Xenharmonic Wiki

Tuning ranges of regular temperaments: Difference between revisions

VisualWikitext
Wikispaces>genewardsmith
**Imported revision 518800726 - Original comment: **
 
Overthink (talk | contribs)
autopatrolled
2,676 edits
 
(57 intermediate revisions by 12 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
There are various methods which have been suggested for defining '''tuning ranges appropriate to a given regular temperament'''.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-18 02:30:56 UTC</tt>.<br>
: The original revision id was <tt>518800726</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">There are various methods which have been suggested for defining tuning ranges appropriate to a given regular temperament.


Given a rank r p-limit regular temperament, we may define a tuning range by finding the [[http://en.wikipedia.org/wiki/Convex_hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit [[tonality diamond]], when this tuning is defined. This is the //nice// tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[http://en.wikipedia.org/wiki/Monotonic_function|monotone]] weakly increasing. This we may call the //monotone// tuning range. A tuning which is both nice and monotone is a //strict// tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a //lax// tuning.
== Diamond tuning ranges ==


While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [&lt;1 0 5|, &lt;0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form &lt;1 0 5| + a&lt;0 1 -2| = &lt;1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and hence no strict or lax tunings.
[[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] defined some important tuning ranges. Their "valid" range was defined in ''Tuning Continua and Keyboard Layouts'' in the ''Journal of Mathematics and Music''<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2008). [https://www.researchgate.net/publication/228091827_Tuning_continua_and_keyboard_layouts Tuning continua and keyboard layouts]. Journal of Mathematics and Music, 2(1):1–19.</ref>; according to Milne, this tuning range was Sethares's contribution. Their "purer" range was discussed in the technical report ''X_System'' in the Open University’s repository; this was Milne's contribution. Today these are known as [[diamond monotone]] and [[diamond tradeoff]], respectively.  


</pre></div>
=== History ===
<h4>Original HTML content:</h4>
In 2014, these tuning ranges were discussed by Milne, [[Gene Ward Smith]], and others. The "valid" range was proposed names such as "lax", "monotone", and "normal", while the "purer" range was proposed to be named "strict", and the combination proposed to be named "nice". After some churn (in the edit history of this page), valid/lax/monotone/normal became "valid", while purer/strict became "nice", and the combination became "strict". In other words, "nice" and "strict" for some reason got switched, and for some time and to some extent on the wiki and in the regular temperament community that's how they stuck.  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tuning Ranges of Regular Temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;There are various methods which have been suggested for defining tuning ranges appropriate to a given regular temperament.&lt;br /&gt;
 
&lt;br /&gt;
In May 2021 Milne agreed with a community effort to revise the names to be more specific, descriptive, and closer to their original meaning. So the original "valid" became "diamond monotone", the original "purer" become "diamond tradeoff", and the combination of these two was left unnamed.
Given a rank r p-limit regular temperament, we may define a tuning range by finding the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow"&gt;convex hull&lt;/a&gt; in &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;tuning space&lt;/a&gt; of the tunings with one &lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;eigenmonzo&lt;/a&gt; 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt;, when this tuning is defined. This is the &lt;em&gt;nice&lt;/em&gt; tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow"&gt;monotone&lt;/a&gt; weakly increasing. This we may call the &lt;em&gt;monotone&lt;/em&gt; tuning range. A tuning which is both nice and monotone is a &lt;em&gt;strict&lt;/em&gt; tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a &lt;em&gt;lax&lt;/em&gt; tuning.&lt;br /&gt;
 
&lt;br /&gt;
=== Examples ===
While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [&amp;lt;1 0 5|, &amp;lt;0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form &amp;lt;1 0 5| + a&amp;lt;0 1 -2| = &amp;lt;1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and hence no strict or lax tunings.&lt;/body&gt;&lt;/html&gt;</pre></div>
==== 5-limit meantone ====
 
To illustrate the diamond tuning ranges, let's consider 5-limit [[meantone]]. This is a 5-limit temperament so the tonality diamond is {1, 3, 5, 1/3, 5/3, 1/5, 3/5}, or octave reduced, {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. It is also a rank-2 temperament, so any particular tuning can be specified by tunings of the two generators, which we take to be the tempered 2/1 and 4/3.
 
===== Diamond tradeoff =====
 
To find the range of diamond tradeoff tunings, we fix the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged interval) to be 2/1. Then we iterate through the 5-limit tonality diamond for what to use for the other eigenmonzo. (If the rank were more than 2, we would be iterating over subsets of the tonality diamond of size ''r'' - 1, but since ''r'' = 2 we are iterating over single ratios.)
* 1/1 is the 0-vector monzo and so it is always a (trivial) eigenmonzo of any tuning. We have to use a non-1/1 interval as the other eigenmonzo besides 2/1 to define a tuning.
* If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean.
* If 3/2 is the eigenmonzo, that is equivalent to 4/3 being an eigenmonzo because if 2/1 and 4/3 are tuned pure, then 3/2 is automatically tuned pure also.
* If 5/4 is the eigenmonzo, then 5/1 is also tuned pure so the fifth is 5^(1/4) and the generator is 2/5^(1/4). Therefore the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.
* If 8/5 is the eigenmonzo, that's equivalent to 5/4 being the eigenmonzo and leads to the same tuning.
* If 6/5 is the eigenmonzo, then 12/5 is also tuned pure so the fourth (the generator) is (12/5)^(1/3). Therefore the tuning is [2/1, (12/5)^(1/3)], or third-comma meantone.
* If 5/3 is the eigenmonzo, that's equivalent to 6/5 being the eigenmonzo.
These lead to three distinct tunings:
* [2/1, 4/3] or Pythagorean - 4/3 and 3/2 are pure
* [2/1, 2/5^(1/4)], or quarter-comma meantone - 5/4 and 8/5 are pure
* [2/1, (12/5)^(1/3)], or third-comma meantone - 6/5 and 5/3 are pure
These three are the possible extreme points of the diamond tradeoff tuning range, so to describe the whole range, we must take their convex hull. In this case it is easy because they are all collinear, and Pythagorean and third-comma are the two endpoints of the line segment. So the diamond tradeoff tuning range of 5-limit meantone consists of exactly those tunings with a pure 2/1 as the period and anywhere from 4/3 to (12/5)^(1/3) (498.045 to 505.214 in cents) as the generator.
 
===== Diamond monotone =====
 
To find the range of diamond monotone tunings, we need all the steps in between consecutive members of the tonality diamond to be positive. So 6/5, 25/24, 16/15, and 9/8 must all be positive for the tuning to be diamond monotone. If we denote the octave period by ''p'' and the perfect fourth generator by ''g'', this yields the equations:
* tempered 6/5 = 3''g'' - ''p'' > 0
* tempered 25/24 = 3''p'' - 7''g'' > 0
* tempered 16/15 = 5''g'' - 2''p'' > 0
* tempered 9/8 = ''p'' - 2''g'' > 0
These are all homogenous equations, so we can divide through by ''p'' and rearrange to get restrictions on the ratio ''g''/''p'':
* ''g''/''p'' > 1/3
* ''g''/''p'' < 3/7
* ''g''/''p'' > 2/5
* ''g''/''p'' < 1/2
Of these it can be seen that the first and last are redundant, and the overall diamond monotone tuning range can be summarized as 2/5 < ''g''/''p'' < 3/7, in other words all meantone tunings between [[5edo]] and [[7edo]] (480 to 514.286 cents).
 
Note that, since the definition of diamond monotone only depends on the ordering of intervals and not on their absolute size, in theory any amount of stretching or compression is allowed. For example, ''p'' = 12 cents and ''g'' = 5 cents is technically a diamond monotone meantone tuning, as is ''p'' = 12000 cents and ''g'' = 5000 cents.
 
===== Diamond tradeoff and diamond monotone =====
 
In this particular case all of the diamond tradeoff tunings are also diamond monotone, so the diamond tradeoff range is entirely inside the diamond monotone range.
 
==== 11-limit marvel ====
 
===== Diamond monotone =====
 
The [[mapping]] provided for [[Marvel_family#Undecimal_marvel_.28unimarv.29|undecimal marvel]] is {{rket|{{map|1 0 0 -5 12}} {{map|0 1 0 2 -1}} {{map|0 0 1 2 -3}}}}. We don't know the tuning of our generators yet, so our [[tuning map]] has variables in it: {{bra|1 ''a'' ''b''}}. This means that our first generator (the period) is 1 octave, the second generator is ''a'' octaves, and the third generator is ''b'' octaves. If we left-multiply the mapping by this tuning map, we get a parameterized tuning of {{val| 1 ''a'' ''b'' 2''a''+2''b''-5 12-''a''-3''b'' }} undecimal marvel. Or in other words, all tunings of undecimal marvel are of this form.  
 
Applying this to the steps of the 11-limit tonality diamond, then, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for edos 19, 22, and 31. This is the diamond monotone range.
 
===== Diamond tradeoff =====
 
The diamond tradeoff range is a quadrilateral with vertices (given in terms of frequency ratios rather than log base 2 or cents) [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]].
 
===== Diamond tradeoff and diamond monotone =====
 
The three vertices with entirely rational number values for the approximations of 3 and 5 are not in the diamond monotone range, so only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is both diamond tradeoff and diamond monotone. Other examples of tunings that are both diamond tradeoff and diamond monotone are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the diamond tradeoff range.
 
== Other tuning ranges ==
 
The diamond tuning ranges, though they have historical momentum, do not preclude definition of other validity ranges for the tuning of temperaments. The topic of tuning ranges is relatively subjective. Milne himself has described the diamond tuning ranges as "convenient mathematical fictions", and proposed that the reality would be to define some sort of empirically obtained range of tunings over which a sample of participants can correctly identify that tuning's intervals in the way prescribed by the mapping. But realistically, that is an almost impossible question to even ask of participants, and relies upon all sorts of a priori assumptions about categorizations of intervals by their ratio, which is quite possibly an entirely bogus notion.
 
[[Paul Erlich]] has proposed other tuning ranges, such as the set of regular tunings in which the temperament has up to 2×, 5×, 10× etc. its optimal damage under some metric (such as [[Kees_height|KE]]), or a set of absolute cutoffs on damage applied across all temperaments, though there could be no objective value for such cutoffs that would be both amenable to the entire community as well as useful for the entire set of regular temperaments (including extreme cases like [[macrotemperaments]] and [[microtemperaments]]).
 
[[Dave Keenan]] has proposed the range over which there is not a "better" temperament that maps the generators differently, for some definition of "better", likely taking into account both error and complexity.
 
Others have proposed the [[step ratio spectrum]] as a helpful way of thinking about tuning ranges.
 
== References ==
<references/>
 
[[Category:Math]]
[[Category:Regular temperament theory]]
[[Category:Tuning]]
[[Category:Terms]]
[[Category:Diamond]]
[[Category:todo:simplify]]
Retrieved from "https://en.xen.wiki/w/Tuning_ranges_of_regular_temperaments"