{"id":96,"date":"2012-01-12T21:13:32","date_gmt":"2012-01-13T02:13:32","guid":{"rendered":"http:\/\/bachpeople.wordpress.com\/?p=96"},"modified":"2012-01-12T21:13:32","modified_gmt":"2012-01-13T02:13:32","slug":"appreciating-16-comma-temperament-part-iii","status":"publish","type":"post","link":"https:\/\/bachpeople.com\/jp\/appreciating-16-comma-temperament-part-iii\/","title":{"rendered":"Appreciating 1\/6-comma \u2014 Temperament Part III"},"content":{"rendered":"

(Originally posted in March, 2010)<\/p>\n

By now we hopefully understand that the Pythagorean temperament is really a stack of pure perfect 5ths (Pythagorean 5ths) and can bring out melodic beauty, and the temperament based on the narrow 5ths, each tampered by a quarter of a syntonic comma (mean-tone 5ths), can bring out chordal beauty.\u00a0 Then what about those so-called well-temperaments like Kirnberger, Werckmeister, Vallotti and Young?\u00a0 What is the 1\/6-comma mean-tone that we hear about lately?\u00a0 How different is that from the Vallotti or the Young temperaments?<\/p>\n

Before we get to the discussion of 1\/6-Pythagorean-comma systems vs. 1\/6-comma mean-tone, let\u2019s take a look at the 1\/4-comma systems.\u00a0 Among those well-temperaments, Werckmeister 1(III) [Die erste Art, Num. 3] and Kirnberger III are two popular 1\/4-comma temperaments.\u00a0 The main purpose of those temperaments is to have harmony-friendly narrow major 3rds in the \u2018normal\u2019 keys but without the wolf interval.\u00a0 No split keys required!\u00a0 In a very simplified sense, 1\/4-comma well-temperaments thus have eight pure 5ths and four narrowed 5ths that are each tampered by a 1\/4 Pythagorean comma.\u00a0 The unique part of the Kirnberger III, though, is that it actually features four mean-tone 5ths (each tampered by a 1\/4 syntonic comma instead of a 1\/4 Pythagorean comma)..!\u00a0 But as you know, the syntonic comma isn\u2019t the same as the Pythagorean comma:<\/p>\n

Syntonic comma = c. 21.51 cents<\/p>\n

Pythagorean comma = c. 23.46 cents<\/p>\n

23.46 \u2013 21.51 = schisma (1.95 cents)<\/p>\n

The schisma is soo tiny, Kirnberger just hid it between F#\u2014C# in the cycle of 5ths.\u00a0 Technically, therefore, the Kirnberger III temperament has four mean-tone 5ths, seven pure 5ths and one almost pure 5th.\u00a0 I understand a lot of keyboardists like this temperament as it has the same CGDAE as mean-tone (and no wolf, mind you!).\u00a0 Well, those well-temperaments, having a lot of pure 5ths and some harmony-friendly major 3rds at the same time, don\u2019t really stay at one place in the melodic\u2014harmonic character spectrum.\u00a0 In simpler keys, it belongs to the harmonic side of the spectrum, but as you move to the keys with more sharps or flats, it becomes more melodic in character \u2014\u2014 I think this is important.\u00a0 I don\u2019t think it\u2019s just me who had this notion that the more sharps or flats you have in the key signature the worse the keyboard sounds harmonically…\u00a0 But if you understand this character-shift from harmonic to melodic, the way those preludes and fugues in the Well-Tempered Clavier were written do makes sense.\u00a0 And remember, even when the temperament is on the melodic side of the spectrum like F# major, many 5ths are now pure; depending on how you present the triads, you still can play beautiful chords.\u00a0 I think the Eb\/D# minor prelude and fugue from the WTC book 1 (BWV 853) sounds totally and utterly beautiful in the Werckmeister 1(III)..!!<\/p>\n

Now, finally, we get to the 1\/6-comma mean-tone.\u00a0 You can get this temperament by having eleven 5ths that are each narrowed by a sixth of a syntonic comma and one wide 5th in the cycle.\u00a0 Yes, just like the 1\/4-comma mean-tone, you still get the wolf, but it isn\u2019t as bad as the wolf in the 1\/4-comma mean-tone.\u00a0 According to my quick calculation, the wolf intervals in the 1\/4-comma and 1\/6-comma mean-tone temperaments are 737.64 cents and 717.92 cents respectively.\u00a0 I understand that the 1\/6-comma mean-tone, just like the simple keys in the 1\/6-Pythagorean-comma systems, doesn\u2019t have any relevant intervals pure, but 5ths are narrow enough to make some major 3rds workable for decent harmonies.\u00a0 There are some good things about this 1\/6-comma mean-tone \u2014\u2014 but I think they\u2019re relatively minor except the one thing I\u2019m going to mention a few lines down, and I\u2019m just going to let those advocates like Dr. Ross W. Duffin do the sales pitch.\u00a0 Now, what about tonal characteristics..?\u00a0 It\u2019s not there in this temperament, as Dr. Duffin himself points out.\u00a0 It is still a \u2018mean\u2019 tone system; unlike the Pythagorean systems, here diatonic semitones are all the same, and chromatic semitones are all the same.\u00a0 That\u2019s great for fretted instruments, but this is just a \u2018somewhat\u2019 harmonic temperament without much character.\u00a0 The 1\/6-Pythagorean-comma systems, on the other hand, still can shift from harmonic to melodic, though its spectrum is narrower on the harmonic side than the 1\/4-comma systems.<\/p>\n

Practically speaking, from a string player\u2019s standpoint, I\u2019d prefer a 1\/6-comma system, because the open string 5ths in a 1\/4-comma system are too narrow and a bit uncomfortable to tune, especially in an ensemble situation (good luck if you are playing the violin-family instrument with a mean-tone keyboard!).\u00a0 But within some different 1\/6-comma systems, as long as the width of the open string 5ths are the same (C\u2013G, G\u2013D, D\u2013A, A\u2013E), I don\u2019t think we should be too picky about which temperament to choose, to be honest.\u00a0 If you\u2019re a baroque violinist, I know you have those experiences where you had to tune your instrument quickly without the aid of a keyboard instrument in performance situations.\u00a0 There\u2019s no way you could tune the narrow 5ths quickly, quietly and accurately without a keyboard instrument, but you adjust pitches anyway, and you get by.\u00a0 This is to give you a perspective: around the pitch A we use for tuning, a 4-cent difference between two As in unison would give you about one acoustic beat a second.\u00a0 The same 4-cent difference between two As at an octave below will cause about a beat per 2 seconds.\u00a0 The higher the frequencies are, the more beats you hear (we treble instrument players do get to realize this every time we play, don\u2019t we?).\u00a0 In simpler keys, the differences between Vallotti and the 1\/6-comma mean-tone are usually within a couple cents, and when you play with a cold left hand, your inaccuracy can easily surpass those differences (maybe only mine :-D).\u00a0 If I\u2019m to choose a system for my ensemble, I\u2019d give a little edge to Vallotti; in the Vallotti temperament, you can reach the most melodic side of the spectrum in F-sharp major and E-flat minor with pure 5ths; in those key areas the sharp mediants and leading tones can give Vallotti some tension and nice melodic contours that 1\/6-comma mean-tone cannot provide.<\/p>\n

\"\"<\/a>
Vallotti and 1\/6-comma mean-tone in cents, rounded off to the nearest tenth:<\/figcaption><\/figure>\n

There are many people who don\u2019t appreciate 1\/6-Pythagorean-comma systems out there.\u00a0 Some say they are too new to use for Bach, and others say this temperament is too much like the equal temperament \u2014 nothing is particularly beautiful.\u00a0 Although Vallotti claims to have worked out his system by 1728, the first time it was cited elsewhere (that we know of) was in 1781 in England. \u00a0 Well, not all historical temperament systems were documented, and it is not inconceivable at all that some other folks from Bach\u2019s area and generation had come up with a Vallotti-like system… And I believe that the authenticity and legitimacy of specific temperament theories aren\u2019t really relevant to actual tuning system in current performing arena.\u00a0 Also, if a 1\/6-Pythagorean-comma system is too moderate and not harmony-friendly enough, no other 1\/6-comma system really is, including the 1\/6-comma mean-tone.\u00a0 Granted that a 1\/6-Pythagorean-comma system cannot give you pure 3rds, it can give you pure 5ths with their Pythagorean melodic quality in a certain key area.\u00a0 The 1\/6-comma mean-tone cannot give you anything pure that\u2019s relevant, mind you.\u00a0 Why do they dislike the 1\/6-Pythagorean-comma systems so much, really?\u00a0 I think that 1\/6-Pythagorean-comma systems are the most well-compromised temperaments we have today that still incorporate the character-shift that I find important.<\/p>\n

Sometimes people ask me what I think about Dr. Bradley Lehman\u2019s new Bach temperament based on the suspiciously irregular drawing on the title page of the Well-Tempered Clavier Book I.\u00a0 Actually, after skimming Dr. Lehman\u2019s explanation, it does make sense for me to interpret the drawing that way (it\u2019s not really a puzzle).\u00a0 And it also seems to be another ensemble-friendly temperament; it has five narrowed 5ths tempered by 1\/6-Pythagorean commas, three pure 5ths, three slightly narrow 5ths tempered by 1\/12 commas AND one slightly wide 5th tempered by a 1\/12 comma.\u00a0 It is basically another 1\/6-Pythagorean-comma system with yet narrower character spectrum, but when I get a chance I\u2019d like to play around with it.\u00a0 One thing is certain already about this temperament though; because of the fact that this has only three pure 5ths, the Eb\/D# minor prelude and fugue from the WTC Book I in this temperament won\u2019t be as beautiful as it is in the Werckmeister 1(III)..!\u00a0 I\u2019m sorry, I would still want my WTC in Werckmeister! \u00a0 : )<\/p>\n","protected":false},"excerpt":{"rendered":"

(Originally posted in March, 2010) By now we hopefully understand that the Pythagorean temperament is really a stack of pure perfect 5ths (Pythagorean 5ths) and can bring out melodic beauty, and the temperament based on the narrow 5ths, each tampered by a quarter of a syntonic comma (mean-tone 5ths), can bring out chordal beauty.\u00a0 Then what about those so-called well-temperaments like Kirnberger, Werckmeister, Vallotti and Young?\u00a0 What is the 1\/6-comma mean-tone that we hear about lately?\u00a0 How different is that from the Vallotti or the Young temperaments? Before we get to the discussion of 1\/6-Pythagorean-comma systems vs. 1\/6-comma mean-tone, let\u2019s take a look at the 1\/4-comma systems.\u00a0 Among those well-temperaments, […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"gallery","meta":{"nf_dc_page":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[15,5],"tags":[17,25,69,77,88,104,105,111,119,120],"class_list":["post-96","post","type-post","status-publish","format-gallery","hentry","category-early-music","category-essays","tag-1-6-comma","tag-bach","tag-kirnberger","tag-meantone","tag-pythagorean-comma","tag-syntonic-comma","tag-temperament","tag-vallotti","tag-well-tempered-clavier","tag-werckmeister","post_format-post-format-gallery"],"_links":{"self":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/posts\/96","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/comments?post=96"}],"version-history":[{"count":0,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/posts\/96\/revisions"}],"wp:attachment":[{"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/media?parent=96"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/categories?post=96"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bachpeople.com\/jp\/wp-json\/wp\/v2\/tags?post=96"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}