Tetrameantone: Difference between revisions
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The '''tetrameantone''' temperament is a [[nonoctave]] [[ | The '''tetrameantone''' temperament is a [[nonoctave]] [[regular temperament|temperament]] [[tempering out]] [[81/80]] in the 4.3.5-[[subgroup]]. It is similar to [[meantone]] but repeats at the double octave [[4/1]]. It is generated by [[3/1]] and, like in normal meantone temperament, four of them make a [[5/1]] plus two double octaves. | ||
For technical information see [[Subgroup temperaments#Tetrameantone]]. | |||
== Interval chain == | == Interval chain == | ||
In the following table, odd harmonics 1–5 are '''bolded'''. | |||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
! | ! # | ||
! Cents | ! Cents* | ||
! Approximate | ! Approximate Ratios | ||
|- | |- | ||
| 0 | | 0 | ||
| 0.000 | | 0.000 | ||
| | | '''1/1''' | ||
|- | |- | ||
| 1 | | 1 | ||
| | | 1896.239 | ||
| | | '''3/1''' | ||
|- | |- | ||
| 2 | | 2 | ||
| | | 1392.477 | ||
| | | 9/4, 20/9 | ||
|- | |- | ||
| 3 | | 3 | ||
| | | 888.716 | ||
| | | 5/3 | ||
|- | |- | ||
| 4 | | 4 | ||
| | | 384.955 | ||
| | | '''5/4''' | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 2281.193 | ||
| | | 15/4 | ||
|- | |- | ||
| 6 | | 6 | ||
| | | 1777.432 | ||
| | | 45/16 | ||
|- | |- | ||
| 7 | | 7 | ||
| | | 1273.671 | ||
| | | 25/12 | ||
|} | |} | ||
<nowiki>*</nowiki> in 4.3.5-subgroup POTE tuning | |||
== Tetrameantone on tritave == | == Tetrameantone on tritave == | ||
Tritave-repeating tetrameantone (3.4.5 subgroup) is made by cutting off the 4/3 interval range of the tetratave scale. extension will start at 3.4.5.14 subgroup. If we want strict no-twos subgroup, will be 3.5.13. | Tritave-repeating tetrameantone (3.4.5 subgroup) is made by cutting off the 4/3 interval range of the tetratave scale. extension will start at 3.4.5.14 subgroup. If we want strict no-twos subgroup, will be 3.5.13. | ||
| Line 150: | Line 129: | ||
| 13/5 | | 13/5 | ||
|} | |} | ||
[[Category: | === b15 & b4p === | ||
[[Category: | : ''Temperament being restricted in different directions: [[No-fives subgroup temperaments #Superflat]]'' | ||
Subgroup: 3.5.13 | |||
[[Comma list]]: 531441/528125 | |||
[[Sval]] [[mapping]]: [{{val| 1 2 1 }}, {{val| 0 -2 5 }}] | |||
Sval mapping generators: ~3, ~325/243 | |||
[[POTE generator]]: ~325/243 = 508.099 | |||
[[Category:Tetrameantone| ]] <!-- main article --> | |||
[[Category:Non-octave temperaments]] | |||
{{Todo| review }} | |||
Latest revision as of 09:28, 29 April 2025
The tetrameantone temperament is a nonoctave temperament tempering out 81/80 in the 4.3.5-subgroup. It is similar to meantone but repeats at the double octave 4/1. It is generated by 3/1 and, like in normal meantone temperament, four of them make a 5/1 plus two double octaves.
For technical information see Subgroup temperaments#Tetrameantone.
Interval chain
In the following table, odd harmonics 1–5 are bolded.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 1896.239 | 3/1 |
| 2 | 1392.477 | 9/4, 20/9 |
| 3 | 888.716 | 5/3 |
| 4 | 384.955 | 5/4 |
| 5 | 2281.193 | 15/4 |
| 6 | 1777.432 | 45/16 |
| 7 | 1273.671 | 25/12 |
* in 4.3.5-subgroup POTE tuning
Tetrameantone on tritave
Tritave-repeating tetrameantone (3.4.5 subgroup) is made by cutting off the 4/3 interval range of the tetratave scale. extension will start at 3.4.5.14 subgroup. If we want strict no-twos subgroup, will be 3.5.13.
| Generators | Cents (15edt) | Approximate ratios 3.4.5.14 | 3.5.13 |
|---|---|---|---|
| -7 | 253.594 | 7/6 | 15/13 |
| -6 | 760.782 | 25/16, 63/40, 14/9 | 125/81 |
| -5 | 1267.970 | 25/12, 21/10 | 27/13 |
| -4 | 1775.158 | 45/16, 14/5 | 25/9 |
| -3 | 380.391 | 5/4 | 81/65 |
| -2 | 887.579 | 27/16, 5/3 | 5/3 |
| -1 | 1394.767 | 9/4, 20/9 | 729/325 |
| 0 | 0.000 | 1/1 | 1/1 |
| 1 | 507.188 | 4/3, 27/20 | 325/243 |
| 2 | 1014.376 | 16/9, 9/5 | 9/5 |
| 3 | 1521.564 | 12/5 | 65/27 |
| 4 | 126.797 | 16/15, 15/14 | 27/25 |
| 5 | 633.985 | 36/25, 10/7 | 13/9 |
| 6 | 1141.173 | 48/25, 40/21, 27/14 | 243/125 |
| 7 | 1648.361 | 18/7 | 13/5 |
b15 & b4p
- Temperament being restricted in different directions: No-fives subgroup temperaments #Superflat
Subgroup: 3.5.13
Comma list: 531441/528125
Sval mapping: [⟨1 2 1], ⟨0 -2 5]]
Sval mapping generators: ~3, ~325/243
POTE generator: ~325/243 = 508.099