50edo: Difference between revisions

As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice; it is a somewhat sharp approximation of [[2/7-comma meantone]] (and is almost exactly 5/18-comma meantone). In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo for which the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It is also almost consistent to the no-21s [[25-odd-limit]], only barely missing consistent mappings of [[11/9]] and [[18/11]].

50edo is also quite strong in the realm of tertian harmony for a meantone system, as the errors on [[7/6]], [[6/5]], [[5/4]], and [[9/7]] are all balanced to be roughly half as flat as the fifth, meaning that this set of thirds taken as a whole is minimally out-of-tune given the damage induced by meantone. Though it fails to approximate [[11/9]] well by virtue of not having a perfect hemififth, it inherits the excellent [[16/13]] from [[10edo]] and additionally has a 1.2{{c}} flat [[13/11]], providing even more qualities of roughly just thirds alongside their more complex [[fifth complement]]s.

=== Odd harmonics ===

{{Harmonics in equal|50|intervals=odd|columns=11}}

{{Harmonics in equal|50|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 50edo (continued)}}

=== As a tuning of other temperaments ===

50et tempers out [[126/125]], [[225/224]] and [[3136/3125]] in the [[7-limit]], indicating it [[support]]s septimal meantone; [[245/242]], [[385/384]] and [[540/539]] in the [[11-limit]] and [[105/104]], [[144/143]] and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension [[meanpop]], it can be used to advantage for the [[coblack]] temperament (15 & 50), and provides the optimal patent val for 11- and 13-limit [[Meantone family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo| 23 6 -14 }}, so that in 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.

=== Relations ===

The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "[[Golden meantone|Golden Tone System]]" ([[Das Goldene Tonsystem]]) of [[Thorvald Kornerup]] (and similarly as the next step from 31edo in [[Joseph Yasser]]'s "[https://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").

== Intervals ==

([[Enharmonic unisons in ups and downs notation|EUs]]: v³A1 and vvd2)

| [[5/4]]

| [[14/11]]

| [[9/7]]

| [[13/10]]

| [[33/25]], ''[[55/42]]'', ''[[64/49]]''

|-

|-

|-

|-

| [[50/33]], ''[[84/55]]'', ''[[49/32]]''

|-

| ''[[44/27]]'', [[64/39]], [[33/20]], ''[[80/49]]''

| [[12/7]]

| [[7/4]]

| [[44/25]]

| [[16/9]], [[9/5]]

| [[20/11]]

| 1056

| [[24/13]], [[11/6]]

| [[15/8]], [[28/15]], [[13/7]]

| ''[[21/11]]''

| ''[[40/21]]'', [[48/25]], [[25/13]], [[27/14]]

| Downaug 7th, Dim 8ve

| ''[[52/27]]'', [[64/33]], [[35/18]],<br> ''[[49/25]]'', ''[[108/55]]'', ''[[63/32]]''

| ''[[88/45]]'', [[96/49]], [[55/28]], [[128/65]],<br> [[65/33]], [[77/39]], [[180/91]], [[196/99]],<br> [[99/50]], [[240/121]], ''[[336/169]]''

| vD

| [[2/1]]

== Notation ==

=== Stein–Zimmermann–Gould notation ===

50edo can be notated with [[Stein–Zimmermann–Gould notation]]:

Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.

=== Kite's ups and downs notation ===

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

=== Sagittal notation ===

This notation uses the same sagittal sequence as edos [[57edo #Sagittal notation|57]], [[64edo #Sagittal notation|64]], and [[71edo #Second-best fifth notation|71b]].

==== Revo flavor ====

File:50-EDO_Revo_Sagittal.svg

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 160 106 [[1053/1024]]

default [[File:50-EDO_Revo_Sagittal.svg]]

</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

=== 15-odd-limit interval mappings ===

{{Q-odd-limit intervals|50|15}}