submajor Second- and its interesting properties as a type of supraminor third.

| Greater Supraminor Third, Diretroptolemaic Diminished Fourth

| This interval is of interest because not only does it have 13-limit interpretations, but it also has usage as a 17-odd-limit interval, and all while being easily reached by stacking three Ptolemaic Minor Seconds.

| As one of two neutral thirds in this system, this interval is the one that most closely resembles the [[low-complexity JI]] neutral third, and thus, it is frequently used in much the same way as 24edo's own Neutral Third; on top of that, it can be stacked in interesting ways in this system.

| As one of two neutral seconds in this system, this interval is notable for being one half of a possible generator for this system's superpyth scale.

| Ft>/, F#↓↓, Gb↓

| As both the approximation of the octave-reduced thirteenth subharmonic, and ostensibly one of the easiest 13-limit thirds to utilize in chords framed by some type of sharp wolf fifth, this interval is used accordingly.

| In addition to its properties as a type of submajor third, this interval is also one third of a Pythagorean Major Seventh in this system and is thus used accordingly.

| The main thing of note concerning this interval is that two of these add up to this system's approximation of the Paramajor Fifth, thus facilitating the formation of strange-sounding triads.

| This interval is none other than the approximation of the octave-reduced fifth harmonic- that is, the traditional 5-limit major third- and thus, it one of four imperfect consonances in this system, and, unsurprisingly, is used accordingly; however, this interval is also the approximation of the Pythagorean Diminished Fourth in this system, which sometimes leads to interesting enharmonic substitutions when building chords for purposes of voice-leading.