No-fives subgroup temperaments: Difference between revisions

This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 5.  

Subgroup: 2.3.7

[[Tp tuning #T2 tuning|RMS error]]: 2.523 cents

Subgroup: 2.3.7

[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[Dominant (temperament)|dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.

[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents

Subgroup: 2.3.7.11.13.19

Restricted to 2.3.7.11.13, this temperament is a no-5 restriction of 13-limit [[Ultrapyth]]. This temperament was created to yield [[blackdye]] tunings where aberrisma-altered 3-limit thirds become tempered 13/11~19/16 and 14/11.

Subgroup: 2.3.7

Related temperament: [[Meantone family #Squares|squares]]

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents

: <small>''For the full 19-limit version of this temperament, see [[Tetracot family #Byhearted]].''</small>

[[Tp tuning #T2 tuning|RMS error]]: 0.8727 cents

Subgroup: 2.3.7

Related temperament: [[Meantone family #Septimal meantone|septimal meantone]]

{{see also|Gentle region}}

Related temperaments: [[Hemifamity temperaments #Leapday|leapday]], [[Sensamagic clan #Leapweek|leapweek]], [[srutal]]

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents

Subgroup: 2.3.7.11.13.23

[[Tp tuning #T2 tuning|RMS error]]: 0.6265 cents

Subgroup: 2.3.7.11.13.23.29

** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]

{{see also|Heartland}}

Related temperaments: [[octacot]]

Subgroup: 2.3.7.11

Related temperaments: [[octacot]]

Subgroup: 2.3.7.11.19

Related temperaments: [[octacot]]

Subgroup: 2.3.7

[[Tp tuning #T2 tuning|RMS error]]: 0.4277 cents

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 0.6937 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 1.032 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 0.3891 cents

Subgroup: 2.3.7.29

{{Optimal ET sequence|legend=1| 19, 22, 41, 145, 186j, 227j }}

Subgroup: 2.3.7.11.13.29

{{Optimal ET sequence|legend=1| 19, 22f, 41 }}

Subgroup: 2.3.7

[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents

Subgroup: 2.3.7

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

Subgroup: 2.3.7.13

[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents

[[Subgroup]]: 2.3.7.13.17

[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents

Subgroup: 2.3.7

Related temperaments: [[Breedsmic temperaments #Hemififths|hemififths]], namo

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents

{{see also|Heartland}}

Related temperaments: [[Tetracot family|bunya]]

{{see also|Heartland}}

Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquiquartififths]]

Subgroup: 2.3.7.11

Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]

Subgroup: 2.3.7.11.19

Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]

[[Subgroup]]: 2.3.7

Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]

Subgroup: 2.3.7.11

Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]

Subgroup: 2.3.7.11.13

Related temperaments: [[Schismatic family|tsaharuk]], [[Hemifamity temperaments|quanic]]

{{see also| 5th-octave temperaments #Slendrismic }}

In slendrismic, the period (1\5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])² = [[8/7]] * [[1029/1024|1029/1024 = S7/S8]], which is a significant interval as it is the "harmonic 5edostep" in that it's a [[rooted]] (/2^n) interval that approximates 1\5 very well. The generator is [[1029/1024]], the difference between [[8/7]] and [[147/128]] and therefore between 3/2 and (8/7)³. The temperament is named for the very "slender" generator as well as as a pun on "[[slendric]]" (which it shouldn't be confused with). One can consider this as a microtemperament counterpart to [[cloudy]], which equates them.

Related temperaments: [[Pental family|hemipental]]

Defined as the 320 &amp; 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.

[[Tp tuning #T2 tuning|RMS error]]: 0.0164 cents

Subgroup: 2.3.7.13

Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]

{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}  

Subgroup: 2.3.7.13.17

Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]

{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}

Subgroup: 2.3.7.13.17.19

{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}

Subgroup: 2.3.7.11

[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents

{{see also| Pentacircle temperaments #Parapyth }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents

Subgroup: 2.3.7.11.13

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

{{Main| Heartland }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents

Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]], and with a generator representing both [[3/2]] and [[16/11]]. It may be considered an [[exotemperament]] by some definitions and not one by others. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.

[[Badness]]: 0.185

[[Subgroup]]: 2.3.11

[[Tp tuning #T2 tuning|RMS error]]: 1.237 cents

Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[neutrominant]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].

* [[neutraltet17|Seventeen note mega chromatic scale]]

Subgroup: 2.3.11.13

[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents

[[Subgroup]]: 2.3.11.19

{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}

[[Subgroup]]: 2.3.7.11.19

{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}

Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.

[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents

[[Subgroup]]: 2.3.11.13

{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}

Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. [[27/26]] is tempered out rather than [[1053/1024]], so [[13/8]] is a major sixth. These temperamenets intersect in [[7edo]], where major sixths and minor sixths are not distinguished.

[[Tp tuning #T2 tuning|RMS error]]: 4.367 cents

[[Subgroup]]: 2.3.13

[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents

[[Subgroup]]: 2.3.17

[[Subgroup]]: 2.3.85

<nowiki>*</nowiki>Wart for 85

[[Subgroup]]: 2.3.7.85

<nowiki>*</nowiki>Wart for 85

The dog temperament is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19 subgroup analogue of [[Pelogic family|mavila]].

[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents

Boethian is a diatonic-based temperament that makes [[513/512]] vanish, so [[19/16]] is a minor third. It can be viewed as a 2.3.19 subgroup analogue of [[Schismatic family|schismic temperament]].

[[Badness]]: 0.000374

Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.

[[Badness]] (Smith): 8.998 × 10^-3

[[Subgroup]]: 2.3.29

{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}

This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].

{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}

[[Subgroup]]: 2.3.7.37

{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53d }}

It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.

{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}

[[Subgroup]]: 2.3.5.7.37

{{Optimal ET sequence|legend=1| 5, 14c, 19 }}

[[Subgroup]]: 2.3.37

{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}

{{Main| Reversed meantone }}

{{Optimal ET sequence|legend=1| 5, 12, 17 }}

Subgroup: 2.3.7.41

{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}

Subgroup: 2.3.7.11.41