Alternate Tunings
March 1 , 2009
Not much progress here of late. Actually, the animations currently titled "Ratios 1, 2" started out as studies of Just Intonation but rather quickly led me into the mathematics/calculus studies which is the current focus of the musemath website.
General browsing brought up the following which seems worth noting under Alternate Tunings:
Unequal Temperaments: Theory, History and Practice
1978 Revised 2008
Claudio Di Veroli
good quote:
"Theoretical/geometrical [fretting] values are useful to get an initial idea about the place and shape of the frets and related consistency issues … The charts … help the player to mind the few notes that need a bit of 'tweaking' while playing. However, the final position of the frets for accurate tuning has to be found empirically…"
A review by David Bauguess is very good :http://www.pianoworld.com/ubb/cgi-bin/ultimatebb.cgi?ubb=get_topic;f=3;t=004424;p=0
" Four individuals were particularly helpful and influential: Paul Bailey, Owen Jorgensen, Ed Foote, and Claudio di Veroli.
... (di Veroli)... is an excellent researcher, an early music performer, and harpsichord tuner. (BTW, his Ph.D. is in statistics.)"
***
Some notes indicate the following may be useful here:
http://www.huygens-fokker.org/scala/
May 5 , 2007
These are some preliminary ideas and source materials I am looking at to see if it may be possible to create an animation or two that will help with some of the tuning explanations.
My approach - "What is common to all these tunings that can be demonstrated in a Flash animated sequence?" "Can I avoid innumerable tables and long written lists of fractions?" I do not expect anything new or unique - spirals, circles, progressions, string division, etc. The hoped for improvement would be in the design and interactivity.
I do not advocate one or the other of the various systems, the music will have to do that for itself. Personally, I am still using and listening to 12 tet, perhaps with a bit more attention to detail and some increased interest in "Non-Western" music.
I rather like the following article:
http://music-cog.ohio-state.edu/Music829B/roughness.html
Bill Hartmann
consonance-dissonance
http://music-cog.ohio-state.edu/Music829B/notes.htmlMusic 829B: Consonance and Dissonance
...Tuning
There are four main reasons why modern scholars have lost interest in the question of what is the best tuning system. First, in the 1930s, Carl Seashore measured the pitch accuracy of real performers and showed that singers and violinists are remarkably inaccurate. For non-fixed-pitch instruments, the pitch accuracy is on the order of 25 cents. Yet Western listeners (and musicians) are not noticeable disturbed by the pitch intonation of professional performers. Secondly, on average, professional piano tuners fail to tune notes more accurately than about 8 cents. This means that even if performers could perform very accurately, they would find it difficult to find suitable instruments. Thirdly, listeners seemingly adapt to whatever system they have been exposed to. Most Western listeners find just intonation "weird" sounding rather than "better". Moreover, professional musicians appear to prefer equally tempered intervals to their just counterparts. See the results of Vos (1986). Finally, the perception of pitch has been shown to be categorical in nature. In vision, many shades of red will be perceived as "red". Similarly, listeners tend to mentally "re-code" mis-tuned pitches so they are experienced as falling in the correct category. Mis-tuning must be remarkably large (>50 cents) before they draw much attention. This insensitivity is especially marked for short duration sounds -- which tend to dominant music-making.
| Just Intonation | Other Temperaments |
| 31 Tone Equal Temperament | More on Alternate Tunings |
| Mean Tone | |
| Lucy Tuning | |
February 26, 2007 - I have been reading up on just intonation but no good ideas for a clarifying animation are coming to mind. The following article might be good for the more mathematically minded. This may be ephemeral, the home page address includes the words "removethis" so, if interested, be sure to save a copy.
http://math.ucr.edu/home/baez/week234.html
June 12, 2006
This Week's Finds in Mathematical Physics (Week 234)
John Baez
Today I'd like to talk about the math of music - including torsors, orbifolds, and maybe even Mathieu groups. But first, some movies of the n-body problem:(ror- The following is about the only part I understand, if I understand correctly:)
"...there's an unavoidable conflict between the desire for simple ratios and the desire for evenly spaced notes, built into the fabric of mathematics and music. Every tuning system is thus a compromise."
Other ideas in Just Intonation:
http://en.wikipedia.org/wiki/Limit_(music)
In music, a limit is a degree of harmonic complexity based on pitches’ relationships in “just intonation,”...
http://www.windworld.com/tools/fretnote.htm
You can define a just scale by giving the frequency ratio for each degree of the scale relative to the tonic note. Thus, a basic major scale in just intonation over one octave might be: 1:1 (the unison, or first degree to which all the following ratios relate), 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2:1.
By selecting different ratios, an endless variety of just
scales could be devised...
The intervals created this way tend to be unequal -- for instance, the "whole step" between the first degree in the major scale above (1:1) and the second degree (9:8) is not the same size as the "whole step" between the second degree (9:8) and the third (5:4). This unequalness makes the business of fret placement for fretted string instruments quite a bit more complicated.
...if you have different strings on the instrument tuned to
different open pitches, the fret locations under each string will have to be
different; the frets cannot cross the neck in straight lines. Many makers and
players have experimented with fretted instruments in just intonation using many
short "fretlets" placed here and there under the strings, or using wavy frets,
or even using movable frets and fretlets...
Encyclopedia of Microtonal Music Theory just intonation
[Joe
Monzo]
Rational tuning systems with a prime or odd-limit higher than 5 are frequently called "extended just intonation"
(ror - What are lattice diagrams/interval matrix things anyway, should I care? Do I need to delve into such topics?)
Doty, David,
Just Intonation Network webpage
What is JUST INTONATION?:
Just Intonation is not a particular scale, nor is it tied to any particular musical style. It is, rather, a set of principles which can be used to create a virtually infinite variety of intervals, scales, and chords which are applicable to any style of tonal music (or even, if you wish, to atonal styles).
http://www.rev.net/%7Ealoe/music/just.html
Just intonation employs rational intervals between pitches. This tends to reduce or eliminate beating when several notes are played at the same time. Brass instruments and some vocal harmonies use just intonation. As there are theoretically an infinite number of just intervals (with a practical limit that can vary with the acoustical properties of the source of the sound), parameters can be chosen to reduce the number of intervals in a scale. These can include
establishing an upper limit on the size of the factors,
establishing an upper limit on the size of the prime factors,
exclusion
of factors based on acoustical choice,
establishing a minimum space between
pitches,
etc.
Most just tunings use several of these
parameters...
(NOTE: A number of just scales are listed on the page with
specific sets of prime factors, as labeled. - ror)
http://users.rcn.com/dante.interport//index.html
Dante
Rosati , a teacher of guitar at Julliard, among other things, provides us with
some mp3 samples of his compositions on his 21 tone just intonation guitar. http://users.rcn.com/dante.interport//guitar.html
Here is an excerpt from his page on just
intonation:
Harmonic Theory and Just Intonation
http://users.rcn.com/dante.interport//hartheory1.html
Just as prime numbers are the building blocks of the natural numbers, so are they the building blocks of sounds...
...In the harmonic series, even numbered partials always generate octave "duplications" of some lower partial...The identification with the fundamental lessens still further with each successive odd number introduced, while even partials always echo a quality from lower in the series...
...Each partial is also itself the fundamental of its own series. For example, the ninth partial of a series is coincident with the third partial of its own third partial (3x3). This shows that the ninth partial is, on one level, redundant, because it can be derived both from the original series and as a projection of lower combined first-order and second-order partials...
...The seventh partial, on the other hand, cannot be produced from lower partials upward by any method. It is "new" in a deeper sense than the ninth is, or the fifteenth. Nine sounds like it belongs with its parent: three...Seven is, on the other hand, an original and unprecedented emanation from the source, as are 11 and 13. So primes mark the beginnings of new lineages of number relations...
From another page of his:
http://users.rcn.com/dante.interport/justguitar2.html
Just intonation is the system of tuning which most directly reflects the stucture of the sounds of a vibrating string. There are certain intervals which are highly consonant because of the way the partials of the constituent sounds match up. The most consonant interval is of course the unison (1/1), and after that the octave (2/1). Then comes the perfect fifth (3/2) and the fourth (4/3). These are the only consonances that were recognized by the Greek theorists and they form the basis of their tetrachord system...
...All ratios can be thought of as intervals within the harmonic series. Thus 3/2 can be thought of as the interval between the second and third partial, 4/3 as the interval between the third and fourth partial, etc. Thus, the harmonic series, as exhibited by a vibrating string, has encoded within it all possible intervals that can be expressed as ratios of whole numbers. Roughly speaking, the smaller the numbers involved, the more consonant the interval...
...Therefore, in choosing a set of ratios within an octave which will make up your vocabulary of tones, you start with the lowest integer ratios and work your way up the series...
...The question then becomes where to stop and how many of the ratios to include. The more ratios you use, the more sounds you will have at your disposal...
...However, there are a few things to consider.
First, frets can only be so close together before it becomes difficult or impossible to play...
Also, there is a limit to what our ears can hear. There's no point in having two different ratios in your scale if, for all practical purposes, they are indistinguishable...
...So, you can generate a scale using rules, but there comes a
point where the reality of execution becomes a limiting factor...
http://www.justintonation.net/
The subject of alternate
tunings (based on other than 12 tone equal temperament [12EDO -Equal Divisions
of the Octave]) can be researched from this site. Carl Lumma's site has
recordings of what he finds are some "Essential Microtonal Recordings". Kyle
Gann has a nice article on alternate tunings
Something around 30 divisions of the octave seems about the most anyone has had any success with so far as fretted acoustic guitars are concerned. (From my reading to date, this may be the case with just intonation, equal temperament and lucy tuning.) This may be something of an upper limit for acoustic guitars - but I am only speculating - I have downloaded an mp3 of a gutiar with I think 53 frets/octave, unfortunately I didn't like the piece, a lot, and failed to save it for reference.
David Canright has an article on a 38 fret/octave just intonation guitar in which he expresses some satifsfaction but also some reservations and the idea that multiple interchangeable fretboards might be preferable http://www.redshift.com/~dcanright/guitar/.
John Lucy 19, 25 , and 31 fret/octave guitars. http://www.harmonics.com/lucy/lsd/frets1931.html
**************
http://math.truman.edu/~thammond/history/Babylonia.html
Then; to get to the exact article > Find > scriba
(Note: this review is one of many on the page - I believe it
is the only one particularly relevant to Musemath. Most of the articles are
beyond my capabilities - I am particularly interested in the algorithm of Viggo
Brun for "what kind of equally tempered scale best reconciles the intervals of
an octave, fifth, and third (2:1, 3:2, and 5:4) simultaneously" - one solution
of interest to me- (31, 18, 10) - ror
Some photocopies of papers on Brun's method may be found in pdf format at:
http://anaphoria.com/wilson2.html
http://anaphoria.com/mandelbaum.PDF
http://anaphoria.com/viggo0.PDFHome Page
http://www.anaphoria.com/index.html
(If you are really good at math and physics, there seems to be a fair collection of scientific papers here on or near the general physics/math/music subject (as well as some rather less scientific topics on which I will not comment) Brun's work I may be able to understand with a little time and thought, but there are also science topics here that are quite beyond me.) - ror
Scriba, Christoph J. Mathematics and music. (Danish) Normat 38 (1990), no. 1, 3--17, 52. SC: 01A99 (00A69), MR: 91i:01154.
The author discusses the relationship between mathematics and music from Pythagorean through modern times. His story begins in in Pythagorean times, and as he explains, the notes of the musical scale were then determined by the ratio of a perfect fifth, i.e. 3:2. Twelve intervals of a fifth are roughly equal to seven octaves, but are in reality slightly more than seven octaves, the discrepancy being the "Pythagorean comma" of 312:219, or roughly 74:73.
Archytas showed that intervals like the octave 2:1, fifth 3:2, fourth 4:3, and whole tone 9:8, or any other interval in the ratio (n+1):n cannot in fact be divided with rational numbers into two equal intervals.
However, he noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5. 5:4 can be thought of as a major third, and 6:5 can be though of as a minor third. So the ratio 3:2 is divided as 6:5:4.
Similarly, the fourth of 4:3 can be divided into the product of 7:6 and 8:7, so the ratio 4:3 is divided as 8:7:6. The interval 7:6 can be though of as a shrunken minor third and 8:7 can be though of as an enlarged whole tone. Scriba suggests that the germs of the idea of making this division lie with the Babylonians.
In the Renaissance, the musical scale was modified to take
some of these ideas into account through the work of theoreticians like Ludovico
Fogliano and Giusseppe Zarlino.
...However, it wasn't long before there were efforts to make a scale of 12 uniform steps. The first to attempt to do so was Galileo Galilei's father, Vincenso Galilei. He tried to make each step of size 18:17, though that of course led to problems. It was Simon Stevin who first had the idea of making uniform steps of size 2>1/12.
...Later on, some mathematicians even began to question the division of the scale into 12 tones, with the idea that a division into a different number of notes might lead to a more perfect representation of the intervals.
For example Christiaan Huygens defined a 31-tone system of temperament in his Lettre touchant le cycle harmonique. One source even suggests that this has "led indirectly to a tradition of 31-tone music in the Netherlands in this century".
Leonhard Euler's efforts involved an attempt to reconcile the ideal "octave" 2:1 with the ideal "fifth" 3:2. He analyzed the problem by using a continued fraction representation of the ratio log 2:log 3/2. The convergent 12/7 corresponds to the popular division of 7 octaves into a circle of 12 fifths. Other convergents include 17/12, 29/17, 41/24, and 53/31. In the last case, for example, 31 octaves would be divided into 53 fifths.
These didn't answer the question of what kind of equally tempered scale best reconciles the intervals of an octave, fifth, and third (2:1, 3:2, and 5:4) simultaneously. This may or may not influence the course of music, but Scriba shows how an algorithm by the Norwegian mathematician Viggo Brun (1885-1978) gives an answer.
If the best answers are written in terms of the number of steps in the three intervals, the best approximations are (2,1,1), (3,2,1), (5,3,2), (7,4,2), (12,7,4), (19,11,6), (31,18,10), (34,20,11), (53,31,17), (87,51,28), .... The triple (12,7,4) is the common case with 12 semitones in an octave, 7 in a "major fifth", and 4 in a "major third".
As Scriba explains, the case of the 31 tone scale has been especially important historically. In fact, Scriba tells us that it was back in the middle of the 1600s that Nicolas Vicentino described a "archicembalo" with six manuals with the octave divided into 31 parts; as mentioned above, Huygens clarified this.
Moreover, Scriba tells us that Zarlino and Salinas shortly thereafter discussed the division of the octave into 19 equal parts.
There is apparently an organ built according to the principles of the Dutch physicist D. Fokker (1887-1972) that also divides the octave into 31 parts (it is now in the Teylers Museum in Haarlem)...
Note: the article includes some links to "closely related topics".
http://www.xs4all.nl/~huygensf/english/index.html
Huygens-Fokker
Foundation - Centre for Microtonal Music (Especially 31tet, after
Huygens)
Here is an excerpt from an article written in 1967.
http://www.xs4all.nl/~huygensf/doc/realm.html
On the
Expansion of the Musician's Realm of Harmony
ADRIAAN D. FOKKER
...Approximations. In the case of two irrational numbers there is a straight-forward method to find the approximate common divisors. But the case of three or more such numbers offered an unsolved problem, until some ten years ago in Norway professor Viggo Brun gave a method which he called antanairesis 2), using the Greek word for cognate operations employed by Archimedes...
****************
Fibonacci
Equal Temperament based on Fibonacci sequence - some preliminary research may be found at "Fibonacci".
Had I found the following sites I wouldn't have created my own animations. As a source for Alternate Tuning it does not have much of a priority for me. It has however allowed me to explore irrational numbers a bit.
(Great Flash - Fibonacci site dating from 1999)
http://library.thinkquest.org/27890/goldenRatio1a.html
http://library.thinkquest.org/27890/flashIndex.html
Matt Anderson, Jeffrey Frazier, and Kris Popendorf
http://library.thinkquest.org/27890/mainIndex.html
HomePage
http://nlvm.usu.edu/en/nav/frames_asid_133_g_3_t_3.html?open=instructions
GoldenRectangles
The National Library of Virtual Manipulatives (NLVM) is an NSF supported project that began in 1999 to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-12 emphasis). The project includes dissemination and extensive internal and external evaluation.
http://en.wikipedia.org/wiki/Meantone_temperament
***************
The following refers to practical adjustment of movable gut frets.
http://home.planet.nl/~d.v.ooijen/lgs/meantone.html (February 26,2008 link broken)
Meantone temperament for lute
The examples in this article are for a renaissance lute in g'.
In the 16th century several systems of tuning were used. Some of these tunings, also known as temperaments, sound very well on renaissance lutes. Meantone temperament, in which you have pure thirds, is especially good for lute and vihuela music from the first half of the 16th century. It is not so difficult to tune your lute in meantone temperament, and the result is absolutely worthwhile. In this article I will explain about meantone temperament and show you how to reposition your frets and tune your strings for it. Finally, I will explain how to put your frets back into their positions for equal temperament.
John Lucy has had some success with this system based on the
work of John Harrison in the 18th century. This is of interest to me partly
because I used to be a navigator, all navigators owe a lot to Mr. Harrison. He
knew quite a bit about the practical subdivision of time, an important element
of music.
http://www.lucytune.com/academic/manuscript_search.html
Lucy
Tuning
"...Many still seem to judge the validity of a tuning system only by its proximity to integer frequency ratios and the harmonics, which are arrived at by this (to my mind) simplistic logic. From my readings, experiments, and observations, I am convinced that John Harrison had managed to break through this barrier in the mid-eighteenth century...
At the end of a very successful scientific career, during which
he made three major new inventions, and "won" the prize for Longitude with his
horological designs, he devoted his last years to the study of musical tuning.
This is a subject very closely akin to his other particular areas of expertise,
i.e.navigation, pendulums, harmonic motion, and the study of mechanical systems
with regard to time..
...Up to page 14, Harrison explains that he has been
unable to "solve" the problem of finding a mathematical or scientific
justification for his choice of PI as a basis of his tuning system. He gives
many musical and mathematical examples...
In his book Concerning Such Mechanism, (after "slagging off" his contemporaries), he very clearly states his conclusions: The Natural Notes of Melody may be derived mathematically from pi. He gives no experimental details, except that he used monochords, and clearly understood and criticises WNR (whole number ratio) logic and practice...
"...A true and (short, but) (*crossed out) full Account of the Foundation of Musick, or , as principally therein, of the Existense of the Natural Notes of Melody:
Wherein is shown the Absurdity of such imaginary stuff as was first crowded into the Scale of Musick by the Ancients, and has still been retained therein by our more modern Writers, in order to make the matter seem more Reasonable or Philosophical..."
http://en.wikipedia.org/wiki/Lucy_tuning%20
LucyTuning is a form of meantone temperament, derived from pi, in which the fifth is of size 600+300/p (= approximately 695.5) cents. Its main advocate is Charles Lucy, who discovered it in the eighteenth century writings of John Harrison.
This is just a placeholder for now - Well Temperament, and things I have not thought about since reading Campbell & Greated many years ago may fit in here.
So far the rest is just a copy of what is already on the "Links" page.
(Note for Guitarists - if you search for "alternate tunings + guitar" you will find some open string variations that can be applied to the standard 12 fret/octave guitar, if you search for just "alternate tunings" you are more likely going to find microtonal guitars and synthesizers and a number of other instruments and systems of music.)
http://www.absoluteastronomy.com/encyclopedia/m/ma/mathematics_of_musical_scales.htm
http://pages.globetrotter.net/roule/stimm.htm
Temperament
Resources on the Web
http://www.xs4all.nl/~huygensf/english/temperament.html
http://www.tonalsoft.com/index.html
http://www.tonalsoft.com/enc/just.htm
The company has an
extensive free encyclopedia - focus is on microtonal music but includes Sumerian
tuning math and frettings for the ancient Arab 'ud ("al-'ud") and other choice
items.
http://www.windworld.com/tools/fretnote.htm
EQUAL TEMPERAMENTS AND JUST INTONATIONS
As They Apply To Fretted
Instruments
Musical scales can take many, many forms. Some scales have
evolved in an organic cultural process, while others have been created by
theorists. It is often useful to describe particular scales mathematically,
based on the relationships between the frequencies of the pitches that make up
the scale. Different sorts of mathematical reasoning can be used to arrive at
different sorts of scale types. The two most common general types, from a
mathematical point of view, are just intonations and equal
temperaments...
http://fretfind.ekips.org/index.php
free fretfind tool for various tunings
http://eceserv0.ece.wisc.edu/~sethares/tet19/guitarchords19.html
From
Bill Sethares; The 19 fret per octave guitar is discussed here.To get to his
home page for more alternate tuning material : http://eceserv0.ece.wisc.edu/~sethares/index.html
http://www.patmissin.com/tunings/tunings.html
More than any
sane person would ever need to know about tuning harmonicas.
http://www.terryblackburn.us/music/temperament/index.html
Terry Blackburn has a simple approach and common sense attitude
towards "alternate tuning" that I find worth knowing about.
http://ccrma.stanford.edu/overview/
http://ccrma.stanford.edu/overview/pastmodeling.html
Computer Music (technical, academic - I understand very little, if any,
of this but it certainly appears authoritative enough for those
interested).
http://www.medieval.org/emfaq/harmony/pyth.html an excellent historical study of pythagorean tuning.by Margo Schulter.
Here are the sources I am currently using for Indian, Chinese, and Middle-Eastern music which can be sampled on the web and have some academic discussion as well.
Indian music
http://theory.tifr.res.in/~mukhi/Music/music.html
http://www.chrysalis-foundation.org/Bharata%27s_Vina.htm%20
Middle Eastern music:
http://www.maqamworld.com/
http://www.classicalarabicmusic.com/index.htm
It’s our
commitment to present as much information about Classical Arabic Music as
circumstances allow.A lot of good links.
Chinese music
http://www.silkqin.com/08anal/tunings.htm
http://www.silkqin.com/index.html
http://www.silkqin.com/12more/links.htm
John
Thompson
http://www.chinamusic.org/music.html
Los Angeles Classical
Chinese Orchestra
http://www.ibiblio.org/chinese-music/
The Internet Chinese
Music Archive
http://www.chrysalis-foundation.org/China%27s_Ch%27in.htm
http://www.cechinatrans.demon.co.uk/ctm-psm.html
These
sites have some theory but no music samples.