Lemba: Difference between revisions
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'''Lemba''' | '''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 the right [[eigenmonzo]]. 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, and ratios involving 11 are not approximated at all until you have a large gamut. However, ignoring the 5 and 13, and focusing purely on the 2.3.7.17 subgroup, it can be highly accurate, with a total error of less than 7 cents in the [[tonality diamond]] in the least squares tuning. 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. | ||
The name is from [[Herman Miller]]'s conlang name for the temperament. | |||
See [[Jubilismic clan #Lemba]] for more technical data. | See [[Jubilismic clan #Lemba]] for more technical data. | ||
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Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77] | Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77] | ||
Gencom mapping: [{{val| 2 2 5 6 5 7 }}, | Gencom mapping: [{{val| 2 2 5 6 5 7 }}, {{val| 0 3 -1 -1 5 1 }}] | ||
{| class="wikitable center-all" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! | ! Edo<br>Generator | ||
! [[ | ! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval)]] | ||
! | ! Generator (¢) | ||
! | ! Comments | ||
|- | |- | ||
| | | | ||
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| | | | ||
| 229.179 | | 229.179 | ||
| Golden Lemba<ref>L/s ratios are always precisely Φ, and | | Golden Lemba<ref>L/s ratios are always precisely Φ, and mos scales are always precisely 2Φ</ref> | ||
|- | |- | ||
| | | | ||
| Line 243: | Line 245: | ||
| | | | ||
| 234.485 | | 234.485 | ||
| 2.3.7.17 subgroup least squares {{clarify}} | | 2.3.7.17 subgroup least squares{{clarify}} | ||
|- | |- | ||
| 9\46 | | 9\46 | ||
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== Music == | == Music == | ||
; [[Claudi Meneghin]] | |||
* [https://www.youtube.com/watch?v=2ziAZx03KF8 Lemba Suite, for Two Organs] (Prelude, Aria & Fugue) | * [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for Two Organs''] (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning | ||
; [[Herman Miller]] | |||
* [https://sites.google.com/site/teamouse/LembaGalatsia.mp3 ''Lemba Galatsia''] | |||
* [https://sites.google.com/site/teamouse/lemba-gpo-test.mp3 ''GPO Lemb''a] | |||
== Notes == | |||
<references/> | |||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
Revision as of 18:19, 7 February 2023
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 the right eigenmonzo. 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, and ratios involving 11 are not approximated at all until you have a large gamut. However, ignoring the 5 and 13, and focusing purely on the 2.3.7.17 subgroup, it can be highly accurate, with a total error of less than 7 cents in the tonality diamond in the least squares tuning. It forms mos scales 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.
The name is from Herman Miller's conlang name for the temperament.
See Jubilismic clan #Lemba for more technical data.
Interval chain
In the following table, prime harmonics are in bold.
| # gens | Period 0 | Period 1 | ||
|---|---|---|---|---|
| Cents* | Approximate Ratios | Cents | Approximate Ratios | |
| 0 | 0.000 | 1/1 | 600.000 | 7/5, 10/7 |
| 1 | 230.966 | 8/7 | 830.966 | 8/5, 13/8 |
| 2 | 461.932 | 21/16, 13/10 | 1061.932 | 11/6, 15/8, 24/13 |
| 3 | 692.898 | 3/2 | 92.898 | 15/14 |
| 4 | 923.864 | 12/7, 22/13 | 323.864 | 6/5, 39/32 |
| 5 | 1154.830 | 48/25, 63/32, 96/48 | 554.830 | 11/8 |
| 6 | 185.796 | 9/8 | 785.796 | 11/7 |
| 7 | 416.762 | 9/7 | 1016.762 | 9/5 |
| 8 | 647.728 | 36/25 | 47.728 | 33/32, 36/35 |
* In 13-limit POTE tuning. This tuning is very close to 26edo.
Tuning spectrum
Gencom: [7/5 8/7; 45/44 50/49 65/64 78/77]
Gencom mapping: [⟨2 2 5 6 5 7], ⟨0 3 -1 -1 5 1]]
| Edo Generator |
Eigenmonzo (Unchanged Interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 5/4 | 213.686 | ||
| 15/11 | 221.016 | ||
| 12/11 | 224.681 | ||
| 3\16 | 225.000 | ||
| 13/10 | 227.107 | ||
| 11/10 | 227.501 | ||
| 13/11 | 227.698 | ||
| 8\42 | 228.571 | ||
| 6/5 | 228.910 | ||
| (2 - Φ)\2 | 229.179 | Golden Lemba[1] | |
| 21/13 | 230.253 | ||
| 11/8 | 230.264 | ||
| 14/11 | 230.415 | 15-odd-limit minimax | |
| 13/12 | 230.714 | ||
| 5\26 | 230.769 | ||
| 10/9 | 231.085 | 9-, 11- and 13-odd-limit minimax | |
| 8/7 | 231.174 | 7-odd-limit minimax | |
| [0 63 -20 -20 22 -6⟩ | 231.250 | 13-odd-limit least squares | |
| [0 17 -6 -6 6⟩ | 231.294 | 11-odd-limit least squares | |
| 52521875/177147 | 231.298 | 7-odd-limit least squares | |
| [0 66 -17 -23 25 -7⟩ | 231.399 | 15-odd-limit least squares | |
| 17/13 | 232.213 | ||
| 12\62 | 232.258 | ||
| 129140163/1500625 | 232.418 | 9-odd-limit least squares | |
| 18/13 | 232.676 | ||
| Φ | 233.090 | ||
| 7/6 | 233.282 | ||
| 7\36 | 233.333 | ||
| 9/7 | 233.583 | ||
| 4/3 | 233.985 | ||
| 21/17 | 234.274 | ||
| 234.485 | 2.3.7.17 subgroup least squares[clarification needed] | ||
| 9\46 | 234.783 | ||
| 17/16 | 234.985 | ||
| 21/16 | 235.390 | ||
| 11\56 | 235.714 | ||
| 14/13 | 235.851 | ||
| 11/9 | 236.851 | ||
| 16/15 | 237.243 | ||
| 15/14 | 239.814 | ||
| 16/13 | 240.528 | ||
| 15/13 | 247.741 |
Music
- Lemba Suite, for Two Organs (Prelude, Aria & Fugue) in 8/7 eigenmonzo tuning
Notes
- ↑ L/s ratios are always precisely Φ, and mos scales are always precisely 2Φ