Wednesday, July 18, 2018

Meantone temperament

This is a hand out from BU choral workouts. We were practicing singing half tones.
My question on Facebook: Anybody knows how to sing half tone? Does bA and #G sound the same? Why there are different names? makes me think they might slightly different in pitch.

Answers from friends:

Jim Binkley if a system is based on just intonation, Ab and G# are different pitches. With Equal Temperament - they are the same.

Ralph Knag Jim's right - here's some more words: "They are only the same pitch in equal temperament tuning. In previous tuning systems, the black keys were a major problem in which ones to tune them to - for example, do you tune it to G# or Ab?

They are not the 
same note, and they do not have the same pitch, because they belong on different diatonic scales. Ab is slightly sharper than G#. It was a problem back then, because keyboard instruments back then could not play all 24 diatonic scales without having at least 1 of them having wolf intervals. For example: the key G#/Ab was tuned to G#, and the D#/Eb key was tuned to Eb, because that's what they used more often. G#-Eb (actually a diminished sixth) is supposed to be enharmonically equivalent to a perfect fifth, but it was so out of tune that it was dubbed the wolf fifth.

Today's keyboards are tuned to a compromise system called equal temperament, in which each note is the same distance from each other. The problem with this is that thirds and sixths tend to be quite out of tune, but because of its versatility, it has remained in widespread use."

David Badagnani In 12-tone equal temperament, fifths and fourths are within 2 cents of just intonation, but major thirds are 13.69 cents too large (400 cents as opposed to approximately 386.31 cents for just intonation). That means if you play a major third on a piano you hear a horrible 5 beats per second, something piano turners actually use in order to know if a piano is "in tune" or not. The 386.31-cents pure (just) major third doesn't have beats. The meantone tunings used in the Renaissance and Baroque periods have thirds that are closer to this ideal, though somewhat compromised for more distant keys.

Good string players and vocalists do slightly lower or raise their half steps depending on whether the pitch is the third of a chord or a leading tone, for example, but, unfortunately, many musicians don't pay attention to this and, as a result, some choirs sing chords with thirds that have the 5 beats per second, something I find intolerable.

A great invention from the 16th century was a keyboard with split keys that allowed for the differentiation of the enharmonic notes (such as the A-flat and G-sharp you mentioned). You can hear how it works with some actual music of that time period in this video. Chromaticism (coming from the root word "chroma," meaning "color") is really much more colorful with these gradations, and it's a shame that this original knowledge of intervals was mostly lost in the Western tradition, outside specialists in early music.


quarter-tone staff

Thursday, July 12, 2018


旋宮轉調 (五聲音階)

宮    商 角    徵 羽 少宮   少商 -----> 慢角調 /黃鐘調 /太簇調 /大呂調 /*碧玉調
Do  Re Mi Sol  La Do  Re 慢三絃
1     2 3 5     6 1  2 遯世操, 八極遊, 崆峒問道

商    角 徵    羽 宮 少商    少角 -----> 蕤賓調 / 清羽調
Re   Mi Sol   La Do Re       Mi 緊五絃
2      3 5     6 1 2         3 瀟湘水雲, 陽關三疊, 玉樓春曉 等多首曲目

角    徵 羽   宮 商 少角   少徵 -----> 慢宮調 / 夷則調 / 南呂調  
Mi   Sol La   Do Re Mi      Sol *慢 一三六絃
3      5 6     1 2 3         5 獲麟, 遠遊

徵     羽 宮    商 角 少徵   少羽 -----> 正調 / 仲呂調 / 清角調
Sol   La Do   Re Mi Sol      La standard tuning
5       6 1     2 3 5         6 梅花三弄, 高山, 流水 等多首曲目

羽    宮 商    角 徵 少羽    少宮 -----> 夾鐘調 / 清商調 / 姑洗調
La    Do Re   Mi Sol  La Do          緊二五七絃
6       1 2    3 5 6 1          秋鴻, 搗衣


5  5 1  2 3 5  6 ----->  慢商調

1  3 5  6 1 2  3 ----->  無射調 / 應鐘調 / 復古調
                                      胡笳十八拍, 大胡笳, 昭君怨

2  4 5  6 1 2  3 ----->  淒涼調 / 楚商調
                                      離騷/ 楚歌

b7  1 2  3 5 6  1 ------> 側商調

2  5 6  1 2 3  5 ----->  泉鳴調
                                     *慢一三六 (一絃低兩律)

1  2 3  5 b7 1  2 -----> 間絃調

1  2 5  6 1 2  3 -----> 離憂調
                                    慢一二, 緊五

1  2 3  5 6 7  2 -----> 無媒調

6  1 3  3 5 6  1 -----> *碧玉調
                                    慢一四六各半徽, 緊三與四同聲

7  2 3  5 6 1  2 -----> 玉女調

正 調 宮調/意 -- 梅花三弄, 流水 等多曲
          商調/意 -- 白雪, 鶴鳴九皋 等多曲
          商角調/意      -- 莊周夢蝶
          角調/意         -- 列子御風, 採真遊 等多曲
          角徵羽調/意   -- 南風暢
          徵調/意          -- 樵歌, 石上流泉, 洞庭秋思 等多曲
          羽調/意 -- 烏夜啼, 春曉吟 等多曲

外調 正調以外皆屬外調

1, 以上所有調整理自明朝[神奇秘譜]與[西麓堂琴統].
2, 列舉的曲目除古怨, 陽關三疊, 玉樓春曉, 與胡笳十八拍以外均出自神譜與西譜
3, 以上有*號為重複出現,但內容仍有所不同, 如碧玉調在神奇秘譜與西麓堂琴統為
4, 羅馬數字為方便記憶唱名
5, 慢角調, 清羽調等名稱是以絃名為主, 如慢角為慢角絃, 即是鬆第三絃. 清羽是緊
第五絃. 與五聲音階之宮商角徵羽無關.(請另行查閱古絃名)
6, 黃鐘調, 大呂調, 夷則調, 南呂調等名是以宮音在哪一個音高為主而定的調名.
而音高在古琴絃上, 一條絃有三個半音的範圍可變化.(請另行查閱古十二律名)
7, 宮調/意, 商調/意 等名是以首調為名, 即在正調調絃的情況下, 該曲圍繞在哪一個
音居多, 那一個音即為主調音/意.

Made by Peiyou Chang