Lemba: Difference between revisions

Line 1:

'''Lemba''' is a [[regular temperament]] which is a natural extension of the [[jubilismic clan]] and a member of the [[gamelismic clan]]. This means that the perfect fifth of [[~]][[3/2]] is split into three equal parts, each approximating [[8/7]]. It also means the period is half an octave, and repeats precisely a tritone apart, tempering out [[50/49]]. A generator plus a period comes very close to the [[golden ratio]] phi, which means ratios in the sequence 8:13:21:34:55 etc. are also well approximated, and any one of these can be made just by choosing a suitable [[eigenmonzo]] (unchanged interval). The combination of these factors means many composite ratios in the 2.3.5.7.13.17 subgroup are both well approximated and accessible with a relatively small gamut, giving you a strong selection of chords to choose from. Its main weaknesses are that ratios of 5 and 13 are conflated by the tempering out of [[65/64]], favoring 13 in the better tunings, so traditional major and minor chords are strongly neutral flavoured (supraminor and submajor), and ratios involving 11 are not approximated at all until you have a large gamut. It forms [[mos scale]]s that are always double a fibonacci sequence number, at 4, 6, 10, 16, 26, etc, which means L/s ratios remain well mixed and clearly distinct many iterations down. Related temperaments include [[~~Gamelismic_clan~~#Baladic|baladic]], which discards the 5 and 11 to improve the accuracy of the 2.3.7.13.17 subgroup, and is very accurate in all of these except the 13~~. Or~~ [[~~Echidnic~~]], which has a slightly sharper generator that sacrifices the precision of 7 and logarithmic phi to map most other intervals in a more accurate but complex way.

'''Lemba''' is a [[regular temperament]] which is a natural extension of the [[jubilismic clan]] and a member of the [[gamelismic clan]]. This means that the perfect fifth of [[~]][[3/2]] is split into three equal parts, each approximating [[8/7]]. It also means the period is half an octave, and repeats precisely a tritone apart, tempering out [[50/49]]. A generator plus a period comes very close to the [[golden ratio]] phi, which means ratios in the sequence 8:13:21:34:55 etc. are also well approximated, and any one of these can be made just by choosing a suitable [[eigenmonzo]] (unchanged interval). The combination of these factors means many composite ratios in the 2.3.5.7.13.17 subgroup are both well approximated and accessible with a relatively small gamut, giving you a strong selection of chords to choose from. Its main weaknesses are that ratios of 5 and 13 are conflated by the tempering out of [[65/64]], favoring 13 in the better tunings, so traditional major and minor chords are strongly neutral flavoured (supraminor and submajor), and ratios involving 11 are not approximated at all until you have a large gamut. It forms [[mos scale]]s that are always double a fibonacci sequence number, at 4, 6, 10, 16, 26, etc, which means L/s ratios remain well mixed and clearly distinct many iterations down. Related temperaments include [[Gamelismic clan #Baladic|baladic]], which discards the 5 and 11 to improve the accuracy of the 2.3.7.13.17 subgroup, and is very accurate in all of these except the 13, or [[echidnic]], which has a slightly sharper generator that sacrifices the precision of 7 and logarithmic phi to map most other intervals in a more accurate but complex way.

The name is from [[Herman Miller]]'s conlang name for the temperament.

Line 9:

{| class="wikitable center-1 right-2 right-4"

! rowspan="2" | # ~~gens~~

! rowspan="2" | #

! colspan="2" | Period 0

! colspan="2" | Period 1

|-

! Cents*

! Approximate ~~Ratios~~

! Approximate ratios

! Cents

! Approximate ~~Ratios~~

! Approximate ratios

|-

| 0

Line 74:

<nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].

== Tuning spectrum ==

== Tunings ==

=== Tuning spectrum ===

~~Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]~~

~~Gencom mapping: [{{val| 2 2 5 6 5 7 }}, {{val| 0 3 -1 -1 5 1 }}]~~

{| class="wikitable center-all left-4"

|-

! ~~EDO~~<br>generator

! Edo<br>generator

! [[~~eigenmonzo~~|~~eigenmonzo~~<br>(unchanged interval)]]

! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]

! ~~generator~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

|

Line 290:

Line 286:

== Music ==

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria &~~amp;~~ Fugue) in 8/7 eigenmonzo tuning

* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning

; [[Herman Miller]]

@@ Line 1: / Line 1: @@
-'''Lemba''' is a [[regular temperament]] which is a natural extension of the [[jubilismic clan]] and a member of the [[gamelismic clan]]. This means that the perfect fifth of [[~]][[3/2]] is split into three equal parts, each approximating [[8/7]]. It also means the period is half an octave, and repeats precisely a tritone apart, tempering out [[50/49]]. A generator plus a period comes very close to the [[golden ratio]] phi, which means ratios in the sequence 8:13:21:34:55 etc. are also well approximated, and any one of these can be made just by choosing a suitable [[eigenmonzo]] (unchanged interval). The combination of these factors means many composite ratios in the 2.3.5.7.13.17 subgroup are both well approximated and accessible with a relatively small gamut, giving you a strong selection of chords to choose from. Its main weaknesses are that ratios of 5 and 13 are conflated by the tempering out of [[65/64]], favoring 13 in the better tunings, so traditional major and minor chords are strongly neutral flavoured (supraminor and submajor), and ratios involving 11 are not approximated at all until you have a large gamut. It forms [[mos scale]]s that are always double a fibonacci sequence number, at 4, 6, 10, 16, 26, etc, which means L/s ratios remain well mixed and clearly distinct many iterations down. Related temperaments include [[Gamelismic_clan#Baladic|baladic]], which discards the 5 and 11 to improve the accuracy of the 2.3.7.13.17 subgroup, and is very accurate in all of these except the 13. Or [[Echidnic]], which has a slightly sharper generator that sacrifices the precision of 7 and logarithmic phi to map most other intervals in a more accurate but complex way.
+'''Lemba''' is a [[regular temperament]] which is a natural extension of the [[jubilismic clan]] and a member of the [[gamelismic clan]]. This means that the perfect fifth of [[~]][[3/2]] is split into three equal parts, each approximating [[8/7]]. It also means the period is half an octave, and repeats precisely a tritone apart, tempering out [[50/49]]. A generator plus a period comes very close to the [[golden ratio]] phi, which means ratios in the sequence 8:13:21:34:55 etc. are also well approximated, and any one of these can be made just by choosing a suitable [[eigenmonzo]] (unchanged interval). The combination of these factors means many composite ratios in the 2.3.5.7.13.17 subgroup are both well approximated and accessible with a relatively small gamut, giving you a strong selection of chords to choose from. Its main weaknesses are that ratios of 5 and 13 are conflated by the tempering out of [[65/64]], favoring 13 in the better tunings, so traditional major and minor chords are strongly neutral flavoured (supraminor and submajor), and ratios involving 11 are not approximated at all until you have a large gamut. It forms [[mos scale]]s that are always double a fibonacci sequence number, at 4, 6, 10, 16, 26, etc, which means L/s ratios remain well mixed and clearly distinct many iterations down. Related temperaments include [[Gamelismic clan #Baladic|baladic]], which discards the 5 and 11 to improve the accuracy of the 2.3.7.13.17 subgroup, and is very accurate in all of these except the 13, or [[echidnic]], which has a slightly sharper generator that sacrifices the precision of 7 and logarithmic phi to map most other intervals in a more accurate but complex way.
 The name is from [[Herman Miller]]'s conlang name for the temperament.
@@ Line 9: / Line 9: @@
 {| class="wikitable center-1 right-2 right-4"
-! rowspan="2" | # gens
+! rowspan="2" | #
 ! colspan="2" | Period 0
 ! colspan="2" | Period 1
 |-
 ! Cents*
-! Approximate Ratios
+! Approximate ratios
 ! Cents
-! Approximate Ratios
+! Approximate ratios
 |-
 | 0
@@ Line 74: / Line 74: @@
 <nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].
-== Tuning spectrum ==
+== Tunings ==
+=== Tuning spectrum ===
-Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
-Gencom mapping: [{{val| 2 2 5 6 5 7 }}, {{val| 0 3 -1 -1 5 1 }}]
 {| class="wikitable center-all left-4"
 |-
-! EDO<br>generator
+! Edo<br>generator
-! [[eigenmonzo|eigenmonzo<br>(unchanged interval)]]
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
-! generator (¢)
+! Generator (¢)
-! comments
+! Comments
 |-
 |
@@ Line 290: / Line 286: @@
 == Music ==
 ; [[Claudi Meneghin]]
-* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria &amp; Fugue) in 8/7 eigenmonzo tuning
+* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning
 ; [[Herman Miller]]